From d02144624c5d27e3d88b68a1cb72a2379009b75a Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 23 Mar 2023 04:21:23 +0800 Subject: [PATCH] deploy --- .DS_Store | Bin 12292 -> 12292 bytes .../counting_sort_overview.png | Bin 0 -> 106222 bytes chapter_sorting/counting_sort/index.html | 26 ++++++------------ search/search_index.json | 2 +- sitemap.xml.gz | Bin 646 -> 646 bytes 5 files changed, 9 insertions(+), 19 deletions(-) create mode 100644 chapter_sorting/counting_sort.assets/counting_sort_overview.png diff --git a/.DS_Store b/.DS_Store index 56b40731d6940a4ce093e83cff5be911c8d0123d..48e8be2d8e7d8963ef53a1accc6ca2dccfec4427 100644 GIT binary patch delta 1136 zcmdT@T}V@582J8|fy|!x9!*dV>7LJL z!uC>KM#mlIxXmn=Q}r-pGUAsAzL?zbfWq`+3O(@7q$hVSb<$dpH%Th=Kx;)ao~?%Q ziXcf&Gs6{g0P+?5Ry2UcDTtYa)DNn0nk{&b^6%2p+47pE06lKrc$cVh6pc8k$?^Ix~DP4*90 zz>a~yyjTghowa;Z_EVwr0_m{YNs)k$&Qti=EXXd2JNoKdaX7Ne2e({$QY2u?HG*5u zfaE@2_M)6^Vc6Ziffn79_>BaKx(-ekYN9it!%x(;fYGBTwW&SKT;sWl=kCKHuVmIX z*rhZuHRKIKz{``ZP{t+%Mqdp~c#T+CD!?2y%#|Nue%|MVpW3h+k1o+W7U8qc04X0& zZaR-KW8P$A6s@WLI@_dOB?|2H@5Tp~f%iM;rqtorub-ePXe&iQO`sKE+TVu9BI!V5 O*?;dBS> delta 1136 zcmdT@O=uHA6rN7g$&zlCO|sdfHPj@{A1Ja7!9!F`Bo=KiYU)ke&{kWqNfz`_1WS~H zP*7u^ry>$56)I{&BOXLBhaB2V5W%7sDF`A8@dpYbBF<#nZ0bS0di&n@eeb<_^Jb<~ z)2aN*zGQLiYU{g79GyiM(%I3(@I-bPA;f(h7iQ`_LI$)4jWJyv?@5ki(<;@G@uFCY zI}OH?kvqU~TUBo6cf+J8;4*?Qq8IK{xU@i_58jF<^6T0-?FBVa(V-8HSQ7Y22VA#= z$?d1iFl!A#&B8aL32d2#j5SP#7FCwNPW7sl*TcOyhahM3Leyq%RZf{G_CdyGBJ#!+ zmM~>gbm)WE)YVf!ph{urlmu+ML}3I5B@=01nr8_aDW<~+e2}8JxB%ZJ7sTv5_vztt zYO2KRg!(5@u#a-*=2|sJ(e8qf%x@=ty3KG}?k6wrF&C`MPN>Sf9`<6TA3_cnIrNP+ zf<8yr7R)%@Eit&^IL@suZ??!gI;G~tH47!hxnUd`ntw$X*4Sb?oA~RNn~VL!D`1a- zK-t~_cilUA{N7NXyiU4gC(#J_=pMl3MTmJ6ZgoY_E+^&*z#Wf?Xaqd+?4!hXMEZ65Go?Jz49^@kzk z=Sd@sVT3>kbitfoz~W>bu291OKFR!izzsi|!xX+;q3_rZp95yd2Y9mP8f3 zseheq(vFrF&>q}{BPW0ly6C3#%ReJ&UvE>;3p5MxIM|10lFOk;+kbj`GLuB-)7jr@ CW`8#T diff --git a/chapter_sorting/counting_sort.assets/counting_sort_overview.png b/chapter_sorting/counting_sort.assets/counting_sort_overview.png new file mode 100644 index 0000000000000000000000000000000000000000..eb5708b2079eb27da8bfc32db19f6b7a885aedb9 GIT binary patch literal 106222 zcmeFZWmsIx5;lr!a1HM65Zr=GaCdiix8M>iXmAPc?t>HD-QC^&PWImC`<{EU@4x%= zz?xZ$VXf|(lCG+^t23ZPG?{tpUW=+?>=7|ma7afzGflvsFmCzmLCQP}yISlnrDs1*vx1~uCkL|+Iz7&Z?9@m}q4Z>4wdOwz^4 zu>Y;-K-E7W2NeyaiARGq-a_V*5?V!OFkr{ZLT-+28CsK|2gAmyTCeo%P^0&WTh|6m z?|+2-sT)^ZSJIK6VDRx34$qAEti5(yQ?fE5C&Ptw)_Yx~r z*;!b#oNE^KxR%)VrpZIF)rAErDd!jJAD=&x^-`K9i3(|N{Tp|hV*HVFBIsNu^QHJv z8K8|B%N)l*jGV}ltvE3-X<^eNSV#y*!|)u2h995%K3G~KO9gNb3uLM|FE=Je`pxNT$(LqvLJKOX`!2lpx8!|{jVUHUsZhq z6=hQ@W~hd!$oVFi*+2jHkOUl87$DFcXi$afazDsAP_ed3MN@60gQ{ZL=oB~UQ5T@Q zQV%Nq3$Yj2pv1C$g%jlu&K5d@D1GT*>N?8#v-eaSVPBG!TqFzrWi0Mwh0yM`I97@a zlym4T!33o{5w0*Sir-79><;=DGvI=yno>cjtx}SV3{nE2g{~1{a7^$p-lakNm(h@j z9yG^nR!Uu=se2C-^h6PHerY(ap_DD^4|~whMv<3ZnY(Uc;HqHl1WRgxi&NeIz?OZ& z_=mdt*Fk7Ts)}~fz#6TNIGE55jbfuHNMq73sc^)jE%JZqF*X9E z*V-zWmd+VYKiNt#Ro3YLV_-V5Nd-MK(P_*07lOzr1wrW^K_+?s?7#*_9>l6AuKrK6 zm5}uZQ@;Yu_4rd1OUT(Eb0MTw#D93v*mMxcsvU5V>p!WiBNhZLLlB=6`a?m9xIp3b z2cg6E;{MQkb73G6xY1?7e|#%JM=&LH(nL{{IuyKYI8-i~3KH{XdKPXTbR1(Dxrd@joB+pDg-+KI$JA{QrM2;jfTp zsSXeOffsBbhH$;k0`2KrZ|5hcsgr@Ja!UvI5{CE_Bf;w{EHUsRJ3a4I+MrBE}@Zux{FX0`m|y%#NMu`;^*K zb#B0K5;hdT6Z!p zPX1RU*AxbV&WL!~fG&l77|C!Q^jEyoxeN?f+WEk&0q4lbM_B8oy;i0rMO*i=dF5)V z!nu&cq(4af8)A$Ig7o+47WEj)Ll>+J$}H}UqqD$$$?Fy?+!oTFz)%4+O4;;I!;deU z=GSs!SXqkyKs^(`1JgYT4TER)#H5eA5TLB9pEr+yL3L&*x+ z6Rw=!*7NQbxh$SOG-gU>kM(baf$?jY8yiQ?jkRB^k*a&1Z_Y_Uvz>FKVV2K7s@lFyli*TsKsEs!$M<8%?KmfZUyiBT`QQvCwA<+2}l*&8SI@2okS7!*cp z#4X(nwIz6L7L?Cv_R-EDW)2c0ME z3azBI;J86gw`Zk*J9Kw1iPeCjK<;H$;Uc~B4z^RlfJQS{XujDOMKAVW7c)7vr+n@# zDf5{&Oj_O0l`=8JqlA>I5`S6N1U)dgUhVGd22TcA+seCUkZyi!ZIneNv4P_{PPUUW z)IF|zj?a6(`jrf(bDbt~E|0#1dse_OI?P(eRCfET!{bIDo*S!`&z+`@L)$>Bks7Ie z?!egnNU*FOS`O-ZsTRR>_;Di~Mty5s{J1cV>u=-L1?DLy7$??s0?7VBUK%){Wx5!2 z8(WbbGB}_g3YA`{@))?dy(ic0>qJ3l*}eFQ>o_n4yU&#^bV>|@RTxO&QEIl@Kvb>o zyt@zIceVq`yDq|28W}M7u$<$0T{>s?0x82iY?h$X$o(#6kcg-M=@powo!k;2dN0+Rcguh=tZ#p;2Y>&GdRvuHY&e(Yr;OCHuze}SC1S_@TC&+rrLmW4-)bs zWa_B?mHGRmfGy&=d+AzFIUTZo~5A1MX^wJvnzF|7PqmqE1lc&t>-&Zn9H?AIFbkU zY9;SdS!!QBV3O^n<6}n~JDRz67kd94=>7%YfrH-)p#OXOpGl~BD3Av{`YAGhBzY46 z80_7XvB{~tz$z&f zxN2`EKOQ^Do;6cE9}%OBd3pSql{-rbih+Izc$#SeV5(r0F9X=QXdYwcUj~Z+`etv7 zFXKqwAKzHYK8iecQ=JpOt4++S(pXc2W>bVKz7#83j~5KNMCISLFWcbbuf;M%B!-MS9cYpN?v?}rc_C#)6Z zGWi3zgaZ1tZpHYq`A@TNhj{`*uW()kWO53ndPukyZVT6AJN1j^#`5{`0LK9>cvej6 zLMbrTk-#5?3H0pnCEa8fE#SuqrB+w`0Z)Avfj^09s4T@d5gn+hStO~aO~J1FE$%K?fZf3!=z}rKfs#!I7cY|-^c|v5yTE^=fH5ZQH`d!z&lC#S($TAlBWRv-xZ2hS3mxmZutm4g7Wl{c z=yu!ra2*M6p95Nj5!1%@_G_Db8K`MKr}NuSPXzQ_Bn$`^cVU=^>;eLVTBKLRiIikA z*0mp4pWy;FH$$2wI}k(1BfxKW5$MMoKJ*_?Gfa|yn+zT}{2=yR?ERdR5v}ydf?cpc zMA`BX`#@|Uj3$Ko^2L|E^;V>B@avV4S;-icsTUvd`>!=Fmg)R?KliPJ>LZ%UZADxn zdTw`CgcuwG!gyp{yvRbtN>w@KOipdjrpcwEhIK8AM?Y#o3TmpI=(top%#4?dKs}mDQVePc3pWBd)lrJM%$`YTVqq;IG z2A)gHRh(c*rr> zg3R*pk!mn{LYTYHtUuRa4D^`FO^f3-61)16F~*g>B@y{C{QU<%_r>JccX|eb{Yv+J zy+LtIdBYbl23$7f%IV-#pZBKL-FNQu&Xy(>i%C%k5xUV1W*R0bn{$yY>4=g@-R0^J z#FR6SnR3JL(!?orIPvM2m;qUNDj1<9>0=tae`j(}w4mV-Or|0j7Tg{SI~F~KMm9Fb zMCzyCASe|2ekOCg-$zEO_3+;=>N!~SU!+d-pRPuG@0xgQd!SS>@RAwAZ0Qs`amx&Z z^k$DlZ}TvP)5D!}HH$aHA7NQMh*qvVs;UT^IJi_d1$9e##Ie%5Jjw09DCe)QG-$&- z$2pKvB4#{vCy(pJvst`MTyD|u8+|pxG|?uOZG^J>38MN-A~7Y-yu-^|-`W2&O=%Ew zhz1Aj?XkI~efo&ojM}ld%-kVDf9ZL{oXR57`Ax;>RBkKFYspEFb!UK|xWaAf)RGel z=w`TqaCRuxH_ae_Zx6HUwn*8lI1KweN{Y{~B<*h_lIL5BjSTcH3)u@=l-^ipP*s?l ztgo|=Dar+bvY z2-t&{65xFx=FPu(;wM^?DbmO1w0qxg0z1Xb#p_v5ugCp5mB^3h1@GPo&nU8#lyZ;j z!*6PXYN(iLDw3sW#BapDVgfuAA^3eglRJomwtPP3PMPG{w|WozqW@%ld- zYL;rGl?uN_6PE8dlRIP|2&zTwW-hf}{t`PfKm)BNzP)5MdWZr7>#d65!@mSK zVpzX2;;&qFh(UQV)O6`vD*)I_O&)uqziMlL()CC-I5FW4+Za^yR0A1~s>m;ZU@SbM zza4gKcW5c;n>1RgvZz(s_c!od!DfWlZx&bO4eIdhF@Y71t1!2IC};}KY1$c;3!j|T zI;O>?A$D`iTzQ-fKnkW*&gQy+{%(tO3;zu^B0z{=%V|#Q#`ygw=QyM+v}xeA%MK-F zOvd;&;PqSQ*=j2-O=%`!;LLO%8GVb zlG;7$P6-nWveB+tDnB~VP8k*Jtz4OI4vj?Gq?uZ{vzR?o5toYIFd0B9Tn~u$bGUGF z#=He5#p_Fa+8m0WLpjGGjkH9m=iz4qb(4|KnA_ul-xXU#{cO7Lg;_X$z1DInD#J(* zlRO~w`fHH;ooK;m{86Y#k1$U|pslY@kb`cTl5MH5rdJDUIO;5|dlb@SV8c{#*4<$D zsNV>e@gocDS7Y-grFoCbSZ`y<)YQvBa2bmH`AnL+OugAHQS)4j)Jc_3%CBC>0Ms7T zm_!|U+l4oMr>=rW(u`8<5HT_j1rX^s=}DFj?$wtrNr`!_a&-_omVaj1CSrce`SVBO zyzE^9y=UzFXfU6?JC4LUZcwbP0shxWZfGy~c8BMy<$|>k!X3YJKx4Q~EJ529vGu2X zhQm2F(|`}$h21FP>VgYhwb1YNms#brjBsU3uMEz4_Mw=&2D}AJz71Qt=_Dd^c9jqK zPr+<>&BV5%sq_O48{+xZ5x-5>t!gm`hYPv?q8KyT^Sv1uC~-8j6CVxoWT>gD7A=5a zv;fTgG;AK@VWmYtT?`-&P{FXzvMcR;bgGR-tZ_sb?{rm_nK9|%=bc9|1U2(C6|i~e zvtDoLz|z7o637WZ7owvydPI#zC`t`cc17WdR(l~?3Wdx5UA^&{wo zpZTqFs^5V4tEtPbzbsq-?CGJ*4#hj2D{n@mP8g)wgs zd?9LkHd}Aw1@Cn04S2~L4hV&YWO!J~&vOTM23iz5J}w8eWhZ@yznT;{U)|RShDvn8 zW?N0yKD9r(YInQoog6`c(ph$E*VT-9tR2{WZX+hFEq;I^h1+l%FK(X*+HyAUmCx~Y z;AVHAyU9`fBDunU8v5C{YpD1cwN>uSR_D=+tF&UUqoB1kZ}zqb%ary=4V_cT0}7Yl z;Py82JJ;~|vHfj33Bqn!MZOE@q9YuQjo<3tN{(i?@MOzfw~y@g2a28pq902V%q!wry2 zCT~oypLmBaKXtaODD6_7@*|o&ugsam1~!j667Hj0*%%Gq)1Of0B(UK`5x?a;JhGin zySV`(y&vd}47nzNsE$qW;9Azmn?-;2hg}A`(TERG=!FYtz+wTui@R6O;Iej~rBerD z2QErCqb+6X7>HhiJqgjVs-h%;5zdTw77WA%Nd?J^%Q3E0Qn=1M_o*sL5_t<%VZSjG`q&WR2gY|_W0Vjz=es;p{CFJHyHMEa4YZqA1gjFdib!#`o~gu$ zeDVD$SgH|b2P2Wa#iLpVwRT^cNygxq@v4mp1%olbWV6XnO3Tu?CM#!GF_9T@%ILS_ zjbJ-c-B2;`zoR2IGNMS33X7t*RcL?}y}Vu8Q~RA~-17#2!$K%n;D8!zT994r=U9+J=B6fLUvUY^0o3$?;q%~tVI&>x50d;?lH zNS`k1kt&w~vKni2LQb@mWb#mOe$L$oJ5>8_yTHt>6W-IDamTIk3qDM+OV{Yy7()Hn znBmj?l+|UW%XVvYNK&H^|Nd$YG}t%YyUA?UFc|eBv*Szp_OkNiMrr<<>Nn8sJj2O> z?N2Hm82hmo9NbvoJ;!SwGk%pSImG8#z3NBdv#COK`&&6X$T%E>>5jx5C4=E%Qe~_Z; z7YCN`i@fPVd$VGD4BwEbZG1NAvz>6w){Jc<hASc$;k*kVB>Rxft_k-kijv=guB#QbL+So{SSR$A9}Ne4#OtFX)?t04>|;(XY31 z_+EdRXs3%Y9HIafncDZ(FVSl!njX47?xB}2m(aS)!(m!3`bzMe&gytp_$;6T zhykH`0b$8CUT5Bxy{zamy1UsAIV$lU@eO-9F=E0g)CtX8VlTg{)^Ik}o3s+wG4VU= zaP>)JGoA|S4KG3p_%Mg}AkrG)rTM*=G;%d&kAgEs>#{G$h?vn2p*{!oOh46Jrx*I5_cl?xDQJn49caKDs2kfl zZV^OZX;MjYqCW@Q%MUlS9Vup4W>Ex@6D4ruYSDsDB@hs!-o9(y()JEv%1%o0qw$e$ z6j9Fy_dPs79u0h|d@C=m3QrmL{hTW#E+aZ_Ng(g`Q?yWr-;kgsYT%mn&qx>S8!y#Y|kZAJ=5axG8R87bzw$nx0a`9&SyQET(2MiT6j@(``T@MX_Q7uENO?RK8N>!ckdT6O0>C`TbSqs zhlK>ks~PKthp$W>p>ncD4()%-Ovc`P35e{N#8DX* z?=-a+2EpDC9HA5?A}-SLkj~`%BHm0$!$lRPp_d>Rjgy3zQS(0HyQ>KJ(g}Ed^FZoc z$IaY{?FuU{bw(jbq+17qG9ky$sNO?DSD)JFbho71p(t5U z$noNz8q(FoOpejXSiV)~-B}7hqp;9W-xRT3Jc0bL@-}-g$KuQ})#HtLOx%5MJf%&R z=KXu_<>2^tL?T$6If!nIBD$D8|K6%`KmslO)vxxT@xe5_4wVYKP{=YutM{%}f)usj z7fKeN{fcNEr30hL$;7Xm2L&}4aVt>}y&sO$Btil&lUX95bT3*g2E`J2@bXqe#w@CI zJpu?0nY}8#mC^9uunxCMYNZNeor)eV@(t)w@*TLjiSY?0^R~s!5XW#(c)+T&`*gxS z2UCdPWQB>en~#tG?y=PBFP{Pp1Yfr<>lUgE+0dxG@9={LY&4%xVjX|8xrT(o+dLYNZ?@+4_-eRxVLiJ@Imf@&axn!fJ&o2+q<876`juCBrGeo* zK5$TSL^*QgJ@qkjQ!u(YFbTLRvXH#)Xj6U?fa@k*C2vxd@hrpt}z4XB-C{!lVsJ(5Tzx zh>sUS12L78FF^lFo60AvuljON_+qQRQKSdOhZQsX*Dr=7$?6kJj18FTY`V5ZD`BFdspXz}rU3wYvKq0u-X7 za(vnV#QGg{cB{9gP-4e{HFK{a0A&0jM){7S+Cy5(vjcX_pFW94n z%uL%s$Ge8XF;-3@$YC?1W$zZUA)V1159f$;${OP{`NF2_9K8CqU1Fe%;14h811MyR2H;*GXi z`e!VvR08Q=z?_`|hSy_M*VIPWnBp%WR#1Rwf&+r-E9*MJDRDe5n?%7oyYUNQU5K!cS~3RG3S0cvUvuYFxKc zLZ;3@05RwZG8Kp`zBCbOX50sGKr)fMO7PY&l-Vz#es8>mwA=X=C941caWZ*QVQ{~w zn1}>Zf764*(&kJ&>awySaQZu=@RZB8gVDc1`&_nM%(-0lWBTG+WHja2ugDYKDNl#4-a>UrJg)-wq;{p3z#eu&X zhA}flPA30WGE!PO*s;K$0$bd& z_xMFzDe(1z;n>UXTH!RRT)}F!RLs#C-wi6Q#YMYwRn3Gz!|lF3`F1!8@tmW2*vAW> z?*2|`KGT3?{TP6bpEgkXH;C~XFPMeWnf?AM?dL!KQVYGlc-ye*YVdW0{DpDuxO?`a zY@cCg_7h1ULs0zD7L||~ji%LG3gnQ^WY=U@M;9fnEDe{0z7ONoEY}em1N9LHHH(Ep zKZ(dlclw=LD7(HCrfvJeEA}o*4LlF=I?E%E`bNa`^WG{_;Q=3I#xec7S^s-B{6$#^ zI$;JAHz_eVTJum+A9Cl>UY!gu%?Kh)=H3i?c{;UgGLD!Imqg3bcpBM^CpD?x-$0Ab zL776=kxvG=T$Mj4?3Xod)9>DB9bRANrIQ4}Yyziwj>TF1f<}|7=bp4>ER5yRwdcAp zakO-5t4Txe6# z)>hi*=y=_7$KH0k5|L+vj9C|$-({Ci(E#WS?dne+!>Wi3kN=4fI3S+lK`|BID$r3wHZ%+d_ZoQ5!{YifO;KkUi$(Cj|YSSMPA}>m8Q6I24@w%CrDtvD}1OU+m zQu*4Ji=g~)O1DNB{jv>Fr_m50fWPhjqdv z8RpZsc;=z1cKzBJ5-hvSoaW^T8R3SE*r&*%o{nk#r;!P zyF)~j2l959?$vv>G>2I24~21aGiA0xZpo4|YY;liYuBodTFOZLEp<>002t9wxei{Z zd-+!GQC=A){u8~Q+7y4-ayY@oXKU(qyCRg8oh7FoA$3AF}WY&o45#A8T)HFa7VSL16rV3}KAki9b>RmzRjq%!ex~ zVSXz$lO36i+&Z)9Ifs5=XBOl30AJC7(Hxn9(Y6n<##ZsCFW?jO|sn6NL}4cshzAik~$C z49n-ys>G{5Hy8vP9L3?I5hQBqmZ%@!LYycp6XEiG;s!3%uvZdruSqRzKq!!U+bAX! zDp>uT4g8t5q@|(Zrk}nA)3IPKdpz*W_`P0ggBkcu34}*Oo^S)ou$v*6P83Mz5M=J- zBtdvix%?!eZ6y}ui-8|=$&HEs(wB^ij8T#0&19L*Lbu8f2Ph}Tk;$g)rf2|+C|=YY8+N-JDJh#E%B^qO1kO4Rh4 zMGCH#{4n^hO6<-oNVMwTn$_)rp(^aJmzf-__%zDL!1>z&e{wG|LH2ssL-_BWb69Jl zK-xin)9yb)@k9xP4O`sty{fa;M`4=iNd;{{+GuAHla8b_<#0_IK8 zagon5-p8&^`xsd03zO=K7I1Q6%`v+YX!(<+0Q&A46ZPO8s1?tz$hCn>$k;5#Ff`o0 zFHJ2ifH4G>EKlasT;E{w0xiO;T*;$?*(0?b>F6__ZM)zM#LyNr0vXyRt#aRB$g2;u zCl-7)NYPml-sf@G3U!TcA1KVT_ue*T~U_JrgJ&{rCVyF>!E_B=FRZvoI^-3)x~t{c`D_9}Nz)*U!#d!7kUA3(}<8$DiN$7Wm(gpSFH z5qQH6(eXU|c?7^YtJ zfU#J>-Hv@#IsX?Be(MKRa6k4hV2yWXO1!befqYdQsLqUF)T9Z@SXjigV@a`H1Pi#Z zdY334u{VFj9ytFZ7cX=tBCMpk5i~dV?h|*{g23yJAC{f;a91~wYZOzx6da=!*bkVP zm}rNAjMA4-M_i)jM!#Tcx6Qe}P=dx;^}OJ<)@`&F*4y#82pT|sv#)#mKA9+*Yh|VR#45+{$x_HJ9E3JHV{vm61EI z`M%ylw}(<3SM{XQ&%rlkF}v(_rY(*@HN}a0A2?Aa(6r;4_qjstTlCOWfgitq%bQ>8 zRY-1V#`&#{UONBV+a+m?Y`Ot^-SNi6u~>^*!080ya^d?WY&hH4a|gex`Ms^H-{Ev( zq1u^;mGMZ?u$Xqi{9t{@q5NP@{bLr?bMOc*Gb20a+$&SdYWHSI#K66E8Xnqv#^xbZ z2Go1%h7C31!eu8olJhdo+JqHYtK0eA4lk2wX7f$sd|*GEgdjK=azhcMQ~TDmX)W^1f|xBZ01iZ*oTb6 z_i8>XOkM_=?M(AbhF^og5mdRzQ(PKlKayqe$G}JP=<2G=D{zu{2Js%?J&eEK3Sjr^ z+5So)j!`M6|A>hfQEbGiAQu*GAOOhC&COh04s4~6%UlU;bq6lT@lS(+`B--6 zC-&3^U5CHEM(w{Uk~w=P2PSkW^lr#p_23&=r@C%u2#%nZ|D<8fA7fX}rDH_+(c*^V z?yfY2q9D4{(}e?$djNjr%aCL&xP1 z;UVeIXN;<)veIjQPWiU?J%`2*u!LLm^5x6rmPJoD8$S@lIlf4wyNf<0#s$Qc54-af z4w`xy_>FP&nfdbA4i#0cX;S4gQutm%gZW`$(y7PB#ckg5BYpuV@@J+!{)iCOtjET| z#8o>_MnoAP3(2IDy^F`GNzM~sdd^9Z_lOBz?#{cf9!VLUWC$wSK6dUN9Cd&hnL)pE6kuhUl2mWA+UR-^5m+o-PIz3p_w z@ftf)789^F*E_>C3+nqw9mtA^oX z;$uvjGvc{`zM(drXDEHYX=DSA6hcU?ScVQ}k2XOXRPCYs714W0zd^7l6pGQ(PZ2q} zj7VcP9g1lsVW=k`$k9+OYWu@Vt;-rm)B^@kVjtl0N*!A|IWQd(|Hfw$JhnE3%Nl_6 zy>oMH*8st)J557JHWTNogoJDMPnHSQ)!o)r#VWD=$nz?+vcfyuTt;h*qoe{)#bN+L zi<%o7nUnmg!D5FM{Hj%C{n2pMm<Xz6C3WMm&#s-V#~HZLKJF=rybswS6aUc9xQ| zDWj7ML-)G!9Vuzw?=hRC*I}d~60OG4mXNesVWaWZ%n3L$g)(bCbqUo@51c0`vZc=K zgwxEmIa5S2bZSxP;a9POZCc^9)P#;LBBhX&53L}K#Ii3f*>_vTt${*HubvL}i)^sY zg8r4OtdK-epA~udCbO-`j8+@JlJ_D*1~DS%x@|#bCP@@uQ#lD~F=k z*OER`rFlHWxGzSlZDw+_nj|$^JI$zYv^!@xtlfXeds@)^6m=5F!x4iRss4jy3jF3o z+^Nj!Z5s_aYPaL*0UfDsA&}xiq^sWjzF248Y>kQ2+8)W$et=$l96xy6$)9WE3gvoB zcx!bsh{9zn-1Aud&8!gad^nwngiYu>w?_B~#0olc;DB)#sJ(KpLkz_rV4 zWoS;rq~e}iiRmHiwKS;MoDkd@pr*)W0oP1bg+7TOx$yM(kMK1gkz0od=c$S&yGb z7(Z{hSo;{aJ=wPP9&|N%y<+<1IoRu8Tzj;d4InuTR7gbsa6Dd0w#fpR{>rp!d!l^8 z4i}Pa?iA4Ruv{E7kwodKAaFt;ph+&x6ezbS)9^As#R?S|m;Kej{d$zLfjz!Faf^wh zmL$+*Bw3`m30}@H61Fo0jYvL=w9^gM z-e6cTuVs1GEYN91+jYs#r(SBX#+yB)8pnKXJyYJa3cY_fSDqthzG^Wm4NbGK zfz?OF?wI&7Be`!T!&HNcY7Y(CBG?*8b4&)xhz4^%?Prxj$Y0ImE>{hy1s(d}=_xCh z%+}vI=rUI-|QN zAh7Rda;9;!1IkM?=a2{)H}gr_dU65?*R)wZV!1}fr|qD*2_Rsg%s1{G)qWRqn!7$5 z=j9c5wVZt3t~>r)p&oU}VGH`;0*Z}$&`fo?}JRn$$1)c!noujv+(wbLvI945yO#bNU zXaFpYhUbj_Vf!ALer=kCXQN&XcyhF z_4JrCK0b#&Dx1MdVH@>wpugD=PeO;U$7|i&O+4t+t&ejE3dV;BCv$roFlm5BHX#ir z^yJ!t!~|*2$aM`A5Pv|R$(&i^JlhhUz(>Lalx;46-tZ8m_e;%uMKqW2gfD) zdavUYo*w;W^wZvJYqeABI{^CxTq72(#)WnU3`aR~X!Z`@TvoCWVUFetiJm8vgw+V+g4nAK-5Hbj8DXUT3Qj zn#~*w$z)R5jB7qKswKDtr*H?fN3ZdE+P|iv35<8@ifhJgU2Hbx4`vJK2((#Y#>hy#j zr(|V$N_=U<7IIw$Zwf0uq=+$mORuU%R5+jAT z461;f4+d_uY8?Dp!JzV)DnM2zo7(^R;MbHjsy%ru>X|mScU^Y`eqgyP;R&|&Osks# zQvv2}V2IYb4_rk)w;qS>#L?1hMMlq~hjYTETw2Nghg$@;!P|2DvGG75=<*>S4nEtc zy|hKnl3}_^KI4`nN=Gs3R7U%k>J2>Oiolj4wM&SL_-|?;2({ENk{z z%vMQbGo^h!_^Cfv>y;8IlTW=ragjiJ;irCEgT0Gh+sPOe&ZjlsIA*o`_2n0fq<|TN z7!kmuGOsKw1?aDy94bjq35MU4(ST*Z&XExjEeq6~0!FNx449_Oz-+1#xQP7j33&np z^#Tu+YR$o6uUXh<9?!~~x{!fFFvecP)G4&&gimuAn*H&!agWCPkLTQb9goqu&g4GO_Q5S$tkAmwOxfZB3Wcc+UYk`+e=>yay5B` zgooKj7RF7_Q~59u5+zY2H5W~vZv^<+BONyJU*DGfG|hWMa=()!+`=H73XLo18oyqg z+k3I=dX_tMA>A4mI|6*=GU&CD>c|)6WPDjSF|SRj1lkEsq@B-{)UUb-p;RQy-X!iUFDgprts6 z#`V~CiYlwi{d%4zrJw4;3y$BhWH+ex_;K0FK6>G|H#lij*<<<4dru+rT2$@6 zsdaWnX5bb=f~G*|izNoph7=UIg7IFh45EW%)=jVYHXK+tiNZ_L9cJVB$xdbexH1yQ z_;rIz_zn-&2oX9Z-JyzLS&>y>QLRBIt_p;cg~`%o4O#nxfCQJWNvoIlM8ap= zRD!yS$C$j5nYssGE)Cwsyn?2&w+-iyc_e~e)(qVwO@k+hsE@! zDW2bR`*)NiExTHK?2~n&zTI|yn1}W(FMfi^v!`>Pxft0VjVjcREc2>=+2s;C7xEZ7 z4ul!Fi(=Zjs|_miNTD}i{q*{Sl+RpdLNmPvzXXHy_e(H{8BCLIlAgwo|K6}A6&B`j z`H;Ue3XJ}mXo?+mB1$l<@~wspkT2f@-)afqo(wA9uow77>ESGCgp4pOzE5g)VNeDt z#zqGs6PaMkd@&K0F9Vt8?c5R+8$|Gd3%IL$@YA@aX}7W>iMev9wFd~ zd3tKkM;xmtfU5(*X{_TIaUT_W?pD+ZnC=;xz=y*WC$$qO+$C z^Zoy!>K()5YQuKn*tTsujnlAk(%3d8X>2D=qb6x=+nCsDY}>Z^P2cC;`}p?X`7y`L zthMgzyiSe&-lhTjStSDUo54ldwo0FOj<}s+r3)4*ufZedy6DZPi&~PF$aAv&uE2pz zB*)<7^J*<93}h-k=4O1_L8s+A$|oSTu-e1x&OTu64NHan6jZz0qa3GF4b#`DYFXz z23uf!%ToVi0pHf=mC?`F)G) zXalYOd0*VKeiswjcbge(Va>0Z)SFnJo!lvp)>MI8hp>yk)%gBn9^|;Fo^1DXTryH> z+So;RZ*dy`3jrnaI$cunymmSajjZJTRrHJCE+#}X?);So=pMVoJW>O5My`2;jllVr zPQ}vD$=)#VuG;XM3=y>>t4PF8N>qA!f;GNXuVk2O$_nqSLU@?}v@`kLG;i~Lb=6zr zCH<}T*#_6vSq+Q#{S+63tdGK@PIj1mTk#$;nl4|)TK!stsaL-|;^<#VswV$myrg;{ zY*J0xFUVL*|FyaFD!?%|_1C~)o6@RVPxX{U&@ ziC(v;=k6{TaJ8=-ZIWhx;)uv$9NGjm4zdTc$}#rPQc3Kk8+) zQB-<^xZwQQ8!_c<#dFYp#(Gf}jrt=j?4qG29MMR#!1ehQEsB?+<`%h-Xn|(5E*vSD ziL1cy#8{wXCgc4|@iaa53uL9fbF5RZPd|w}JLP}iNb57;Yh4wbkfyf-Xu~HLPPW*@ zJ38ficWb`8`jx@UNLW^d?>rvT5C2o>BB)9!Bde-BaMd}adXu%D1%BT^+Sg>vit3-} z?-49o%-KET`*{mP-6R76nFhAduNZ}D2*S}rwv0H1PwN>Mq*f=1NMWp1$F14g8$J@G z*?EW&e~=x1Yk#K81^aj8BE)s)LGG>KFHP>b+Yh=}q3W9jIn7|# zp)ahFcu%vRu6(|zCF37#*nCT0GI$S^nw{hr6pF8B=KI2R^ET0aj{V^C5x4_MdQh2_ z$?v_kbxmU1QPOH<|FM!bc+%2$81qx{R;w+Ix1sO}X)OyNi_Zsej6X=M7Xw`eP@WxJ zGT>_s%$#F=J=GKscm?>xPxf)8hu0U0*?IvR@v)abj^Y3u*b}k58s>8QSq9&QijBG1 zrk0M0(2GVspQJQ-G+q2}G5@C)b;_0Vw3L0UJ_>Rnl`E`-NjRuuov7DrARKk0stZ=o8z zYSrVPSi+OLu(4gBJX1-xaMM5>2g1_nQ8Xp&2BYBH=$Q;={_Y>2|5IU?qwwztd$h?a z9;f*ipa!L8j7$p*72*9{G>$;b^~F;&Me)*1dJJBX!EXNSDo@3k0`*1PXy6|y0e6G;)L0Q`%?{XRs(fZXRz~GMa z<@0H#jh04HBIWu*gF?ph%=K<+LP5HmfuzmM!sc5FmBuvEe^*CX0Yqi_dOcYQ!doek z%@}r(K`V~E6a(kaL@Z?@xq2;|VaJU=##`{7f)DQ_tXrohG#egrA~x0_Zfo$Kt}D5Z zRf{Q-I|#llWz0Jv}7)qTdmOKpJmHQio^ z_4Rw$w5sc%gWWokmv-DVT;d-b0`z@*>7xW}+L|MGiMyYS(uVt>1?XzRdoooBuwTJ; ztiWr=LPE^DQgYdc0s+f7;@9PT8&cBT*1Q6o6TxATQqLypbof{gS#JK_1eyN5J?7P!;YVfRzf{p3CW zA6vTuG@B8jJ+k6gjM7_-?W$}OA5HPg*%%M`d<%`NuTL+8tX&EsRvxi!O&tYC*|3ko z?A97NrGz^zL*tE}yAPPFY?z400)nlF#Mbo(Je;vv0>A7V#-uhqDOr@@64^iW(k+0= z&wnFK8DsWM(+%w&Rt@kE{_|D+4oD05-I*yBSOlgaZkh)5tvgXvCi6Trv(a0wLWF?u zi+2hBtZfxFzW}i6nOEaSnnQDSeb7eB$5~9;jMwXBu`7nt2;*QSUvp7w?Mv7g;h%-d zQ-iM(bhG1KJ#Ve?%OV&L@t94M293!X{U1^(_vgOFAHHnBirVeYG5C(^vNrB+co4WG zKez<}p1$|>&P8+{^_R0L%fd!#=dej)Vu9ASg%OQTaBt%M{fsi_3sM2QZdEBFmAal=u()Jdjkn_*W6sukThXr4k0(QQ!srBK%tp)5&(U&hrK6mh? z7eShRQCK$)6&Iqa86akjpv_(596-~x_-r+hM<~+%MIZ_ zmnD?}HNZ-5^=$T8mvN>0T5)c6KY5A0vYm7z zd$sU&v(mQ^8=N<-ia@@&O(-n3&GaVjaev26O%6$WJ;!iY$a2*9ue)rD=D!Veb#G7} zbd*KHiQciuUQ5;-bvcwXmo&sgaY3iF@?%FWNwyZaoR(SM_0b+d6sB8oAGt2G7lrdK z8VS(ZMfs=zJA8wdB*iJ?H%9fj1#O!B)m!c>yU5P!h4Vh4dIvNHo5nA-H$7!ZdBg!^YLzZEUB;A zj!S({NA3iOUVHuZz&xQtv?*wMNpmF{t|2VYj^YRI9Bzn%j<#JCCaBH)!0+{b^$~5U z)N{Vv$N7+TMzc!sKN)UEMv%VH0<%>4TbmCaNg2Y@#%_t_$tv`IXcw{=`Jya6&wL+K z7`&$_)O}dyJh!@B%;)}55h;GE9Vq;k(ADEI@CHlOPzg)3jQFT6@?ITla2e&oZKj55 zbF{$xf5Aw-uDffwqk`%9s`n<(arB~v-XY-Ta-n0LK~Kxz=elogmXX2AKFFxn>RR0p z%1A(%wbTX)pt~UEWLcuxMdYxap8nrHLGC~n1fy2b@?3UMJ;4+;M32S$8r9|>$379a zMPHRAF2Oat?ah|MGH1<#M%L3_lRFiTu#y1~9ytTykr`C@INL+K(;{^l=w;%;>>WA~+kE-JkI2U^12e4S?St4ZK~ejpEqG}Fzu6l^X_9i?6s>T zBlXBzS4z|>+}5fVGyeYpCef&FVlU%`JzX2h+!Vb_>!OAir;#0Gi%rk(Ijf&PD)Wsc zC(=O(gg{VF`bebP&Hj6*j}mui?|3(xL~a7SbHsIs5S&e4czxE}|uvYoCd0{eY2%I+pKfi*b!tpOFjqtK9VP>yhe3m)Im(_}L}cLT9cG43FzUbwTMv zoAGETNVApC#`UXYHJcl^Ze~V%I~COQD)jGUGl@AN9WhzUKxcsV@#Ua`46^I1qDmSn zG9%UhJRrZ5z-au8)4g7Bk8d%%-NrzxmYjeaXfVDYfgf$iz#o(bj{`{UJbVq`x6t}> zw%o8|SlGuqErXTJONHt6dr69<2^Vp=vML)*y{RtKVAriY-($nh_`=4td*l0S`a3Ki z)Oge=xEkE;PrXi;TL0F)q3f%5T)$)!)?1#+>+7>`+|#sbaWHf3jB?U~bc5MGZbwSi zN@o40-Fddpnt^3Gh%s>ehFskNH077^jQ!}QqEn9QN!ekyY-d+s#Z18!{h@($Iw*bb z7sowq3f{_Ho00fmGsdvezulc~>IRj9nJTbT!f5ok=+Nzp<_v|fhIgbj@w~ahXESO- z;7flT#zSSpgxe4;HmU+DKIXXsvfcC5rEnuQ2xTc#mN}I8S&xda;lr_4OeC|}i+E1q zACIi;fldM@aTvr=0>X>U8|odTSfo?+y}kB&PtwazD^Re<6gJvbx=T?bL!yAOeBJB$ z=V$^&SW7-U>q5&jNG2g#a+W(eY6*UkL1Sg@HT&PH7aI`cY*L4t`qZ#&2Qsci4wy9R zT=SU?qahe^bNwq-iw2)OO>yaJTMpYM z#vm{m4X?(t^ckm$Nc|i<)(z6iU^-5FjZV}2O&OM4`g{p^D^y%TbPIhY@Y631mqKwl zGDbT|$m->S^nN;c^=v#X*tU$Y95CFTJ6>)u=E60%(&;zUWLvo%FBw+4=rO(B0txpi zoC2CwuN<^4%0=UH6CE)R6$L@Tvz>aM2ajrfbvgn}ruqgk*!7dV=vA}`g!E$+p!9d=X7 z_Q$S^H#0LcFE1|_8v{50flqyMFk~d$^Z+m*RM^=sa9Mm%3JCgN&9DI_8`GG_yknQ@ z{KHVw+?*~kC8az~@GPYcE?OQUTvMg5cDB&st?sVNFUVDb@Rt+ACG|G4lOX9|--!1> zP2eI&w_5BmEM=UPbR<3DH=NMgS6u}S&?y&+9m*R!>rtK0scyNk{)z&1)!yJ)|K0NM z6-7QIicp&xpiwxXjkCJ_4!Cq$B5->yWLve{#`Y+8zsgQw@GnUeXn=t>!EKNYd!Yu# z38%pdn+60e5H7)B5KN!G?&}xzNdk3{EOIu>wu^N)PT2uyv%LbXDsixp!cVT)W>+1U z!E5V9&LsEwUFrT$T8rjXNd+(I5I8Y9M-HnxiYM406mtG4haL@bnRwpI4#_67$B9I%Av{_{p)$JI>z_@>&EMqku(zL znfbncve9F7?_hoLm7lS->`MGzL(JjQx@Bvt?QhxER|BF-Z z#Nw+8i&wzteiEICfBE{y#x*79t}cyfZP(n*IzY<~hv{Wtu2t?iE3z9ZMMs$gv+DT>aN0S8^rLAnx?YhQlFjUQjKnTo>?95V2_9> zvNiE)VQ)$?0n|PkbpO&<1_!rp8+`;74m!DM<7gQLF4ImPjHNy{JdW+WO@1TVxsZbpI5AD?p~f=qW?8bz`SvlzxL*GPpv;RV(44 z{`Y!WJWM;1!|uw9JRugEchB9{z=o}hLT#n%IP}K+N@R_`$c2+1oc3)|mosJzA&iLL zPN+LzEi=GmX(6c@_7$1=aHrniYqsHv(Kowo?a~`DZ==&^zg3{ZiPO#PVfsrEaW>ZH z{2(hkQ&;|ZX3c4#;c&~Gvxxnx^>kopNBTnxaMu^@zVLF^+;!?GCEYa*@8gawdgq6};8|e)NE%K!mw}A3kto zF>C?!ZE{2*ySK>4+kec=(q8gr$yQHuEJ6A;Z#5zQ`MD)Y#Kgh3j<>z-M;)z7Pb;~z zmO;A7wkBwRtE1B$t8V{W>0IRT9bL7n^6O1;}0@>`jyOqe0vV;-Hx5utS3-6uo1||M@aaBLv_k%RvV@d|T;{PC4Sgv>V z9~oF&F~n7wZizuiug8ICWgEA;PlMq?wj!WIL~d?|Z-5CHHJNP`=Kl9e1WY$PGvPw# z_1@b$yf!IGmGxW;x;sKp6_szILOKlhz2)q<+-|)WFB6|JW090-X@=oPK45)(sH&uPX`xqtB39n;`iqJaJbJL zqt7b^To#vd-L8I2Dc82iv#gd|&0s|l3NN$t-#m(ZQFPZnBe~lV4RgFrm8iGZ>3HjZ zCy!buBO<|!f>bV78s(}NU!R`-^j%nqFU@sl33-2;+Bvpf9ooJ-o*jxIri_U6Esyqn zeYo}pw!~UnTdy|RBf`Nsy0gx+1{IrMZgj}V$TZsGQm&CD7EQ#pL{CmOWV*BMMxb=+pCq4f=MAGrQeP?SW)I;|e>&0K9uzfg zW)q!_x9@rLqO!iCGV%n^u$t?OW zAou?G{^!SIS8o;_Jre>SA0Mgj-SpoCT2HKO?+d54GiW-F>sRySM*Wg^2S()H~em{mH!Z z-muZYFYGgpA(m|a56`$@3gv~@{b3@8wkVhY##a=Z+eI$rd9E!BHUfuq<;&INPw&kx z;NIM`i|)(WQq=XiFTrzL|0ZKpw&D53^mcgwC!hC7D={AQ_L=nHb;o1zv^N zV7aczwUUfZVF=e%`2DRecobu1;@*A*bz$*3EKxIlYFUJ61;}^JWHA}7PqU~paDq99 z& zFX?peVJZnuH`;1y8>aPwA`fscp~*0sxaHaBt3bedYokIsPlxwo67n|HVud#3UB_uv zTL~2Ttv=8}pMJ|@8+7@6-|rU#_tSaoPu5!McaaBj7WLgT%l;mWr1_zrMUM_-BFcM$EDroK8@S?croLDQ+^ZXl>1;|Ho~1kwW^fwm0*q zhgHBctt;-{+rCy3jtMbDY=_gvuA%Z z2RmQy6~LW$II?xpX1Z#Jd{s95F0e`EOLDuwNuI_VOS4gt=mI%wsJEM~do%}Cs#`o? z{%A>ta#x(IwgKw>oZTi#X3d#An%UUReJ}Rqiqtuoy%&$#kX=dHrxh+6&deWQkaxJ6yBQ)2|)dwhHx z8rR^sXk=)g(_L(@$I$X{NW2e>^1&0HgD^;Vj*si^ zqkZW8P`?J*zS0AV&;GpE&HB4G?@yhoxGIMr_IqVXZ#N}(nm4yv2li`|}?s&+Hl346fIGpaj) z7!OC7qCK_oA>)g(`Pr{WBRdoJvXH|-LMRwW$r{;x-`1Laumm!HF#QDxF5tlrPt8F_ zjKl92I=s_1MAO~&!1X4d=O0|~Sor)dC3X;N7{fT=0v5G3Z9%C0Y+4PWN*dZ#b(L$q zXzRS4Jc3RIId&2ZZ4GP`Vtz9r zG-+lP6_ti~{QAi(A;i&fK1I*~kU)h13lGPAVa1KD&2*BaEy;|vk=?;~mjjCuVuK^% z{&fq7!W4pJ#{k74!q}c)YAg#nxis$wbA!1D%Hru4sHL&Rn2FGg29U2KeZL58*#QAL)EeC3s|krSk z6j}sbPUb>~xFe1lLETVcfu5ofAzb;`Jm+e{hq$=&&d+eYnd$5lHs%&dLDlRr=vs5!5S1Ms?*WqApKoT2k9^^IM>>YI65NT zgaNHw=nT##nv9x>cnf&zvrp^1)jeQ>v_;8yQ&+`EE_uuiC)ufTBs=toR4^{PVFf!R z<01V1*Lt_>`~3maeKbGA$N;uf+U@lvTF+RVm19x-7@@s(XHX*_XI?-=7SAK!d^EgVye z>U$S@KWCa9GVZ<-A_HB269jJ(s z$*ivStyDa-sw3Fg5xSq2Cf$$LNQs9iREDS?2(8_RQ2SCvXGGY`0JYfj@%q-6mgv~EFA1wu& zzIcAaob+Dth|2|vEiBM1dFUcm0~ZT@r>ZUDj?ohdW($NO8Kvk|xO1QF}FdP6i zYA2OyrC{IC4yCxz1rswpdIwJMjnRKB9D_Ys%}n>+0lD>1m zU$dF+36)NWnJ&|RcY{nrQU1vJ$pCy=60!U~Nn=U1Z7Zr5Y?RF(4M8wdOetc(2Wt-g zUG1i}TM|Opc#jDvBO9**W2(s%LjACt1f{mt+nhQP7Dt)ujK!}Vig&OBAE>US zebVzQ1yh_OxG@U^w9}O)Mp?U<2C3PW8mwpch~WG~?*Up`c-*~%>FTD-j8Klt&}!mx z2;j@ISU;bcA{6GCJ>ma!fWW`KB0PS(jni6UZ?`0PKrKJb14r`XnltL=_kiA$`Eo`y zFh)z0uCaRX$kCgxqk^1T6jw;?4=c8F^-iL`CephVEd1LkVdBW`duNH}mp3rK}FrLIwsY=|&mmS5`IKRv6zq=;wP8b@UVu`C9 zmp12(`|=Y?-keg>2DhxWHV`fJ5q!?Y1k!@12byOg$YoVTU1uB3uW^Xd3q<XQNkKBOSHiHk{PFc}5EP`(IKfF3BFz0ZW!rkzf zkL@cfi}5bL4LE&=%G&hpN}i?~^N9FH=@>2age$D1{QgJPOCq6TxrpJ4hWx(tr}545 z6Y~5%*N(ClBRM$FPPu=2bGgND!5yofCn2AOrfQrj9Rc~k3T-bd>mvG1i08vhk=NGO zk&ns#ER;Khep*8$;NI3wm+9w3W!z9k zF%mP2mb1m$^%ms(1=tTuhTzifr4G3SDHe9rSEs4*9j#~>t7Zs>{;%$z+K~hNKKeh! zh@&>5^uOW3zZsG6jO{|JDloCcAAR4fw^XvSiP$sO9D$f3_Sb2JW>!Ugf=UkrNXHQI zIc|xv^P-?%37ad^*(L>`u#Y5u-qdNerw!0q;o(}DfAWkHNkx&0WK|fkWgPVsMb%+s z!ugR%>tVl0R$wek!!hUD?0n;N=K02yO^JlU^b>&voGqLKPAeRo@!gLeTNinh0IVu3 zy^2|TR@8h3tiRf6X?A&*_M|tTJS0&S^=_uaG1KrYG$R&Rgh~Grfg&8&BULEM_a_(D z@Z{iis(0_fH5#p#N1AFuZDP#*em(qzfG=&Y+d^cZa<9YeKT-|gc;M1;AL+u})-1v^ zKVD3wChipOhVJ=PSO}X;Bro`-tnaARZ(wwzX9CoJbzTojTSptNl{)N7w+S8WUc8m8 zv*gjG&yp=pKf*pb2xaH&^VgMw$axev-gb zjBHDhm1WlkQ(dkIS5k#wrZ=qU4eS>jE_h)QXm9YJ{TPf7`N$JPe{ zHbg`6#KS{ZUFjve)h+~w3He%@i-iW5`i@2h#4KKRW(u;{r`~Bxv>WE*K9AN28 zLk00T%LRbuZJ5H?syb2?nk8&;=Yn2m#H2xS>;jODSv}!LtwdOx8TA;`-g^R zaX!@}{95`s^bVZ!m^Y>uOjRKBFsE~;60Z`Hoto@M7#aC;!My9uAY^mdmt6Pt1s>3$ z(!RHjkM=>sIqz==rKHp;jRbrTY~?H1!uT{R`LX5hq@St2{)5l`LN9|X1yn8(y~%z5 zy?hK-0OCP1WF153i)~lMp=zZK7nojUldf(>QJ`!oxRxq}$^O`L_xm<#g(_?#9EaZz z@$UHi)IDlY(mMpXYS;92bDcxS#~$$BKhX)OlzX>P28-!g$WOuXIo2qz%C!wtO=^_K zpZ6eL<-HpI`+Pq$6yvo_Ss;TSy_2?XFw!njMX6lQuoxFyFZA2d$a*c_s+!-4Z`_Ob zGSajr7`Z%bDReCA(H(H@{^$W5y`Q%~r(XJMU|hJHZDe-@rN*Sik#3~Ujld=YVPyPe zT$hzYS$wC_U>Gwf#oawhQC*(3f(CH7kdb181D_UD}VrQUs8un2|&3GLn+3K+aI+5SXhAaqleWDrm!t=kQ&@y82O> z&?yzj{8tFymWZ=-jfN7YE`vV>_n3h(e?H(<$t?R>!1(p&2~nkDYIqiKS#x2VWuC0|XKEn~||sXs*Cm&62HDpvyj2h(h?xp?DxFdr2wnpW2+r z!r6)&B}_0aL9riYf;>|-dO>oDOodp#p~1+N@kQxo=Yvc=ztzNmh;1IldOWJz9yP){ zxG*!OWt<1RWD5ca>7H)5zSA9O=7TVDmPuL<)CxphM6d&J5@Da_?r1l#nlnS3pEL?k z)>gV$Vw$pB4Ca5JuSD1r%%gL1TCiXadvU=g%~-x3i^&g*N{nh?Jr#Zn9epV%C;)Um zhsh~2?)4=}dygWeKRgKSX@z4xf zep8*C1n%Nx+~ua7_LoS+F9lrfUrod}HfIh_qdA*0*A|2;eDRS_zOPpaG&z{Qh2^UBc|@=-^nd60 zu0f>Z$8@Y!P1CWuFzFB74%9w9;2@vA#GQuEp`rm zO7b`rfJMVx#K#Aaj#x!Sjn{q0+5*r!{s~wwZHF$1W9io_QOpv=KZ20JDw@FKOkl!2 zg5yL=0au7S!;OO7(gTrZuP9#N0X<*>=I(nHbfSe25`T$CovDf3RXDmaju;adz*nwi zXaqS7dZwBv2C*km4~BfUe!zG2zex(Rfl(bS1bAYs{zB1aeWRjz90o&Ve}*|X4VS+m zC+NLlI5aIDgG2?70oxeWI^Gqulh)e!PVg&ImE03C-FRWbcuJbox4-oF7c#WLJz21J z|4$eLj;H_A1>i@68U^(*+Jx0FrTULpJ6VGBo%uz{(8B-l2esD(_So>PB$4<#j))Ct z!pVwy!oMBB_h{+6_72#g?^?!W3a%XVP}ZmYQfCQ&|E}0rmIPYf)ZH~eB2!>l5jbz~ zwBu?>2EfqGc1c!KfR&|iTuGl)i0kYEiYR#Ypx<0@qX#Hy75Mh}Wk*K5fVcN$Z%XQb+Wi`eYZk^eEbu;wHgqsNK^i-5nUl!2>Y3K> zV+cNyJp@$BtljA&SJ=kj*cIu2EK=Xr=@f1;8qQa)S{tuz>&HATW_HBCuzcvE0dr)G z5?1ps!N?6eRHHuo{w5%Nn+L0*%0nnWX@Szdj=S$q!vMoWdH^lYk%FRXikbY0%wfgU zgV>h%!+m|hc+hx#CwLw-TWIJ}Tfe~n)=kdf=7<`4Y(p=jO3;PVFOA6Zt8GAh^f*H- z_H?K70$_TAs-Ms#;bGzr6TII13I(A9Qrr8tliEyF_O=G7Q~Kq!j_c{>Z(#uNB20t~ zh%;nF)L2Dsf`R5mq$@KmWY&@*&2)_V@BuFplU@(YmV9O12^v(q2b!WC7-01Vd32_2 zRS?}HEu|ww-@XQ$va^8K$mZ0@eU;U<^lnNzJjSp%kqwE(OQ*QaV+YZ{%2NLP^#n*=< zUeVY`W7<_60=-FXPJHSCH-g6E`r2RAfZh!$`8`q#cZrox6`?rys( z_Q`W4pm}RUMM2lU#{0BjLa3*2BCSU5J|X$k>hRzwXUj65p8$*8Qr{*%(Ut>(9=-G@ zU8fPGu*rvnCpI*a?6VfDtg_4kQ^GJVLS_k)7gZYKnB94@)p`r#;@oxRtpt!3cYt}# z*8ymgC6Sj^chEOfMaUYJr|y1kh!G)c?T9!48yG=gOsS5{mku@sachQT0F~qX4D;U0 zB}-_E-k$6QMr%e-2mOL|hX_``0nLyZ>4%Ryj9e958QNzSM&?{MU}8Sb{7Tv@Ax<1z zt=0J@GjPScqeARzL$DU&f&xY?A=xEzmjz5~>Ud#gR}>ay?c>PJ;wMca*pi>fT4*XQ zCPoI2#fAr_<3cwEnf0_=oP8#!;u>;qP*#tq4h~+!BWb;2{+><`tX17LPkilDrLg-{K914wV9|dL>>xpVrqu zve3GGq&d>ZUqgA;oXEG^nte@maq|7s^;?YgMJyHlFoB1G2XTvwjB`!B(8cE?st|EI zN$@Nfe(_pkH;PF3(C1PPT8@b9*6c9@cYlD!@+^tA=qQB?V0fS6{cw?mSUBc8cgi&m zCd|h>+{Pcqru@-D>0U_^{cV#Mvg5Lem=kQ~cEYu#P+(e#v){~Mf&f@Z>1WvxfBnF^ z9n;Hb)*vu{1;f`ClMRBTkKdW<6uRi95m6ctLJV!p`8TRcLeyR`Uht9T#-+MTjzMAO zy%}&w@&sV=1&s2|Ezo(vhJ`tQLgd46_nn}FwZrEikQvss%|EfgW_E_b zC|1&Du;P1LG?&YUImj{s#wgsYjuu|-a)%XVnF55LRU!}GcqFgfq6DC-8Q*}F0*7yr zm{4vffv_aX8-=_dUzGbcp<*z;_y$mGScoO{g0Awi@&9v`wIC%zCbD>Xh^CW$gI_uo!aEO|j(D$)g(9V&ysOycSAo)**Qz&Z44*L1F~xS?g5RqF%+qZv&K zeUOZ(_KY{$h|1pUK!A1*%})V_N)^V-&M|Fsxd=}xoH;hJh0A^U~rF&613xf zM#RE1qEF~LC}#p!@40hI3h6VYrO(@m?3wyo!Hnl_$Vjn>F20m*j?{ykpF3`1ca+`C z*M)XF@(?bwxsA|u!TAAcHOMfeD^%}9$03$7Ps=zoQMZ*f9Y7qgc<_Ky%MUVkoNqq# zC?=vUTR!xYreOevai@ALOadUBcl+Oun=I*NBK7`i5%2)IgAx;)^HrSFt|5@ zc~BjEOCj0h;lv(S5KjX~XT(fg zsf_THQ|s}3D$N0PTYui;JJcutLBYdL+wui2G^ zSAsP!=y;8&F`AZeiZhOQW4*COW;{SN_9oKC!k@3O1#~w&y4jcR$0eT*uU092wU#Q) zviGzPjqk;q`M9~3fR>{(XKRE${hm+N-%YjD{9R^?0-LP}VxOp3^>9cwIynCcO+#O7InBWGWitAFrT|MYF^{~JeqaMN!dQ*_iJskH4@|HggUiUXIY{YcNYf#ktYErm>p6yKbz8qitz=Q$5bv<#d)eBe zCISN9&(M=@qo@VZ=|qDgbN}WcxtY_16n_Wb0udkBS4wbPR4Qq7M#Pm!XJgD!+0Y^X zwDITdS4Sf^u7DzNX|5dGDY4DxqKp>R zsC+}(ky=WH-^XY>AM<3pm$00)OmgI+cke12+(Jb5r(0+(%c(`F33=Wq_!k$i53LP< z-)&4H&W&_o2k?5^Kiok^wVz?^nd&HN zDEsTUn8jzyJk7>XEiog;%_nS7)mzc5rf|o|FTx0dEEgGk~s) zkZUe6{)h|EJ@DhO9TfZaq}IE>-;cenwwOcFbX#A#5?hBn;6tDKIA^hV|Sy>Tr#5E zP(Sg7ZRY4~lsV{&uH(BgQ|htnq)J;|d)~i~Pu;U}gPpy27mO?ayAucrrXKMe_*zHX zy_&XP;&>czzZWjHAOp7NL0*W(HkQpt%b5<94ItNjK^@w)iDvqN{|U|tk(HOb150Y4 z|4KSL|4iFb2i6Q3@1n_aOK-ytu6{nx1*(GEEh>vd%}<-How!Kp#s8>q3JLvto=b#1 z{uLLW-I@~PXyRelkhh6h0dhaWc^j^(QSlE0&s1yK2mFQR<*LYt`dAVod=)#GQZq{a z4(yc_T>qoVUFoctGc;_i#QY!ueOEVrsL&t}@Nam=y^~jpNH8=}$OW6k?rS}Aoc7@c z&&puX@GL)px@WZcd3jVU7)y{iKQ(Qrj(?ry%O$fEz712xew&GqdqyXZG(RBJHk1l2 zYD@=LU`?iAt{IO2QC6wW)XX)_wsk4PAq9VsZctitpGD0TYDdtTLeOaxv1J0xbSQ_s z15#;yu-uxkBp4qc6!93{m3KfwhrAFKgFz3^YY2>4^*#$CJ_VtiF$s1&>T-j7BF6)w zJm`IL;t#{C%T)9+hO~ZXbaN_NdW3=dv35RDn3sG*6W%&+Im`;qKgi7#_T>jlgg{Rj z-C;`P4DH>KEH4_TmctYw%5&D;8~~;+Gn}S2HD|D!hS{h}P_faE&3vgc*{<1vc#O~d zwn5!V9ImRD2c_5s1(u`)2d$-tzl?bMLs99;mTWeK-F}7%Q!)c0c2L$Bfpj+Pxkbh@8ucwaCg=ahXWIuw7oMl#6@Vl<9Dm)u z*;S9BPtdGkRtSHK6lVIdluTHhaE>X~%%!jXZFMYTZ1eHMk56E|M^75$w$R1^F7<7B zdXEv`zu%tk%L+QXJ(LckOxo*V6KfHQHTS=(srB74w#2ZQ+qT4Hv_Lur{C`}X1yEM) z`t=_`x}>|Lq`OlD>26M`iS*xiA|LijVxHxLj00Y$qgF-3?{i1h-A z?lK3)Tw}Fwu#~isVYSlmv%$VtH`^DqNxW24_7d8B*8G}NH$3kg2Q~*cjyA42T2^v< zKVPr2kfx^s0s*7s%kYLOrS_(s(Y>iLOU~c@_?KT2>xXt#sISE25ckrJw>3$dwI#RI zH@pe&?Vk=Tx#Ck>wadN2xDD1QULM&_DEDk`B81<>mOiF5^ z#nM|c(NCTzL>IXijoF_@a9}D&^t&uFMP?(WV2;TMSu8xc;E6A%N0L~mbg3)}JmKaM zgoLd;hbr&cwm0jBz_Chl-5R~TQ__M`NdIhT~Wt{21!4~otPbwF`?_i{ExMi>tp0vTGIKwobh`C4 z*E&@31NT_D8c2=yqz$GVk?0t2vr%8Ou5DMEi+0R5^M^Q(nibwXPQQ|!3$#U0UUA$R zwlCUyuM2-a$tT4haEwiiZ@gZg3Qb_Rwp^@GT2UTT$3p6itPOdv8J`{0a^J!}F0~rY z$}lLR;(V!xFJ_S&cHCBU+**Nt^xq`4DTnD{vE3B8=cC20vL?$n|DEHTpzGNMO9sB9 z)g%^4$m&}U9@F^@;*ABt5=)H_@tc~naM#NJk#15Ev3^y6T|eS+J%=t9x5dFgy|K$s ziJO`Lg<#^g-3qcrL}`b-JA+D>v-D*Hg(PoI5}aSU^e zxhoWKF~!>_*l(<8uNId3bFL~vC=~Of5yPify>tK<1S<$5`I&rSVa@9)Ttb?j5kc9` z;jfQ2BXz(U+GX*;RBD{=&KMQcco^Knz-y?6?AT~?nG}t_F14#CQ!8{O;IxFF{hc(8 z*&OXAOCAddSGnYzQT>um@ZSR9`}#4WBGWq9K)bTKKnal8c;Q!iUz1O1l>+Es5RE8B z-y!d3MZ>CFi5bFT^`cWRGaKzCIpx8W5sBdu916f$*4Kkv>x4wwT0@WMEw*faqIZqGL)a zAUO!lk9S>9UH{(iS_B|Jk+e^O8&)CCh>$KaziSkQnz)Q~slQX=f8aA!^-hm5?Tv+u z-mkrerF`NQ`|9?99g3(7`#tVu$NrB-#sr&O#-?{G`H6o1P8}h6a79W=x`p~fiQya6 zaobHAR1=l{c&tm+PyjqKt>m~gsd;(zRNQcj^x4m0g89uq18xr$DE_1Il~pB370cdM z-tSg-JO?{oIvYI#^||q&m9*5j8w&@qBaL#kn#&PbaDIXLPu0fX(z&ea9t&4ZhV@TA zly!9^c;pt3xk?12|J=nOQq|EexerR9Uv0S;#7B|Ef>2+)6o+cWP$aBnM8GRD{7|TT z8^20J&jlOX2olvio5yvT`Wd(u3w?RFAvMgnjRxms0K^7p>Vi{`qPDV{`i1V0m z1#@{(i2~@LoWr^t3%2yWjf^GKIHx~Z;Cp`YGOD6=a-q3}K#R^aU83frTv<#od^d@^ zDZcy#i{@ffS)eX*&3C;D+pC61emC0@DPQ$ZoB|(|$ zMBL{!7+Kx-;Y$lTtbXAf1=L@;8P3UcsltD^`rH{d=*$k&N`)UX(MYNEnZZ|nO%GJ| zxN?!*O^K{*-$>b_Lscm6> z@Tj}@hjz5PfFe#oq5L)>8qrYv@w*+t@7;t#mGQ66#p%H57!0H`FlOe*T?n=sth7Ge z6_A|x{s`p2atWc!ht3POA6v?>C;6Q8ajjJCd2r&@Cf4GqPWhcTh^>Pf0Bz4UhMS%$ zYF$NGt&)?d_aU@}S*`T&F3vcek*;^$LbrYd_yHPXY!lG9hE=JJ&%{_-tx zL(P>Pr0+fMBxhVD;aFR*`ABP9&DGT#s}Vpq?k?_>#yTZd%hMUg~KGGwE|8vavC||CGVTbG%M)TpR`X-d9J>~562cBk7q!_ zSpf|d@hIOyddZojz%Z(-{i)n_j(Rx4@Q=H;EekrFLRXs?VYBJdv~}3$Pn)a}z4>da z!+f3tE>{KtM^<#Cn<`CCHVpU&)s1;fS(Xx8h6TsdO@>0MFV!!5PFW{b7Otd8a&Sky z{8V1EDo(u>cHryUfaCYc@9A;Y8bqkPYWMqc3_jTUY<{A`D~E|L zC(Nnrv(}GpLsmpYP{MEhI;;1EYJyc)0fHXMhYignr6phK9bJ%pV2xrqdcoy?{qtoj zmHCEN;gsQtW%rK}g83x7ZOXN4poX^L*ZXvC84-h>OX!k7rH~>2jEUXtI>J zV7F20x_}(dB8KoH+VzOJkYW#Mbe6nxa>`P;L%iptF|^9rsO4@!XX+j(h7?#tU)lhF zfE7=m>BN2`xn}h8NjT&jk-ju$>aWadUEkV@fyHp)l*8tAy@~Z$l^?t`z9&VoaEf|e z+N$@?p)8AUx#m;es4{RZ@f|7$8~xB+eU{1Y{qEPL4uo<*Vao4YYn*|boS_ezQrj%T zW7H})Co?#|ye~E7p@()m`~+T}MP$!1ZSUnW%DB$&>7u=#{7Ha7I4r|bqT;N?T=r_w zi@qQ^;l)J#x~^Qx8I|Wis5|s{=l~qW7v)1j@s1YFK)+HHYEuO8j8Y?eawlSQ7c>Jc zg_=)*J9`+Ju7{&h>RSp>vjQ>~W1WY~@X^qpEtk`qwfVF7>t#5GEGg_VuTxV8TN-)` z^4e#!G)UK?pCAX2XxzUT{1+?$yhe7WoqagEz~kBRNiSNwzq`+h5ZEL~LMbV`(vxC0 z4p%l26{PhOZMbf=aLgVxPnX@KNmAg~WE}cX6XyA9X9wNt*v&h80-qh!Rji{hW4vsj zPUi-Cp2RZ`H0>i?w{-Hkh#F_CbX#AzJY(wfYS-Gi6|3H4+GAwjCYs!LMFT`!v8h+n zZY&NLh#FpH*906~{@SFdB?kr99o&mfan9Tp@iFh-24zxUm~@+z5DW*C&=H`l&IQQ3Fy?YiB1P4w3 z>llObi=)>Oyly_)r@6ha&CG^40Wyre*G??TjK`%12gt>m-sCRTXi9sHcDdWYEz*09 z#ai$RubJ)C2yOP1ry=s23YXFJ`ThpejlmG>R9tG@3wT$aBQzhTcPSn8K>uS zPTSI%`K*!2R(^ySb3;cnwT%iyr;;vO9r^pDk_~2E-vp^%ft8DlN3$!v55|yL(3jYY z;TeCiB}s8b2lp0K9IhsT&Btx-M{cLP)m->(ZnO0xV>;P1b5^4D!`7$8 z#eW8sH$adf+b}hTUe2YlXTQG_;Io&2eOB9=C>Tk3txpXCXv|ML-{V-zVfm6EEPeznotb2ff=h{|y+K=1FBMEQ3IXhrxTjXTLhi0kB%s^CYi zN6kY@*zvGIb6A~d#~{w2z0)X#s>)L(l0i$DJ==9?NTx=H#dLb6Le(JFu$aM+94ZS# zN3G##Ug|Mme4U%9kY`a>dsiafKOQS$A#3IXs02M*>z@VefNgzu;{;Q2?Yh6ft@;y2 zdRpO!qmj|Up@)lbpRt#xyTgyC6_lK_#b^to;D~i0;^z{*v{xZIlHM|;YE^HQ7l;sT zt4kuvWXSk_5{%*1=0729qGTqBo?e5B)-e0mtMR_gT>XX*9CZB4U)$3e524f)5fo5& zc^=Y%{JcWj$VK0E^Kzt88eV0#J0xN4SJmEdX8h!QU9|SGUw5aBneMDL>ijk$96|_> zZ$4XnkGneO$LNQ}2)kO1e=$CrW+=_&59$z|(rdP5!v8E~+VB1iDQ@G17SXll6f|Rl z$1N5+B)rGS{BgJTM-nN%nvte6OG%y>Ba+UzQN`~~m^Fr1Lh`Il572mG`~}vtlTy~i zAFmHsSuXP2IZAcsbj)`?!f;cOnw~Yal^xaBw;x^O9+v}>3q&sgp{K@=HgkmT@2VX; zy}Ju7#t-ffkwJtDw2eva_;QAW%HqWPH8=@#>3lO~Rxi;vvZcx<*0)uD9r*m`l8RzN zHW($Ab*Qr-o4Q3S%c}QhQf!0*HqQ->OsRJz(f6!qB z^-}b{*4rDoIXzYuZkz}$weZ9ZsvvD&+bQavaP=b{tSo;rgW_znHw@h7cJscA-#RF& zsOu=tia+{Rgj-S#Vf9B_H%rblXvk(=^0iMZ!fyLTPL2}bIuK^To&rjmf^!i z0lo$fQcG$ygj;(mEl+%PtmM@56BBE7ujmH9czF-&&l)`DDwTyV5!7T>so1u;!8h5F zt(Ya;c?l^7s??WS;(Wy3RZ#9KjL6S# zZ7{d*UnIF>GcQvyso+&94$u2n>HTNOV=&ZKw>5bH)IYbc*H(Co zvqng{(MYPRC-LbaZ%@6G|Bz2<{I{3W{ob67)|$inmrS)zkI%y$!gO#B4@a7Ka{mml zQZTGYppVM>;j@jIIY8#ovHHAOB4&WLGONfn=%ia+UE^@;*{_?1y^xz0aCYnLYwBs! z*X))n)aal@UBV--FC7`y*)7?T{CroGoF`&#{+24Mj-WTJNZiu;||% zOEjNXBuX2KFmorp=t=p~U_(3D5^EgX$6*e?DKi!_hvC%8>+ieFOlGr}}0wTOu zR#ws)4^vZ9Z;s~B&?}9{L%wV7_11A#(cE7AEm8m^ojUW9(FEl(b)^dJHVUca{hL1i z1lA-;pt4VKECjGM;uFik@aS6P8tqPfu74SB^)sRrrK>N+RDkt#b47W=hp+ z8VMF-oozwK>Q2t=Y=!?E34xuM6_*jgmDPRmgS|*t`_}^0YXj|D14O@;_B#Tn#a)17 zsameR^llEqJ_}syC^0(uk`!Ozl)v_5JmojZ4**fJ+)yQtN%ITN$3M3tNB}rSlb(1s z>Yth&CJMT5R<+q8c?(&}WT{ri%w;iK{*|h)9bHpAL4i{#cWbm9;o+l=l+51qv_|y- zSHnzBnST08BQphKqk>J4f}8X}khdGk&<=xu zptx$6o0^z-zY$3E=>Z@~U0pGLUD@Q8$pcXUENW`EsGq?n`FKOTaMn#9*4|#R(9wIZ za*jJHFE{%_x#Bm$Ag;XtOck_3$mb^ex}OuNItJ(!NKyQ5*Zc4ct-58+-H)YzZx%Z; zuV+|K#$eJzdAS!FN5|$%jqz&3SgUv2Q{&w?2F>|z@xn_|673}$ z3ezrg63uQ8*6d*E&B0pFNaRWUK)IEHe(M6M-STL$zANC9S5wMbNeP1#GV9-4`r9f8 zy&605wt~qRY(uE;5gBv=h@i`uJhSqVM!t(o5W_kOUu*p!fAOmQsWM~ zKY9roQO?D|0FuigdlAmZ{SCF!`q}O!1~LzTV5t<9vbNez&9s^oHepMKN-Y7?70nS1 z7s;4)NVV^$HnjGcayq&6V*|;JrSoYFAt{Xl?(ejysE@!GE-tmTZ(x@Y02JC-!DG?$ z+N^g1GV}*<-`N;ubQ710?C|4F92}fw|Crn?dF^;lPfsrAsDznZ(1=0aJ2-0AcHZ#! zTo5o92PZinOp!~1@VUZt&7;I-bmh1-Uz$tFLmrKug%_QDWx5PRS%}{hVY1TtYJbj#xjy$r;c?=Sp->286!|z%TcMb8L z)W-T!;ypA-$>rAGv^_(|igz{H%M25z1ax;UN}btrQHdWXo%l)G*@>-wRNAWAzhf*? zrz+D(Di$l=Tz^?%OeTuH>;b3uh*L&&L;9Wj{nqR<{${tO!Lmd?G}X|6J^69EaE<8b zq$!{#j_H^x%)rB+VHZP&G@A*vp>Sv6S?1Y*dPcjL;>c=sQFOnT%~KEe*`=SlLCFNd z6!&uy6wuA-Pb85}b^Xrr;Uf(Z(`VhAtw;f<%@3L0dt+%5fekB8DbdSU7Z(7P?;GmZ z=Lz6p1(HbsV;&C_KX1Z0R!!y^&(3o1eZt>oXNk(;n?b!en&U~_5dfFA-*1MpTPw|o zt6t2{t7cQHL#zX=aMd9|5TZBTehtMOP34NA+ypF84Dv2xsa#PdKrn#k_HVJ=^D2Y> z+v5dN2E#0c0uPqjfq`#MI(_UKo1J|Mb$6z!V1k^HsI_uimVlnvwJbO;u zrPB~F4gsfaLu4` zXFwX26vr*%vwE_@O(_C1HqPT3&37%sf6^*KbZBi;5c$Loxshs#4nlr(tO%`muVh@D zcmMeLb#3L((y(k4^>+t%8N7-zV|%an@)+}*4(E;yhQ=w%>ehZA>5rom963~(ny}O` zSNGk`9iIAOP95a6+EUi)*FWTcZV=9b`zx4V?fd9!!bFKy|EYu7M-bqnthRm-&fWPT zvrE zZURhYFON40PM^;a5E$bcp$$m+TrYo^Gy~)RsjOjDNE^l=I^?agI_t9me>ef{4xUwG z*CRdKY;BfQRk2(!A7c;_KjthtCne&do&t~aX;}2#>B_Kpm&-f0NtfhlTV4)CD(fvbh?);#TPIUoLW!LX|`x9BIk+fRP=-92cTYn1D-Qcz* z`Sz3hhh#9rY&>0_^U+2yvenYWW>vhyT?{@h2Wc5#eU!FA8^h>pHg zwnBn>2z%Dm*2=8dfRYxEWQ!_WCdxL|kdH>KSS_P1R(xEfI9I0l9=PtUv_%FZB@z;n z)#~yeu+OygwoQx@r?X`#0GS(28XZXJ7SzJ&?sxL`a^`K%*{$gR!oqydvSjg%{LSX= zNaqbTjDsDVQYKx5&S8K>6by{IYcNsJdcG2j)Ps;$@8J#5h(Dj8N0Fr}&A9&oCf0mR z&dfiX8W5(S%o60I%4IL$;B8Xvxo)P;!*MqS$r~6)Juq6Q>IB@R-f;ZP?rhImAh(Ie zX$+4OY%e^EQr6n9xH2rHGVl@}T%a>%>)^)NM}KArx=;LbQUX@(Zh+G7e0K`lu@0 zTDW(ZGY|z%+tTqkKKBbh5t`w6YOOq$w{lxrz29}v^T7nMm_pXAeT7$AoK=Ehp;R+b zY==!)!PR}}iy0*@E-pG=Zc~c6j)~Y1E`VH&5-ZyUAaMf-SQu<>H=Fw$9uIqPJ*VSc zr9B}3%>@n{wItPuNjJd3(aOQ9pG^|mcqE0%j~W3KTHd*Lru}feeIH;wm< zMKtPW>1fu{EIO}?^Hl7f*dJfR1*kG@vpqXMNw$h40#-Y2-&{2uCfTN=|Z}DK0mTt*gVp& z-=6(sjN68F9Jl_f!5s=jD<9uw?|p8rTK&m5Je~XX55mu=88NW=7%5kqWEUuWLDkRf z8g_G5Yb6u?{eUuYASkt@=A}8&n3&ZR` zQrmRd_g9kHSR-)dkD+h=!*yCM{SCindVID&&bZIL@4fCNKI=rvB4}Zw&g@er(nvEL zLhmLJ<_Q+A_1i=KLrH`dSJ!GX)n&m{0f3%=YfIL=h6hJm#!l@@HfDLUorl+zLy_ot zlN}I}0QwWrgj>+mk2IO{1&`VC?e)b0zO&Nl?xS{QCNT`(PXyuT5No)3Abf7{L2wjy zgw#A1GNmD?GB1`&xhdKT@v+xdr5wHSGl)^SKQYtZb)4@mNEn*RufrJ13KZ@Gde*O1=y9Ov9u;92NA=}%Ao*x6}sF)s< zS2%M^6e=l^n0(myc_EOF19+M3Ck1lhB2$nmKa~2H_nVwq7FdtZlYRSRg6osKXUpp> z{8*a;{3wpJI!u=CdTtarpE@`1(oj?|74yuQv&@C3K_-&tE2qTMUIPC^46R$?al+i0Yf|F;u z37(HV2YI#<;{}wrma`R(E3@qWB{DmnqZ;oiJ5vf5F931Bxv0t06aj;;HO|8@_q#y* ziDNN8dZ^-ZD?{N`JuQ-?A}l#lt0MHm1;`B-w|Tm0{hJhenaZ_e>==$K@RoM_?DyKn zjY%f8P%rQ_{icP=EH@f^WtE*LjPo6FhG~s_ z-pFp-LP>+M*QHR@g}5M?%}s8Ff{4oHQ?FW0DFQMZC~Jof?f@q~IEI0cPYIMbcQ)iV zF^Y#n$)`0u%t8cIyMB}MY*^XQtsuV@U2SEU3cSWqxG#B=0d|l3i3eftzE>)4E{BbY z1Bn6#!yR)9E|!DyzMc<*cz-_$z&^CL*~{1N{s3CdsI-1iYFuJ1qyi=8GzcJX z-!)xNOpTNz{(tD_G4`w+eHJa;Jf(z;;$xsF=|25O6%WCt?hQ1gs7Nz#Y6DZ|Z|?_0 zY8=T-CSynxDw6P6FycWwTX-c?^L4H-ekP!IelkyBE~E3~Bi@;4SMTdja?F3#^| zl}sJDU-jK@P3CjusGu16P`4vS-5(S^Z%JYj^GG(Tpm?Y?fZMcN@AOp-o`w@PZ}pdh ziGRq{8AE5xrVq^d6hOZhc^hfYMWcx){XX*GW_?n+Od3xXQqQ!V%qQekcqF+}H0_ZV zq5&p%)?Z96!~G%Jbf+TT(;T2HTOvJx)>JeaagCCnORJdpg_B*p+W8m^7$+K3gpqa6 z-zf$4WK^Z4e{>v?o=v34=94G7HN8R2rGolVe*m<#d0l<@5D{?y;<_ZW($gaReCRkA zrOS?|{2-nqEkIJ-L!&W^ipxFH@1%KQxi{`vZ4GT(6Tli)cnN3w0x6y0c%Xi*!DyH` z-(p?;4fOmJ;(+mpksZ~1=9z-JUJz<(i+E4FKTstvUc7UaR7AaSKb=p0#A0=LAe%@7 zu1}ScGSXx%C831GC@N)?N!E#jh}W8waWvoW*XQwq0TlFfU7wm(Lo!%#+WfzYsd zT`HrnA_IF1NA@PNgflf!oFoQ59v=cB2o(5OHy>MF&};DRzzsAJ$^P)C_J@HMVT76O zCH-D_mG3k+t!dpcwUm<1rd~&jnL{QpvjI^|@MhA7o^xd4j&S-IIn02hZr9L7GcthVQkc;%`#TN<>aJE~oF@eo#QE9`HMyko|+ z(GlM`RD-%(ovxjMZfTG{Tt?$9Rs2cLao0dvB_TsJXGcIaBk#MTCNZ@~OY`mfPdD5Fquiea#@{2!MgF0>6JQHup z1PonMx?u;dUC9WNrC;sA`9s06-E|vU*A_QoxB?^gVwx-KPBeBP&Th-fs}|C4H4Y=@ zJ$iN@yB;@QJi8C~JJJ&oJiW}VShrsT=b>};C=WTCL-)R|0q3p=6TvkvpM#0p<09ml zZyk@g+d6@tQ~xlXCEt~@(>?t@76aTzw(zBU->mZzOf}{_ABneg0&8CDK0(T?Gfl3h zj2j75k#X_&r5cv|+Bzs!Zj>Fpc&wzIXR8e@)FYcm`YwyTNlJ|ma@-?az&DkgtaO~E zLHB3dPv;rt3Cz19P)y|E7mBm@zq&p>{DQ?();qKqs>5Dh=7yfbTl#dF|mi?Q9ye+I0Hv1CY>Q)RoPhd?+Nr^Kb=Z4r4I=?L($>BG3{xPU z7_4%?(3w(3_O5lu)%e4WS%R7AHK^CEcGzIUcjE(6VrxEeE}#!~;rzm$yEtQfgOYJ_ zKP(y5IhC51pjq~@w&c6kflnW1-=m{jitZ5X=W)&!8&UsCxm2-KkMy@k9D#iE2>hNZDxAXJ{>=Q*i@zG~jMq_1Fq+P6sk=sea z5L&)y{OR~IPC5_xslYF993tCPU0q${Et}c&sE2RUmFbrGvfvyPFzgAVD_EU2|C+`_ z8`r(^KyHdyzI^Vilkqp#UE9Dn(dG(&vbu=7?d0r7YCOel31~EUV5PnEAOqfn*&{%%(J$;Rd_gg5*j6YPHq7K0T2R{yJ>ba4Sg0(To`#AF;l4x)dxt( z-`Gxo{|wZj*Awu?afmSJ;X^bIt4Y+9s>&*za+aNCII}j>Z0k^*J_|zO5|E1skdz`m zOj$UIdOu>9L?)_9#UenYagQQUaUrYRj)%D+lTsAg>R)Q>&1S?0e4-$~VQTkZ%QRBi zN<7e2(oleghPsD3t?fhv-ecEzFMi1fEZDhj_QFcb4>UUr(=~KHl&Bg;zuX>7LoX5Url`c>XX{J# zUO;0ld$%jgb$0SZqglXFMjJaFNd>XY5)P|Gen(ZVw4R>)ICI0cCjb#pm_pZ#^gyaj zp9|ghMI>GIz)3DD&HZ^l>Qwx_ zTtF+p$i27;g2^E#!KS+J#Hj#!EPRwDP)zzJM=EIy);Zt!^QTDh5)h)Q zZxW7Rm0CG*+frubJc*qYrFwI`P^0ItVWIaW%YOJuhaNm#&kL-27 zK;q4KGo4J!uL$Sw9QLpo6=*+hCyqwNPUI6q9Au%ETm0>aKGI*9pUD!YR1+0C0=TGj zVQ0X)OwCiIzaxaqRx;0BxASiyc3mBD{?&O?JqfO*`mfEwfl{J2^;@TZTc*j=iM)HL zb4jXnycyaBy%nI9{I?R>o zw{Bi{AP>C^Zb6oYvm8tiCJMve@b~6SGYNZhWjj13ZD*&_DtTsAWn~@|sTnW3H*j?U zk_UBzTI~0vI}yMeJ;fDC@ys9v3BZ!Ro@#c*s`uwp^U!S^)5(y}D_@L*acZ77|1zxS zVwIZ^%ssw`K<3Rv&*Wgnlk;<}sU-amH^*y)AA0_)(+ln3@D0e;x|fdwZhC`PCeBLm``>~+?W;w)GHrD8jab$X+5m+YTjU39F!H8bQ(@rUfheq`Z8(anC0Y7?o- zhW#e<0hPuxwel0?M?5?UK~r9r4fVc#Vr^9uTyUuc9z*2R03TjMBZ}1cuey6Eq@sm( zoTDimkrq?6kX@*CM}b6MAZmjK3l>P$kGP;PMzBti3ARn&4TKR@I`!HpG!RB;4yK9m z5W7}Fo=qOTV!nWDociSA_Is({h6)k88M$O52@7o}IWn*zYcI&$jCwy?EwjqtTbD#} zqJPDtrKUy+SHcs*_md-2*}$s{MwjcLeYI2jA`ggL7+Cl#Id|JYx2M}uEaeAG1xMrM z*56+aMtOGnV=bR*s_pH@(>Xjj1a40E(4=Wn(bdJ!pB1r6n>;qekDEOMYn<^v+o+mM zlj-~^1a;@q4vmbFgJxng=)ru`E@5G*vDZ7RQVF1|eU2Jg$90m@p=&BUd|IM^=bx@v zN6V0J8`x3^oHR(3c2mXJ>QjnC|Ah#Ck}{g!jY*j!(An!Hp$0%vy()7x1~x$* z{(TYbM>q+)f)W_)rXPc-Lf&Svd2-y|h{Sxru--5iMiu-_LM9UGH$ z?KlLD2Jtm&y9N4*eknF7DGr(ejb8mvyAz>{>vPzk#|(J0ny_$1#DlRYNyullUXAv> zH`MdR9B;$DJSq-L*>i2N@iz{(;DOCKKWYqnV`RH|TuruHWE&z=$VbVF%c_QiW zbhdpa!&}^>;Le(5gVoAxnTA;6r^=5JEs}z{Q5f5Jak_Yf-%`*ZM8FKzz4Scaqmw@g z`BksF9npQ_)x)wRKo~Tl%qw+BD+7D>1|n7MxC*4xq773Z^&{mfvXv~wF-uUNVSg&^ zf(4}eANG5y*p%U0e&m3MaGYNU3b}>c2?zo&A1iS=hxk$3C5#@w62`k6{w;#}TcTnO z#QgqUAAJJynLDn=cwE5ki-m+pl^U|LvP7h|d`UpqqU(6<7J@!%gAQGq*?gtQ`ym$c zAZt-DIv(sXDYn!jo%m@T`Z<+f?Eg5bic_Gj-JD+xh?W_WiJWxB_4wC(cy#&Vnt3I!|@d%q;=d1bt~ zeP5UK2*~@&>g?IN{AVpGp+lwC8e@ZSyP_p38acv{`%{wlezIPOx9`Pm_(~gQ`KZL{ zCw_h9Snqq5@a?t$nxybYhuiM24>dXp72-R69`QT)3gK%i+UYjrRW{1frw`jDvacZ@ zkjA_B)W~F_%Z6$v>b1i@bQ*HYJ~$)602>*Ny%qVScE1>85#@R4V^ejaCugW zcw+&{unnPu*WeGrfe+n>kILfJIv;N`l_OX_oRT+0p?B`2h=^#~ee=z026ZwUph2oG zi-zOEx{ky#`L=IG@Q75hQYW#Sy4ybF`!qfMhZ2K;M$-H*CWg8kg5+xVfofD>^So`{ z#)n&sGBzv1wJ`D4ew;5*iE5Fq2`-i&(<`dyVTB|5y*WVzjnR@y8E@b* zZGtCT#>4Kp{gc5NdBN-`t`KN&;~aBYc;DKu-d4 znloWAu@sF?Qs9~>LFYh~oP4nxH6}hjb7C}^EhiwCt(PNNz?;9kJF#-}>boeBYVW4L zZd`evBUM?D4;hEma?ngI;|Jn8h6IlfG-GOmIg$J!koQ1Xqiz5WfnsC{ z*J3BPdT;yY^#=@;B6Gk5gRUy)Dhgjyq_n5pZ2PMd3{~8C!NZ=c6Z$6D8Y*YRdB*Q` zY$j(m<^5{%-}k+qj4PW=is>BCNa-x#T5>LHFxo9Fx?$;lTzeo?%~czb zW$P)d7oDs2Q<#E4F6S{IA2IAybP1}wFRfUf&{}(>()ZMBwfWKUTF()YU94;KLDvH* ze(3-QvzwnKqkB_}hKHeAvYs$iz3KrFi*r8)6l}(|4FudmMo0CsK?_O{Z94-coQTjqFg$L=3BEj z3NpOle&rx!LSBQ1GZtZ39~7rELH!xQkYux%XZ)r+vr^qi+frt0Q_m@3@K`beVXN@9HHcQluX*yTb!q_?dkhF%E{4C1&6!*j8FN8Nz^#wRcKDa1n;SRh z?e<_yH7w=;kgwXj{btBFk70%BG7_}+hxtgjgssms*n(B^aM|q%9mtu z1#}DI5J>pj)vw;K81B~dvT~Ka*_l5TzZ|roMazPdI*sw+-(^Yc90#J`L`r zrlqC1-nF>9jJKb)=Ah13lk;{aehW*_O3eU*KRT)&r zb=cW3_Xy22x&;0#Ki}z_rfHk7Bc6t2np||iGso|%%hb3~>ZVLx+C=l8nv&OK=D+wE z3_g#0zCig;i^2CUgUT?hA|Ue}T>K#<5}!B<;g?Sx5!Ip!h9Y?(nk`e*60o|3R1Oit zSO?yQZvl|Mn0lO{iHPwXfT|OVVtv$ioBl5o-D1WOI&d9Nh&)XAb6jTY1pdUVbW>3< za7d~iLhxNTa8(6U?l;1|bL0aZhzV5eXneZ+7&OZ3Gf=7J#YW#E$mM0aaDpq1ab`-* zpui%rts7l`qe(b^s|#)242o?(t{xUaXZ?AI@*Fl39p+}+@%m`aOYPzZ@-IDvZV9UG z?=ku-V>Pd;CJa`Z+r|9fCj-IG#{$j5rEFUob-6H64ypS4BE2}1x65bYCNKsTN^0S> zXMsVauGRzm6`zL>pM6G+`{IFqRovi67qhb@VcS4CDCK#{v3#BmOjGFw zCXc1-*DV4rD51g-|K( z_k@g+!Edb}sr{%n`*2o2$Y9YItAl0G3Uw6e7pqMKk3mcym+Ob`=!KgqYb(FTU}D5! z0I9y4vJN%aaw(&JoKsqn$$lAD~S?jEB6U7|DZ*8S`B6LK0? zV)XByH{Uijdi(2wjux7YGp!X`QUa|xFslT6q1jt=zJN>ermCKUkjqvqW1u|h4>DGY zC>ahW3{^EXtdU98I1D8)fm!hHH?6b}`26O#g{&2r#2RX!q(5K&1{>}uoor;4nOZt1 z4Z9s@+uLH!$|T_IZ5+H9?o}A>7;^qvXqv#w4zEb4_}JO?IvRiSu(CF+^Y_X1;e@MB zEOv+Tx<;;`+(LE^d_;K4yYw&uP49L_MMrCn@OY{H)04$1OF{zC0{!S`fz~pY1F<}f zx66!|^pcjA(gqb0s+#*MLn@lc9}*DKKLTIZ$p7eA4zRO-_GmHf8t3s+MsgrKfpHQB z7fNrv9#sBnRMqZ}kM#vE5cw#Zuu^vyP@-t>Wq#pjm-|OoR}(o| zjZ)J-Gdv__B+Qwkl^?E(3vafTC-mI@XArO30E-8~Y5e3mF06w<2M!zbUL zEdDwkUXRS*fkW8Md=wCeS>(HS2nb@?G^H~XET&y&k3IEgfD5wo9&M-eC#{@n^N|X= zs53696TCQ__Je)-=i%{&g7)xz!=|N?4z<1WF}pbjJ^M?e1C4k(J;JbF2gAQW9AT#m zC`OD;yO5S{%+*D)?>p(_Iyp_dUD0Rc#y2z`Z6RB=z7{;`jwbrLM$LxA7N`2lkjuzf zuiD3`SGMpWMv()7n_-2wlxzLr&iG>C`US04m#TrU#txjN1K@gLfwTK^8p`Fo9-u>& zFjZ|ilqKHY_#AuM|4j-wU4Do$;dS7lv1o$vw4)v1^s*iR;Rma(!Ezcm-9?)+%GA0^tSZjvHlPRcf)pNkRM9N2yLCqMn(KuI#N8FuWR)TuwfNY zoFar72?^PI*cVHw)ZP^89&ZrO3Hf^HGEFzSo9GSm7#n}gH0;drZ>%!^q5o4MHR8H| zw{3vayzW$HnFN^xtXt^HP0fw1P|oRVJ7o-p>}Ul&(89Z-}UGIWB})a zo`#lMMQ607T2nM!+z=RlgZ_xYv)9*`CzNJuXUcP+-}ratoJB{oP0KQ8GduMEn+LJL ztB+$E!Ij>ts@Y186#68l1Y{4h#oowa+%*YVPH zCbL5l%e3O`_J;S7;^k1MB19N$6|9P+>RV8pO6l;^%t(dWkD38c21PNN_u9)+vELsp z*K9*4Id)X3lrb1Yh2QjeC&&cc+k)VY_YG#y6>kLAZpa<^)vc}f6&2J|@(y#_YHF>| z3LQA(@JQ^h&$i=Smc=?Yf>Bq?gZ67`H%R)&#>gG+y32x!@v(8s?Yfw`6(C4|3kI+- zuB#YI8BxL4J}eztn>AdIB9dTXq2mKvz9=WRwIsy{S*g=mg?P0}YA^-{ zh98v;haZnq|LuGEu~VQ4SL_Q|=>?w1)uAvbbl=ToKg05QTZvlx-d~Y#Q{}gf0rio& zcx;}9!9sPSOent+`26T7!L(b)>)BCV;2d^D5UmJ|ixt|GDB&UM@R|961T5dE&WEZqLz_W810mC^y=K+TXnEZqL=3^hLb_;sV$UG{{<- zHv;ft=``xh@|NqKqJbWArA}{72=LZKW*IiNe!o^4tQ<0^v5AC*Rt{hSQkr>Q9E`L#nZ#FD&#$zXUV{HVg=n z;U04};YoZVC|5r%l=@c6TO~Y<^#7Rq3Z^`|rCZ$H-6goYLvVKw9w2ydhY;K$xLa^{ zC%C)21b271lXLQ(@7%BM54aRXJyp~^Gt)D>yLb2MwH{`;*ra|?rVV{dD_x<~XC$23 zGS<$cl~RrMEvl-ToKQsP2~(FSaiJSOqSnbraVbF%^3ik+O94V zlFsq#O>uelw7Po(SpYeyMW5EfO*2(1Gj#RjVkEK+pX21%dASevvKEaW>SNo^LsYH1 zhkULR5*=dZqPKDM;Ms6$S@((Jyf^*6qeqM9y5nWLx5*RjO3^SgCKUW()Pu?ZN*C@N zXi1s1c?FJK| z>4i8r79`$jb^}c`3+xY$B&b@1N#4 z5*J?q8LuBv^FXz*!ES@7Gi(zZPr!qHd8BvbgKQ^@28D{VS)_h!GqeEy7b&@ecG|6y z1Za(Qhbh7{TOvhzAA&LF+eY1w3wxkrSxq-XE1f9?tcO6(5|&ERN%B%06{hcN_pVUR zJ0b?8xnsKTp=zD6BC)QBopcT2>DEfBpiuC)XCJPQQNS1~`{tNXcDu!yc4h=BB+ZO$ z`-=0nEyN{C(lQ{b*cG24?Gr9 zY3Y2|+x797I~=*}kfZDPU(=Xou{Hbr207`q5T9|F#HuPp&|b|i z2Cra{kX1IB&{4+lVEQdis67PS_QJGU%6Tsw2z@I6Q9V5h91{x#%d8!shNgCBq*7Y+ zYg<~PR$^dafc%?|vqrmX!5!z*4n1qrZIMDAz&_kU=^l;)7XKAFlP9FJdpZ}$i|N#FiET6`A%q{QspPJ1yH^XrE5w(?kv8?|y2W1=n_4r~4TYx{U^epX+~%95by0(cS- z>oh9p=f}*k(n=c@52Ms~gBy$&T#tGniG|dhdkbo`i1|pYw3JYV4ZBx-JDq8df0Sj- z`=%xZH?G!ef_^gK2^3k3NltG1x;rfB{iNlxCz`=%Z=$kYoNnztoX^o|RN5c5OI zT3Hy5o!kEqx&hKEN(78z@&V?6zccmzHEcx`boGH~Lc+c;0~n!_kr5@SlYwfqm;xIL zV<~i9ctoVQFt9Kj#g+J;4-tTlI!WhhhEY%^@~g%}@y%0NO$~y&VHzhQ@J16Kor_j- zAhp!M_C;l0Z{z$aYAeGyb-=5|=aZ3T{a1Vfp^ptBM(a=R@1GW<`Dv&1jIdR_#&S3w z;)6|%>|Zuyue&|xeq5lE~?!2ycYG*5wGbdqu#Hu0m?U}12Z{H7=@tPnLT4XryRHMl(VhVbV)q%rv z(NE&_p$Kq6pcq}89q>2`Zzi&KRi|5yAAwh;kb>l^;EsXj>3&DtOR4Y82rZUWR}O4Q zh@RLjppmFc5h~uj5A>qLHhQ;NFGp}lX8`7$g2KY}eamEwWt5Q4&Q2q!lzutIX zC4x)uwvotDrz57Gdv9VN-Co;Y&~rU#O`UyAAC3EM7dK1s=7d53W0=dI;VEIyrwwY~GV<@Y3Z5GKi= zcO5!s%Z+~bk^tsE5V76kwtF@B@p>QL^P&TasjUj5>U2!(snjF@r_GF zh`@Im5O@8;7U}@Zl(9q_{Crne2#mQ~$bnfIl@6#R#no8(@vU|H)_xeZL9_4N@lc9;#8yCIoH z62b!k^4kbD+e-o70GR$_zBKEw*nE}npQ6T}-^oQ7B@nI#zO&y8_}JLcjTBg*L0q@J zq39@|A2=co^yFXw+HC)(2VfEfv{foBK==T-zwHk{%$=Orzy41CN&YHj@%b{M24LGq z2g5nD_>M(aIJ<|jD?`l0icG3sTZfL$6icOZQnC2HVAtv!9}Ltt#@=1uuLULdp<;B%Zujyw~g2(8^T$_W%TWD z7rd4fKg{13K|g<1iZ|EJ_hWjS5PFZOnBLyrRkxi;El7;eY$MQ` z&TUC%jLPX$HuLcpK=#uZkJ*Qg++^M?Nk2%@3_s7T_}*R(v%tE}JA*bh3(IIOxg}!K zYE*@Yt75ql*y1h0qrV43YVO;;s61GjO$g{5e;qQ`b#-HCYs#e;N;=mk!oqRUUO1}w zofCv@m>+@`yyPNRp&Jc<+5G!6jo%jBct^8iU$d%|6;;buoB8S5PW6x!%cHboRZVKV zl#*8i8KSpI>CSmjAl*HVQnS+6`gXVtBtMGI`sLrqwnw#uWt}y^T`#{L0li$5@%6lY zt{x=R0=r5jNCU-6y4KPjI27!1jIJc76SVnW!E+7-?dqVvC4uy0AWK8vSe08wgsGSa!$#p73BK`u(1XjVbtswz*yg90PfbeDLKiBOuzWDqq|9cC#iD(coGh>YIW0qp8}Rk{Yo300lnX5#?T zOE0##%QOCJRf4PQt5%zXvgidL3anf48jG?pcB_Naj!#F@45;-;OfCFRNkqL(&65hdDk(hMEn0Z7Bz?m zzDngaTk{UBqNU1?Cm9(7J=*lTsdlbBvVg*oHB!|=C4y#T?{y|Lz zu9(U$oiYmhruCA~nIGh4$ul&0VbN|C)8YceAY<6=0@);mG)~0Mg~kHwKr8HHBAV8~ zo%@TFo|vXH+p{d22LxUXg4??bAEy(O5SiQCo7YY=CXJSZw$x;ylXF7*;~3op0Yi_A zQr~tmDm1#|DV1icIs`ptMS|`Xh#qWciHT{of{FZAlotfVvUG%q!(X7$KD($JGaWyl zJmy)m8s?!~>V&%ZgoBP(77^rJLLwscfDzyIagwcC0>aADn0X|oZu+-r86@S-Zyaxk z#aWd6;eX#1JWNoyO+eE)q(7X(tGK7T z`vOZYExIT;Z6&Dn)(9qOp&RdS6e0noL(K+hK+|%ug8q}8`j0pPnvPO7lzRt%0B!V4 z5XC`t>WkuS71OMycHN>h#|xE`Q1mYv8boMj7MtDOqD6vDQCl*ip9zZ$aMS6RwV3Pa zJ^H_!x|LtX6ZVkiAx~?_9|&Gfc>6}2dJPRmk#8BH2N)Y)sKcr%W%4`#vS}SW6)O)k zu&UWHeac=)5%@3}2#EZiy=y+7-fBK!Rk_enVUl;?(Rigjaf-7t%p!z!`bAsQ!!yGM z{o-f#FQ&z)NfE=a$Ak&)&l1y;B_E>zS%asH7#qE_miSNPDS|oh=>t^fHnH1a5Cavu zf{0PJ(dm`?3pqEkxk_~V!pn(4`L{UTJV@gLvv5@r@dXLKx*93u$+06whFNh{Ps{1p z>c}5v$cxd@&|TX1+^?-UvGwFFEiKvCEYTIjt6qEdSX7c?`Bhtq4ne91#gq{^>6$niPC-lrAl=fE|o zBihzGa?C^2lK$r={uM+46$04?Ohg^I`2SQ!|K~$C5s*?GgGLqtTiE~g%S{B7kq27M z>yrO2i2omtLLOjC`MU%Ed-;A#5WpyLVtl-bi=!$cI5@cH?U7jtWOOVDtVJ4#K-Y4I zP+io2-BGp?1T8Hs-n_`|64psci)tcpo7Pf5>L{THKwSoON- zX!@UZ_n!~=K*rlZYKhdWr2X+!xL=TfE}0=NnEgw~^_b-p zHou_>H1yP~8E(x<-xSPdt37vt#bXQor6lf&1y{^n+`k^hqCa4M!fZz=G!lOgbS;;+ zMnQ$bn{qC(%ma3aiz8^?i`TGoNK@vXoQ%D_E75;1j(?Ia2B2N14zo!u_{4E^h_+sp zN7It6R2or!pf=k6F)+oUP^SS=Gso;Y-==B?$`vab3yRf1Lgw1qFfFuDfj{`cg@k*% ze89vWcJ)7tE&_PwoMC|pfaP>3K)A3+SxbCu_$hI|NWx|6a8xDQfGaYjQ4_v#dJ^>E zW3idfeSss)6uZ2G28c=ZWz$f1g#W&mn_3WhbaZs8Dk}T}0s{Q}7l%GRj*gB!zL%ll zs96;i6;aMcb%{U`a0sSrj_T)I34vbCdej2nb4#McKB?6b=jL>cdlll*y;RtEO7#1y z+VbHU?|eoAR~=RD>BrcfIKNj|>!lh>CZ-887YGQ5o3phQGW}|F;O!kj^Q#gM#>B+j zu5D{0F!M5|V_;yQqw6tOQ^QyQZJ>9#1N^Tc1DMoZK!y1ClCKCkVW-bw3RJr)ctKy~ zZ%aOFtZ^6x^iPQZMSw3%x8cuZ-;faOVmPO^7!bPijmou%MklT3|Fe#iK(29twP}w4 z26O&BCpo>CUGoT??+XR#G>U5pJA#>ITFLrdYfuBt*bFN#-!jz4rdpJ(pnK){#i_Q` zOkN7{##iuu)PHKC{xJ4PRp4v%sWM=ZseZ|hr4j5wNT_U_rz7G;B+0=bYL=N2*65~c z4LK_0d${7z5jsuf*%YGzEW_zqIug)nedr&ki^8G(vM4rJApis{6&aDd-p_*VEEWi3 zdZ3Q;#h9L;`{mc>=4LiYM-1y&sHAk4AmANNdouEXSCX3;6v9##w5+~#rpCswfL*b- z#R`IipgiW6%r(1lGP~GFgEYA|31}%=*GUJUX{6;0W48ae&t(=_64Azkii)b%NlU*< z33`fR2vg^B2FUGF_y7YU9;#f3#*^X<`t~&2H`Ok3=7vw#Ie@W@#bB&O@T5qy1plV>gLpqbJk>JVhOUi9p>YC_$E@~vMRPe)q!YOv&i@&$*-#(#owWGu)H z^|VM9zdM>`wYAsl+1Z(t7bLhfKVvXC@Fbu?WkNiJXP#)3IvWMOe@n@1)w+sDwiY$# zG%Mw!ROEJ;4}j!>J!DiWGU)hl(yl%qH=knD3xxNJ$UA{O_$NhEX3*pDkR1F_ZjS5ZG#!E}5e@&G znFkG7O4}^^Pe|GHf-p8QsjaDrDO^g#!#U#O=O+;3BUb!`3d>CiL^|rWa>sSWCD%F- z?7Ij2A?Rq#byuatbfT{br`F!<^1auyt>xo--=4ZS#ITXgpIEDIGCJqJ73fKkD7FT=E#cHx%S z{L_+3dd=^1$6PHj{H}-3eTGX}GDJWgQg&@>j+&d{t9S}^m%Azap zQNzxr`L(9Ozp=MATdwfCx?%%6Bg6*FdH%=^?Xh)LJPEMRXnuhF#8aSv+_Uwbyb&83 z(0KRot9N9g|14dqbk1b(#dklQTr~Twf%7_Oq-Q(ti_lY;lkV;B$3(_ z30J3=+s04oanqY4j4B(u%(T+u4)VwCn54oL|BvlBSR5k$L=0kW^mEY%=s9cQnn!aF zr<%dxGP~v@C=-_GJs;(_FT07~XSp;QED?5VaSLETr37^2QZ=S|SD%MjT7(^JWx(27 zA6&z(BVlggY|gZGG_=)J9JZZ?s?qyGTt8GQan_09Rwh0p2j| zD`MJh7}!%33?f14b5r~j)tlvFW*se6FG>myE*)>HrKFztVZE3MP0nO(mDL`?T)E{9 zo=4u62G!)*)S4+Xv$kY*Tnh(x>Se{@V}9z`$W05Y6%4|vOp1RE_`kB5SIBnO(`!iX zCNZ!mVIbo|JyTh;`IuSaKGE3zSlr_ojq$SK&El-miE}#PTf;nr6*w%D^e8*s`|yG3 zkq>u^}bF6I_mHvYXesY|6_krZqa+Xhx8zNZSEh7VbuDPb}JL5qbxS+#y^Hn zARJ-h6r$Xr8m-SzeGls&OqjEPWd4M1cvQdX=Zb~V1s3>#faKKqG#!IfJ$D|^)~?sy zzM9(a=@fN;JUxFuLp;8h?mPO!e5owrB22du&IbK3j{wz<<$^2QJl-zpr`>v3fskn#Jc#oABXUR;PQSVomc`EoX6?PL)-glN|~PS zeqXG_mzQxFKEQ_;l9I%UW^p^@mX(dlVDD~iq2l5W)lF!0kJw6gc|b5e5Sp|bE!LjG zK0LN`zB|2?A~)i)Ez;6tZ(cFblslT%lR~pSi|mzG`bksqX-zwJ-VxhuO61^V zyg4WG!`79tDP{-h&>)DkDo4gqCaTeHc8?i2Rqct;!NJ;g-gjm=PKW;``qS!VT<3sx z2@%#J3gjzm;d6Kt z@nhDwj0951OV4UUnmjGIvB@xphMp2~qK%uSE~PMiw9QLDD7{ReNi$o^N@l6absyw6 z(X+RFe}&kiG<7)YFTTs|J@1yIeNpqhA2zEwBPcExTF%_O(+RH~8vBg{ShKw*h7-cd ziCMeAOw9Y(lesPunD*sY5xEN6jehK$@)|@qDYwJ|H*2Unm|jN}C9Stt;=@+t-TmJQ z!5sJL9+u&ZHm?chmQ`&QkIgCZVtsR@rV?z>fpNMA(IiZwi(ZdP2?h~14$+H~)JQ#;*Z8`!+GI&5m8cKEraO_o{SzDu^$ z-}8B_+WJohcyfrSHFSN0#y41Od$~e=O$)>GTHx(_J`*s8_Q30`<@AZ+CV*Xx17T$| z|NNPZ6f04(X0r)p4ghZhu2LlbBn-f*Wj_UnGf3`gX^>rvFBGG&m5jSGp7HEdi;;QO z9-+7pyE8~@FVO%{_XR%TW+S3Q$Mb@sStcD(9R6F4tB4V?1)Jh=q}Qten~PG%gF1Z+ zKbmjXpDd`K>ePf3+hW^05 z+}j*8RwG~N=6ecnc#y?}oz?E7H5r$Xq9Xl?mBQjAhQxtXDurlfSeGbH(>wrd_JJ-d zvpb2dC1D}j_jB1wiRb8x_uAg5ErT=Ry8L1nTAtXC1WlhS?`8RTvCBo0<>n%-d)uLb zmD}c49oa%F`7^6LLY)RQ+~y}|_!#fE1}ewhP9Fca7Tgh5#J8C5R4%Apz>`J8gQwX1 z0fB~wMxTa(cTV>D@&a&Y9sg)@Q>K{2`BSS8YCQon4|!Y&?Vo7RsxjT=+2RYlR|z5b zc&X3B`@1&F3GDpfl5E$y<=tiPLGw|RulFgmntl%1BS3~7+l;}&B!Kq0+`#cX?F12QmgQ!@#xMJ1Wo=uh7reO%?Gt|fcdEHt`C;f&+1fl zj}4t6_T}&`2`%AB^Um+a^8U?W+f%X6-2*R+vf4~>z1!?i<~H1sT=&!FL1MdQdDXQc z=-UsCEe%@NJ~cB3=rYqt0w;G}RnJ?D&W zec-ryN@}E#%-`gHoZ4tdA>Ajw!x^p0w>sCx2H%%zbT0NCgSVm9N~uZdGs*9`Hv9{W z1?;_IFqXf;R>$0>gcI@1g9}?p3yTApGh}bIYZ|0`7fqOzu!XUXgtMVzMg?4THj!<` zHW%48{)Q**+cv&y9&{O{*Eu5U;D-Q&9^cpa;uG{=6N5kLzHEwNLCW3+tVsY@1J4M*=+{|n226)1I9^zdC1%{OQe6R>lzewVlKA7 ze%!b(lAD-?tj;sHI2n=Tltjex#ex^fL!bgL`G9!f;1A}zJG1kCI{k=KErPF@g@Z@9 zbv^$QPv*{jb_l&uur9zL%HCvb>Ih*t5@B({AMUYhZkjBx?@PNE=Yc4K*@2tks+bECcGSwIgB!8rL>l2+=HZr9?vx(^3dP%E% zni@3yCVp(1fxpt7NPEdSy+kk_j-U}lqV8N=Dz9rcpY9B0RVD<%4?nYbQ9wwX`7$9u z5SBd+0>b}jL!Dw90wS|Tzj9!6KF#ai$6frK9~as+YgIgM#S)nqqg8PO74p4=|9e)% z2O%%V-+Kbhc~DSeExdXbMD+AtOU(M|3O zs^nNYmj&CQL|aNZf1l$^t?yF$LV`EbD|cawmJgQ=rz7|RMIx|6*<6C&ISgJWG{xdO zpHJf#qJ}0=y4<)fS5GX|j6E7Kx(H2PKNeWi!@2oK;wUTkm+jqWg@(f|to@R-wPeUb zY;xb#R%NvGx6D!j3HN}2MYK02F1@bGAc?CAe|G76D2=&{P+<9qH4x)D=hGy3BBP8Zn5qd>e^*y5 z>L^lFE%srNiH@4HL#%2A6n3`R-Bd}`K*3mpZ*5xK`gn}y)_WikF}j+xP>qOCk0L!O zC|w0A@MIf?g*Of~1h*L40*`u7w6KaE+Oh-l@Mi>LvCDtHZTOHO$z^_3?Ej{H5p2S9 zkgD^gibxZ3qjGvLYsW7z4QU)J^7H>Btt==JhO>WvD!hT)^)z0;9r4Q zIE~Zh;tCs{e2$3}YnEwRFb`dKd9)g+EtQD?c{iRKVTa;}-_ZL+s`~x##7EbhIUTy5 zz7*2mj+^q!(7?1BPUeO>3&?h=3!X&YH(q`2-p9>-Bya`i=+38}8;RRkavjz8j3~~Y zYH2xF4rVnh3sVATY1O0ezS`DqH7B388N?Bz=s`jv*Lv(-6g}&V^>Per?>iY_Y>i?;xPB4_g>&q&JREygvo_E zs%S-C)5UmzRpU{oik*XhzX|uyUQ-20^=9VqfmBG1~Df)ca&Sds|aa^m_eP2y#n5ml2+DS(7dkK@Egq! zB)0DxgzlOVU!bt7a0oXvN>i1s_JOcozdQj&v3UT*CXegAF)f^81?=tPhgEB{@6R97bJ;V7NvpMeyTfy7Hzq`xSlVa0S=t|zK0@D9(^Un_h(Rwz z7tw=!)1jZXS;*tULXEXl^0EB^NX%ZxE7|Bh`@jEIxbzupD0mLPSM@M3!w<)Z(V(pq z?kuwW`X#b7t^1LtP|QT{WiK(0JTDF~S(x>6^3j}ARquFy2|jH(CZzgB+m(gZ^_TsI znsw#jL45>FXsfNOQ;dDjm04y>_vom%JIt<6nH7x+u&RR0&}i`3Zxzu6NiNIPAh)_4 z9(&+iK_d`5Qr*~CDjD4^C6NqorRhtg<`OTj22q8DrD~m3o0W{1Qb5AYbq{{qDKM z4FrX(^+=0%mb1haYjJWGqRb8BGa-E(A$-10KsRQIhEcUQo25ik8cImKCuYEjn^a3L z(BsoIO;L%YyJqmuWQ_em$w?nV+vv#AWZzm>npqN=!?L@wWt_ar!P(|#y4J;17}{WA zUvn0e3~|0&R%}#VeyhDZi8<}z`(%=;zY37f4t9_Ss(|DP9MIHirOgD0FXQ}d7-stL zZmwH#B>)ChlLz9`#Ag-z+^rxb8~|~Yih-wHnS7@ma34z*E8Gq{i!-NvPUN181#{4= zJr|VViJBYnLm4`Hj#*nNIhyUKCRd`&V#Q7bk%!IIkE0}*W5aZM zL|#0~xFI1PvbWc}vZutUxUfc{36@;vfGQ!_vKqDMd5CPUp9;Jh^MZSJm^VpJ`0S*6n;{1LiTkMbEaq z{XdxKnSSjL8!*Q1@4Pw;1qv^ePcFlVwd&@2>)jibPX?WfQg+2Y4nvUQE{;g@;{Tx$e3(?vJ6&tC}$e1PO_u+80-n&*ORuq-6OC+-e~7F zAnbsA(3D;3|7-@7o6;OMg(-kB6f&B+)~mp3mXS%2<~&lrNJ3LWpwEzpR^&3zbA&l* zYXady2;e1pA)p>;_yhXKRpv>lur5c1-kr}*D-}d?q-{9A2&3#t=bBks_FWSbZB*%F zztD5J`4PLHE&?@#%j>CVvsfR6EQX~ja`TCtZvv86$kdpkmLrJd!!>>IK-YvkxOVg^ z@W;aNt=$$W@p9{7l_0PAd0Fw6tfsl44$bgGtiW4TyHZZ*0Q^YF4-C=z0i>xBOZo*0 zd4M!Lmg}q^9v+^Gt6NQONof7MA&p52E8c`30z}hwi$-*NCWX#ht$_K<>GQVEGsy^7 z`g+qEjKzRTKkF^_y3+?u@osI^^Nr}iX_0oYtadsbq{7$k@2J19?-$lsLq0(WWsQ9Z z2(D6GfX7%{Cl9=9K`Am&*Kl#8r$$&P7pCG|%@&$B4`p9dcjkOzLc`v0m&e)TsI%Ky z;aPXxm=bN{IKVhx(Lhy*c7+~@THx01Dhb-g|L%M5;Zo9(pf%N^i0x~-7r)a@NRIdasc{9o!2yjy*&##6_ZhHn%s9&annui@zW5v}gJgK4zzrPDoc3rMx} ze&+=iR)^Pa5_+F*aXKjT;(m_x(Xbv(eLjR`NxVWIQbVS~5-tgqk!$K37RaR+OS%jv zNC161lj$s~K}R?1_3yk_TP;^4UU!yy6%|Rpefy>=3esq?=e_}KBcQ4D&LIOaS;_Di zE{8WSb#+HDRZ;s0mq*!7+mdi33nmBz$--gqdvz3v)(E03+$~v2Hm!=}H2>iOc!exB z31&y{t zdGHL@Av6&+PkLpvF2KiN&%bSVp96N-^AY?bjbih;pEV8Kq-9ZWi@d>vO(n1CIHQKj zG|+C0;!Fzp8A)UFYSvQ)&ceoMfY~Td~^j& zsaxK^D8&LMBt?!5a5u+b(tEBugc#w&Sw0=Nd`sqc{Y|+<$ z5(|FAc99TZTu@QzE6+sRkj1O1qbj!a$+ZE>QA-{y@|&={bPj2#Lyhkv)a4?K5Oiq`$Ue{ z%e1(r6tuw3RaI++6{S~NCI8~yC0G30n`UFsQEhGB%PX^MKkKWo*4-JJpVN#KbflWr z^lmtcBwJ+>>DXo`WPikois`mIpC;JjGUqZEZ})(b%iRgi#foGGDwsD#GfEt5OM!a0_OCrZX${@jwri{BEiYDn5XA0$^Lv+k5qzA z=zx8+57f6~e?45qj*?yQn)A_kMQMFurV1g_jI_nVUS;XzVIJJF8*c1(JK}{bm@dCT^ksnpMBPNz zYh?zz7}oN~hr!VLnrRD>l!AGS03Et&is`P*x$v4)3*z-|cwZ{K5HXlvf#|LCpAjwn zx~yNCohl2E+Y0@H34H78920v66fndFo12?AImN$wh`4PM{z`F@&rfN2FEC{=k$@`B z(lp&hwhPp0yb-0XuT3e)4(2GRGvQ1W)VVcK7{SVXm?w`T-eN8}@NP{mx#Y9LrXu(O z!7kdxOv64SX!U!35&^9W4d`Rh`PVx(l8H7^k)vR@AB|LOf9N(wOcUW`daYt}irpfe z?b>+mH>sYqlYs_`4`P$+SM;*CRe^Uu>x%6w;=lV=WPoSe%BlbUt&Oc*0A5|=FfKEUnA{deP2 z9+r8(R^Gvp8Lf&K)LDYyS-boH0S1xk3Ih%Mgki&7TQvO4XNhF=N*YQnlyZMx1?J5*M?p;4b8>mQU(_Bddp zelWNEae5Sz(b0FP2|5>8WHNVX;@M?&@;X)n--bE0$i`-_&< z@;>bE4pUb?)&CfZ7T7e@R$-z<|J4WH41rh$n9-I_POJe4Jr@_(W$3rk($a@s${czj!b>;qX9q@^!!Zf-6vx~S8; zx{>JxK_M#)z2JoGcT9Ur0rV2k7VZ`51dvCI;9N=KJ_pzn&E4LMYSBij^^TFWvO!Ez z2F^AiB`V~tR#6qwLo%6(%=+Ya+=;lJY%^KwEk+96UE{wHTRI<93~k%o&P|HX15IM{ z>;Eo65HOU6hwqh|UWrLbc@c`%1qB6WW@Zl&VPVEF>{(dAa+4$@5miCm4;9)v@1}@M z-Bz<|ud3mIUo&p8(0<)U>sZxi&MP%2UocLBVi_}bDL5OY9Tok>2*rJqozu~>@^scg zsQ)c3{Hs0_^t9_QyTglD<1ZlNeWRvOLJJ9`EB|&N+_<2m0ST8AK$b47Ndl2dWJuk* z(?)=1K@VZtW!}z&^K0V4b8+X^5XGb;P`fv+9t;EF*a;DxSIWx-Hb@rkh{Yw67g-{jB-B zzcX)ma5r{_*w8oT9m5#y=|)4Uuq^8%>Lt%78R(ftGfxnj%N+aGbMl~1V=q<3vBg`nrlKRUax{1sZthK3uWf!` z_*o!3JT@i(aFGh;_}80a+W?@Q$q%3}y8gZ_y+4)FEj89)l|U9Ua3}uYob17Nt*}~< zGqyLeb9k+5@+wCGIyX+wsTA?#7QiTD+5TCwO+n&5X?~maRkhj~ePErZv z>gg>YE9P5e$omES1Cb2mo@rh{^lBgKwmTjX<|Fg~LL^}Wr$hNKRE=a=5Ih|J$=R;G zs}fnh7q^0iM#ip105K*kFYFdDSHJ2i40t&tw;2n7wL;h%mRsY((Ytcmc0ZJ-%UG~# zEyDi7p9L`)Z*luRW-A;^au`V(SV&z$RD&dw+?8u76{%p0DQ8rbfe=#JSYQY~6_82+ z$mA#bwd z;JOfbC*O(iL%%@+Fg_@>%}#1Wh@!w58ozpbYzuI}dCkEJzZxS*4!u}A6U-C%`A=fy z@KH9b<@N@oBU_^;-z;iDkn;gn2LT0?;31+xf$@RWkDq|RCVUEk@*;Pu%SQ!YpPa~? z;XjFJy2Qh6s?aU`WfRKy_u=35nN0@7MX+vt;CiVkNWNudTy(R0NzZsH4wOjYaZZBD z#pAizQ7x?7`#p=J|J$z+d;?;o*eVqs0OVErb3YYAtx)MSRN%h#33yGQDEfNep?o}! zp@8!^{2uyy9wnfS)D#Bs4gh#IBOqww?zlZJ9PgDT2rUbu+#I=a7s*@^@!VvrJ---+ zcbQWEJqjQWG5<60n*#fzQvbK0bqSk6PD5p6DVg@~H@Qd~CiW_E4-@J;N_&883ZT(j zS%8WJe=0Va$9jKBLqPOw~^8sQT_8^AJ8=(@Ufbi`}7-Uj^q z3V_awSoiYw2GotF0rDeY7+>IS(S(rKDVfbY>gYzt)xdo>0T{et&~4yiQXK7Oo2*H0;Dg2clqB~u1pN%hq4 z4>-0SildyQU~0PnQ!Cmk1|Gxtdi(Qu2H)0r%k8`_BK}}1Kr)Qp!QHG|*d4DZxKys& z$eYpKKmIYhs0hYRD_hDz8Kp~UaQkR8AP|L++0ZX7?Gk1Zuod3X!p4R#z|rIkRs%>W z0mr$Vtv=?4GsaN^a%4`x>;leZs#S$ZK@~MUy&NZVC$sm{jd_+gRt>?~89>aL>Pw~@ z=cf#;c8TgLAar+a!7@K+^EIOQPY%Km&d}o_8J%kFh9oM1TfJ21`#kQ#@U8(fi6i@5 z#KgoO`NW24$AaP!Tj?~)v}!*`*1Q!LW`1*&(+>FR=GK&zl~qzRzvk!Xr?XP!FNXes zk@4>Cj?)8{hK2^9Yam)J-!4+?tYINSZVQZUt*LT4RB8M5%ht!oM_xe(LQM?+{$NTh zlM)V4R(S;g;aE(c=pmVV?*KZTuspzAwQeMVAW@N#>KSyd!1%&*=liST86HC$Fs2Pq zeKPYyNWZT-PHIh5p-dH`qKGX^qX6@pcjGbBU z7ukoFEy5aTc(ks{3s2w@VQDQZkl065`9vR~k8INeC6!&)-td!apBx73u@ieQXo_ z=$M!V>}dW28{YSX>*c~;-MD<8M{y1h0VwSsF6fIz1Oo2KqNhItDSH!HOrBy89rUGe zl2HUK7+nAm6d1nj)BbwJ{FBmT_zV~pe7-%V_CNt^0<^YDkaig(*jXwpg0!I)0ev8_ z4l9f<84~(5fMLiQ3<()PsdH0+78#iOe7>mtX-mXzucpKbp2P4}waB00+muX(@DI^@ zGClMT1Gl=~50fCtP}(yg7apV^T7KpQ-JZ8yBjx6i)}o`KAw+XFqzZ9i_&myHHXVVk zwc!VVD9Fl=jEx1xW5wQGxx52w?xIN)dA*)mn?dgJ+l4&t2h=!p0xdyqt3_v`&eG#FksTBXnA zDKLkRkAw37(4U0M=>w8v3B#_DLtrwcB*tS|b@cSj`_g$7C)7O&)gs4 zzrf^CG9YILMFyXJ=J#IF%F0lC5VAd%5ti*zvk?&E7iY#q=lFJQ{wP)l=5A)n?*OYQSsC81AsBs$N&l$@MyVF6 z7V#&p5E~wdX@j+|BMf9KJYZGI#As+>Y#+hpmI8}>00qg&pJI~Z=0y%m2~%JldOHKr zTwmg*?#d`Yr2&XYuA%=w9(Jb@rDG$fx+Q;d2JNTSs#-8}G@h1fBdmY1N@N%FXxMjX zoMBKidgTChBRSpD)Cn5=tOagtxxw$&T~h!lP*(|vVI zRqQ02XW7YjP1)5d(qJU0$CeNUI? zuM?ir!o`?;Ga2$UHhgoNQM!ivRRBUMRtCzI2B{8;o$fsBDlbEH%ke=d6f(c6ixURVvoEy&Fqc5MU#oWnnGm z1>6TzX7tFJu%HO8NO&&Ff|xi#-~)mBw6gsU?<&khp{Yuy16{=E=3r_v*GWjOuJ%$hrtTf4ENGjldWXzwSfc$=$CbE? z5_;p=>D;>JQbI~#Aa;qB!+aaDV<1RHJvv3NN&^jDrPo^?eZfLKk@0@kq!e#pO1u4p3~0ono0~l9DS~802wguNjw#hZi)8XH zd_AmhS=$E;k_c3K#LBYiP=KR?za1d3hxl}0D8^=Cxl5pLhMbFld<~=!LE;{l2x79G zfMiFAZ;))WY!YKk?**wCS~wG!R|8DT*=#X773Ah>QxkWwPj&cygYWxWu44M}na?jm zjN6A^F0hmeWns=rKt>G~@}Xi^8K6HP0CRb}OEn=|`BTmEfH%d2%fOJWBcPZW7 z-TfW>-EXaX*P1_I&Ac=3d(PQ=Kl|A=U4IFG6Z)os6RBl5216n>{re9GuXfB!FxEBc z5cS4Bjs}L55%VAm8r7{p>-tSQd|}pVbOm|?Dvp{LqPq89v6sUs_43_N{{#=$!h#n3 z)u|v6(`xD!d3+#j-GaNeq8%a+3YPb^X9?9>r8kESr##0#6}P?CmKGh2v+H3;k?Tkx ztA)8%YcGZc9Z;Ag`&=g?GbO^@46N1RsiEQf_UEr6_>41$F+I*jc4DX*?85YFblzUH zH>iqx7G2z&Y4P9kJNF2BQWk5}>N1UgVE+41+2u;Y>wf3^&BkjWg-O08csNneOM5Qv zp-Sj!6D{3jEaVh}c&1;wAvR9RQ1FTbR>Qxi1IOhHpd+a&^aUe@SlcTL&|6L6y^F@} zdF$1(+(6-TzhvVKn1nH3Yqu7yuA$D*kAr5d^9uUKJ{WXcEUghq9ur+ipbOor#~SF4 z+fP7mgrMiI_CpNw8y)wXjJG2;{dloi$1OLT_*q^I&DWoMPm@I|c11v{Pbk1sFA9wff%7PWd&fhm0 z_R7NA%mEGidIZx?D2tWIh?vkH2K4+zZMFSHEcPaa*X?*&FclLFj@?oS6}Fuu7DS`4 z>W_D%E4nW|w#?OeAQ>4OxMJHwFP)0Mi;M?g^RrmV_E`&GSDZ5O_`;k!9}5kFSK~+0 zduxx~qGLh>$>YTzKZV`cBuIoFhMP*&wJb&y&5StszSA?jKz|FVRYr=s%iV#!sTnwp z%-J4)Z4hR2@SeJ>e`}nXm@2*^EwnoTF846%M%X15MgieSf{rUfi|~784`1N6ms&#^0!|PXHBRs%jB$UL~WpR(PxXw2Q-=8LITuAZmZqEw^liASQy6o@oyi zOW=iOYC~jk?~AnOk*mXjqbuej_(B0j7*P>M*Qa}t2aJ34N9=|7|3vk|KPMZ?ZXEb! z08F-NhWRNCNE*a6VmAQ0AQ>y#Sg5~9gmzl+1+hT3$kBOCcEMk*{NiYrZi2y_yGwgn zhA%eflX-wJ>FqVbueGhU$`B0rjCa`F=Lr21|OeSb7GHjXbWxV&M-`{3@*nApg_H1V?v z{n3_T1W$9aqy09`k6?|tfpfi30yaXGY&!OB=*K#jY{oh;H7dH&ct{AkL|>>3i4wY!4Y7(+Gd0&^ z37N^nZLCe)3m#Rh-DI3EP%iWr=wg)^#0tFkkr06PYa7M6Ulsqkg7T0I%UJOe*UDzv zL59tQ`HSGxXN>+z^CQ0}l9&o+!Ps#$yxiU=52~LUWV*Iq3(nEmb-oasUnv`960vND zB`o%;Sj$r~vUFh)tm)HrHWT>1b9t~ryrdhydB=3%Ttv!As$AqD_+`}JcB=Ypzji=} z#iM3lcM4yFiQ6C7Ru;M%aQvl7P1RLGAu23Fwrz2$I*7>OhMRI_E(qnsdlNLJjegGD z`8|!g8|sOow9aT3K0gqK#gSJpnaFXF7*+~>lllsJ@O81SgEm6?{nT!e*ld}fF6V6e zg)737T5Wv>Kog8sSP-YcuyqLrhzS~j@p=Z@S^=3t_J|MAG;C$RKK)LFkw(^f2l6If zeAqtec)R1#S?Zlcg0%NX55wSdA8tn*ak%I>k@BX_Ix70Wd))5Gm(WET8xv(F7;Ff@ zQhIutf3exbRKiM7f)i+{b#Yg&>1-FO2s$4$=c1!IQ0BBqXR?2+%u1Wm#v`I;qMSL- zKd^|W*A8{jGu03w3Uh5iC=)kjN=IetQ!ztNC#PKxGlkGb`MaS}MhR>PeuZDzkjmiq zJWdppB1P9fS?%h_h=`A&z&o-&CTPc{A;HVcN81e#!lFT0d$T@?j>4Nv%qV!cib`=P z1VtK-Ym{2D`ws6B^v94>E^v}jL)ejpE^6;uNCp(<|zOAb%nSecDDM?J~mE5tnR~ z0DwhlC zm7D-BvsoS&+_h}XIPRhK4_vr<<0$}v^HbeJp)VWUGk)8&AW7a244&@^O8?yu+6eWCnAdvn7+#`QV}pe<*i6S9bi2$$$h zfVf@9Ort5{$M8=%L<%#%KX<+SOXT!dE*~MHcu+QnyJm7BSnf*E*# zm55Gul)(T)4+krio$_f7PKyVbi;Ki@Wwm3r2vE4bFsF~8@Lxk6*=5?#)z~oL3u78= z=8EsATOA&_QFh+1{bV^hEVkc;<-79~8%aIm^?FKI? zU_|#hPyWZLsXjTr^qokP{yr7;3T}5EwILqcrj0O0!B_Vzo26@R{Fn|=@;J>a%%!u_ zx~t|w_6kMI3lUI~hBR!nXUM`Vdc>KjYl6^_wsvmCiz$R7g_R{dnB{qvLLi|RW=WT# ziegLj>r0RC=RpP80jeh>$7WgMj3pxHpW=9q4j)F^Lwi_ApyyLuabAE70j2RewG(QqzOc2;4p*;X4 zit^zeHO}?~7daU-FZf>U_pDh}TiGJNw>9mwr&CD%f^-;^wua7MRu*9Pa^@#)sBp@HUA+@J>qANM?r!i^FhzgFgWd2G4# z!$r%d3a7UvSZ8JCmtkAcUB8@7Y>NDqP)OtQi2Sg*ejZ2b5pw441Cmy~yT~4c4s?zR z?Hul%ezjE=EXF=Itq^paY9PoUahT!#=(^D#2bQxvE6&Cc^(;zC4C=#VLm84Dfx7BAZk}R+C z$7oUZqHP{0x!*^4p6Sl5d$*6~X6b5&2pcP&o=1rV16_0+>mK}eVtHQWaq)Gu>e>ge zAVV@f&E1!EsP~t|NI`F3L#Be0@)YD7u>m^ga67W2YPapwZ&HJUuD_ofHE}u1tiKaw zcAtbNO^cX*EoZuL?4^ykh$thae<{VvceDOI(Kr04y|A=7w5}iZ35D+*^;HU!0erF` zIw?Zqr1eG9*#|N`F33db5bYGDJIQlQeh(eBZ6u=eVUBe0tT}GN!bL5i=)n!!cNBn| z3g>?zrqNa^wQ7?~V~J(pnPL74Szc~0$Wz30 z7dm2UOda+q)Ii}PU|_+*6-G#S zdQC}zEslNGPUD!s;5*&IT{G13YjeNHmN!}Fkxt~53W1#4Yx#-b5j z!*3nvD^)`lSqX`99aGPfnj=fN8Mo4i% z`YQea;82egZX=4H2@XsA$wVuej+)=LY0TYAT~xBY_I5qMQwl-Hj0YkPqA??jMg$S0 zNDv_7(>^x0b~>I!V}MQ@POB*cQl!v%(P`eJ0d;!BtN&f z?BRQPd}eKb(fEtQfT82?t`z*;N~yU&7@c+M zG_d>j>$j*H{5Kb|Opgd1;Jhz(jub2ByBj73;fT%<)IW?9HwS%}LaR&DRr9SHDWl{p zlO$1P^V}+rF38?^3cCxVg5pI_<90;%(i@0cS@qk3rm-Y`t?Ohxj4oIZ?CDxUh2U9O z;4dvoBy4U@osAFJ8FoO#$Mn9ODU*w0fYWMr;rW`tBUjg-?-jgWR>s)4B&fw|pNRib zuMyu3j1Lp7W@MsrW0!u)tq5u$(|hjQ?m$(J{7)l+K#DL&M|N&-__IJ{%Q>dW!)Oxx z8hgFcD0+W}%rQ?CMe+)BWe9Xa4kShTSb&NYii}jB37x7j8XH|3jeIQ=h1?e*(yWD` zz3zQD_TxpepeU)@8;tkJ9qOeGhuk769TDNZ%Be^-DLa;$)zIH(edn*K+_}se znm;kAM~-AaaqfVf)ktB>freC`s9K;LT9>??b#XLRv=Fvn^8pPGyO5hsy75nNSZ#55 z4Qh{iX{@J5KnJzulrDq79#URj;P8|OT8SVNerVM6y%o*zl(CWnT-%j!Y=KAFbeEPh z?+U`EYKR)|Vi~e)%1N+}!w{RxU*$9ddMHD#D(>}iPzs7hkgr5>jRfEN~Si4F>LcJp^wH zEQ+^d^7pjM%t&5W65$JC?Yp8WHnCVM)QGBN1sdgTILosF!**}A+X_DH4oFD!@81xusN&-VXQ z&cMyl0j)N9x{zfBx=MmDWnK9PXnOj@R|J@Enl`ubvdiG1&Gjtgj57XIvA+sYxuN_&oKH9N{_dYO13yswoME<`mZeQLx{Vk+ z?46vHf&ZYG`NBszX459=6P^kQ6mlcO83DMd<9~T@;2IgwplI^%K_(OURW}HNSKzJD zrU^f)pN|^mPGRxi)=4*#bSWGOWJ|i*{e*vuB_IWo(tdy8FhxR)ZgmCOE+FSa;o;Oy z4t=>mLK1J13M4&zrM~+w?QIkftWWU*+N?0h3V>V;ESvDZ2!usN7*dXTRjyVRh&K&8 z2vV1T-Xh>N^{Duwqq7KNrPbC8=Gias2_$7@17kjZ{K$?okV<@cxRCPgn}Ujpv8;hn ziejE@v3A4cb^z3GuD4iNSextXN=iyp4RIU&{rw>3y}#7b0&-(iDnOFSs04xw>$kmi zqvtL*FG2K_r$4QtDUAdb@4pED!*+yW=lk-5bp94jMhbY2Z&K`M%%F4pIcNp_8iz5{ z)1A~1^c}p~EiApgyYoKnqa-Bk_SO=tJ6LSeWJ|57arW@I2bM+o0(8JzUZ&Ts&!4GaiKG9`K~nA?0lF4fi51;8^N(`S+5 z17bI91|43P<7MlGdeAd;uX2Pxh1Zpto{Wj<1Q2cMtY!*Nj}K#gAlaY+g#3F5-HU)Q zQ`~w0C-QiB^(zX$>0)D@ex<|K@K72L_64Y>Iy*gOw$!VFH$De6l6VxMSX9M00R-fx zv$1tJ1e6NPh^v1Bg#vo_OQuM#TT7k=w$%eday@gmlQH-qfDRuM`Z+s*QAHFTZ7gDU z*2Vc5+FIZa*4GlB1_x`3bZkm^JIyzopuI#>;h)8dDzHfW1m0{b+hz$s$fMf%| zA|_e^X#U6RwF<%#8`Hlb7?gfKMItM$zA$nvRQa^3)$;MgjjE+O5tTaL4~|9;*`i^i zAJ)tBc|EQf@9S4qR{GfkjkJDOF!S&<1%?CK~?;m*kRs_H%#o?W4s z&-MU#o#v70s_6q2WVStTo&T=40{Vk{abX#Ayxwuo&|SUuw4a7jI?-q0@87=w!^uug z{qVu2k(Zah;Ec)RL}fj+~4vlaN{O=5&Kf zL`-b8r->7(M9;L6-D)C>R#77nae7T}I zbx&6pG9K$T@M3O`TAmE?^DQhvxL5DT>G{qd95k_P)tT;8$lu5A2z+HaLF3E)nSRN4 z?=o)AA}Htfrppa1EopPE0rL~Q*8~oC9A$+TC1f#<6|@#Y(VTEw6&UsF+93q$XBawH^(L0*dia_F^F$G*wo`Qjg31S{qHzpcMlyT0iD zAp9${#j&w7!hdH8956#S+v2u@4a#)us;9*9o!cwF1s^#g|J^y#4Kg~!{j zx~tM%l7Z^^1~O;cHg@(QA7n5Ej0&+WkSuY1(SP3d&(fW}A|fZ^&6Z36*?bBF`a7yE zI?>-zNG|6ae0b(n=VtK+!J4zj2|6^B$OwUhNg4yvSlIo#fwfS2{ z#;4b%Fq~(8t%np|n{2RX)g`;mX~O|IiLt?R%X;Y8kW3FBUuc~!9l zMD*nj&{md1HBsOkq@c`df4d$u;dgkv+0uOdsPlPkLAw`2X(@Z;gCfkehXOH!dVbG? zvx6fw8ao8NRUhB(LKPMwDbO|o{Weh`IpC)zq~7M7#W zrwdI~&V9V@`M`ae-x)(HUN5D)8o{v`%(6D7nkbc;kHGu)2S;!+!{PkyOr^<}o9cC< zFMyyHMW%3GAm<7ewi)V0@vsK?oKEO(%F(!p2u4*n1(}e`Vxw!P?-FPt3{$mCuM}s0=H{1ZnpIZ*3K;tuz)sfKZYN1avy7GHDsgRXZgNx7eV*Pt zWVYbo_QH>oRW5=yP6zXA60YVQP(Gm<4&!NDVSIP9}-zzY(D zXwxcc;LF+_!(EH413Hb6trW-3-?oKARR^TKx0UP=^mDBf!E>mVCH4u?GHR!L6fh1g zQLhZV2W-Kt4l7Zm>pQ_8EXIR+Aq>{O*j?5m_Ffkgs$4AQ2zmpltY+UhNEVpK2KT28 zch=HFZmY`;>Y2mXyA-Z_it}b5BgL*I&CQXTX-Hwiv^)+cxm{82Wz@@}o!sV>evQXo6}fN6sHNUGucbK!obIsG=qXB#JFVuBCI8WL(RVa;(l@ z69Z+CK*Rs%l=fyZovGQ`D&%Dz)3*sOC3ViwX4LF(7OWI?RQ!+uk)@Nh9`^?wq3+avW2og1S|(5AL>I+_J^P_NJDeR_v4 zmmu6;lb&>?@&#|JQZpKnlF^~@$3PYLRtRR))a2Cc+P2Q$Uxns-kHrHuba#_=OJZRf zn;lZ`>;vvFuQ;Hkle|kF@-T76EMVO*5g*!Tx9IX51h<6QVw6kKnlP2)8cHuR)dCAI)PWlSdkY1^9LhVI@xDgRjd;(|o=*7#Y+5_(=}!$Jf>0{E{bfp8HX zR-p!xpC+S1CB7*E(#I@83ncsTuz!#|E3wk1#x!9_p3@{rg6RCLTOOPUZ`!e&TV+3N z8}g>2@E-fnSl*p1a~i>u+KpAejBT($^#AF+r<77MsGz)dp&VlcqO z2;adTu;78-H;npdq}Zs!<)!BUM=FEXdL8F^2Zez!v_fkp>bKL+D@x0nMndm1@Ln;u zDm#6Re^t3f`#5i-%`H4zoptF&NcgUyqR+X^hd5|)}^->)yf%) zE&3z3e)H__gQGmwNiRz}&5d(3hcd!9pn3j!}3pV?++8nfA+8T+nLW6X&RSUCc86_pM+{bwTAo z4Q#{9m7p(4E??x0v}Sh|$*ylBVQ`(}X)x0CJ{SK9U2GVFKdcaC460G9G)t{o{G6K< zL&6Vj88dyN8gx4O1VW2hz9&aB`=-`)F2;raVrUUm-t#_(f|EN&glq}>{^vI%-irqm zUGC6WMQ*@>V1oIo{*+Xl80;W60Rsyy%}1y4CRxvHZmW+?29naW_sZQyG;d7R&}&=x z9m)Ay@~6BH4~VKQM+@TTip2x(Y2M?Hdm>2I)|bqhPgglR>=jhx9YbC1&6Z{tw9|)} zcLz0|=^5}orIWh9SbsHdrx!#cP|oDpBaP?6Tt2zyRO;^VC&gdP`|~O#(&^W1qG_Rh ze+=2O!|5cM0&<@wf!Ih2)YvGSDx-chXtZ88$V;!`TVh$^7nlfX@V%4%tYyMhk&0nf z)kYa zB_-6Ip;!8senOWi-qZb=l=58?pEnUn+tv49J~4hyZ0I%i*xU~jPX3vGQiE|bSfuUG z%~pw8!;v^?9_G+JV!0RUIVO*sXg2V1NCAtgZ1*hiYQHkZyDu zFS>8M*A6O(x3Wmh_9(j*peeAA^6*%t432%=k@`SP$-r?`XyR%0Sbq~mA62y1ZC7QkfTS+mUGJBo#P7|KRxncpf6-gBS7rX#aJBQ7&WHQ&-h za49mEw_c$LRI`=}Y*$5);U)I9>R-Z;4mLktl%9E})#X~cgee(VQ->xn8rP&_P@e;g zK*(uJ#IsRlY8t!{fz?`z<$2P<`9Yl{ELpy8ai(}3^?ooZq-Pk11u838$$c})WHf0w z!6G?cPN$#o?9bNq?-wyWRfkASzct59Hrc(mc$V5)hGu(pqVkKiL|w_~1N8|9v~D1k zXp8=aq$y+T+aHb5GTe55Nw2Mcd%;C|EiY5#8Q_i!Jy@hy+2645RP}L;i`S?y+0_OO zn$U!c+~#ALytw0>PQ~C%cR>qyj$TL79lH7M^L0T*xfZ4I2C20|e|DALHEr%Uj=eUi z9w{l`=jzHFdUB;yxvcdQD433#TuV#b9i>~!zr#xt%-PK6FaBt9)=Z$-Y;i={eUB#b z5-ebjo)S3;5rpn}1v_J8_BCBbzI_EDNc}zw%8`2K6@r7%*uX3ku5CFpy&SYK!W9rhpRt8aotN_dH)Ax*?>oMuV0{q;PxchSut4#6oR&p*5pQbaBpFTEMSj+6oy&6V5~ z&>VUxd^_JV#Gg1X?UBQoJ=76065sC%*IFX_;dU%)bnY;|RGuKKNXAHVqA=2SaTXuC ze=S&S^0x6$3E|}aPqX=Wk1Z(}*T%_Qcw#uURZo) z(u`A1K-e?dq?TeLMoP25?hAs4hft%r8vFkCz8TO(HQvtN)-%SJ`jV!}pG;7Sl<=}x z8h#}BlgQ)#OvK%I5>z)~%4N8iV@;#jDrOCNk(zG*frKF(LQ{8GWMiAUWIiJvdg2iM zwqpk}2az@y>IySQG&eiU>jxExqf&=8mVuy0zfdB77|(DrDvYw?)Gd0Ht8iBB_fY83 zr8clzdko4W)x$QL59{|D4=pBdH?i7|Tb>?{ov)lVL=mLr|K#)6EPLd*exW zAjjmP#=z=G3B!7sS=n4f-$o5(vWy@P+ss+0uKJQ~X@<@HQij%~Jcok&yF4uo?(1pW(G_HY9=W@t}TOH2$ z@D6(aXDBujwDcQRLfO=sUTTjNErm@m<2GW_Ww1{4gJPhgMbkyJ7)QlP6vKC^=G#B8 zr$jqvQl3^>^71~WqNz=mJcf&^Sii+)R_!*94+}fL$-5LRYzp_{V2x(O8>3;&?iMGF zT3~crzVVkMlTCC}TS>zGyZD(tf_D?`*5c)NC`(vI=NY!1IAVR@8L~#=KP6D{%O=&% z7!(#eaAB5}jx0v(Hj9?T+a--g_Qt5Lp*Kxqi|*;7OA3Mdrwc|gESHg=D?h6Q8K18q zaM^@?{4{r~th9%2WtLrC5?{8J-JbnNYGm1DDaJF3_KBx{Y9fS;aOug@owGqRgwlo1 z@93(g<$u+T!Uh$?ZY-%pC_!(6CUpzPb}fRVq8{_4Jl`Vr3~)l) zxH=s700LD@^(tEa&6hu<4^w(}(phC*$4n#G2*r+dU^PioFvoSoT~X{~rJa7%l4CSi z^MkMK%5VAi~3MW8PEBsg{ehjl9nzIE1gFQQ+=bGSiE-#j;qwmYCTK zW;pESiubAb*2~VTYLe!ta{g81toe4ga2+FdAW6!Qk{qp71?MDiU958Di;=r#4gv{o zllZjN7b~m2He~lHZRaGp96}`y{ch7&Rqyz??bmxj`h*x2GX!058>6+_BSB_X^xtrc#q>8}}ad0cv(e$O~w@@4wXw&j6*f)}7URPXIc?)V?@`Q(j9EV<1V#fiApjh&Z-;T3OI z-e$_%nMEVkzUte=#mRye$~lADL{O`y9m~ciq)?(+6UqFgjW1xostS%!Q-Zo5*G_1P zaUxFSAKfL!{p~Bq96?u(eC0mg#WV55X#&g0@H!j2a}}%P2o$%;6Si;=${KF90L`q3 zsCL+RCVT5Qh5d>4lI49u^rnsHo+jsnj4Y`UE}TlA{X!PusNFHq$@C?}Z&9DIeS%=Q zXkD5KG6kx!Uvqbnm^6A4p}rkRN)2B%uitsX#>L7AACDo<^`>h}H7lT*6k?Q$N};JkZC_gyc!Zj!>s73OmkRBBfTYR5f; zUi;-1{}q=i=D8&~d%~I&*+X%3sh74f6R-rdgj$X=uzsRG%UkmmiK}jDZ%|A;V7F}(?n3qPir0VG0BT77u7$K?p zwwE*tbuOHGji+9dDe>rv^8=dkAWgNTG!)CvNo2}6qR+puprtA5v8}``=?LW~e>io~ z>KDu<^pW3wC*@vB`PzYTmHbJM$S!1+;&g^$HJdlBe?y(OsUrwQ9GS$RmdwfoDDQVf z4xKC9&($qE{LTD_t9KVeA>@nrJ7IE$JEW_jAwECDe<@Z`(F;;+XA_^y2<-p@PP8K1 zU0pk=Jy%;kOZRSeN-if>-$4&+qdn|8!p7q(TUGQQ9D)P)JRyScV{ui|!qN2vZKot; z@(Qdej#9?uaA;0@M8zKa?yISm$KO_N{wXoo(41s_Q4YuGzE;t8ax8+$H-#S5+Zw5- zAxqTfmJ@qUCmSaYfoYm`4fZp@ttt>)fcLbVSFRbdnfIC1oQ~Ux)-~PPZdhO6*z~m9 zO~A6zIPdJItbRManO-5H{&7aUM@M~f%~RH1H;FUB`oT&Jnv$#8)AYNsA+9d`>>VDt zEMHGkm)vRd*JRhAc9U`AG6&cB7s^Cru;=m!eFB9kB+>fMZr(AJf9Z|7qH$C0WtRnI zLQv@*?smD^a-D_h0yEbf1uf=5tiEyg^8=%OQ;M zZpg;#>(|)Y&q7w}J3)AWamD-7lq18+42`aAwlv%m6-7(ye<)Om<{*_ki`tt1S;X%~ zs?Vgl_`UQ?GW)WZv403v2rsr3Th>X@tV(D-1r51*##U;j)klz$o*zp2KHIoPY+ zI=9@V$@q(HpK`Z$Va{puRBEw95~pqU61dz?G%CMg!NH>t4t~>n=L_|8Y#KRd*T5_DJZ71lh-T3or#cmlqD3Q21dOs;g_z-1QznWLa~w1l2u*(h zBX@~aUHa%j({U19Vh~A#H_=}vP4AraZf@7gcx)Gew|R#cZtc1abz+(YB&kslyGL}s z5{el+LvS3IMyD8@wmeJQVPgX$Hutn$zeam@?W3D8aAU|DCYajzQ;mpWQ&6YWexCoj zcu}Ioq;)=+CWhZ*9;4iF*nDUGQm)0lKOyrd9NQlI)Cp-QoL1bXhj_EFlRnFTkma3M zC7e)7pG6vcTFkp3D1Prp2W`g*OhssQiIhgjJ-oU=ZwgU z%u_`fAKffg{Zd-4v#^0-vg;4SPcp-eci*98F9hywzi5v&DzAl_Jh>1l!>B%)jnmQ&9PkE^5C8FoPtT8-jIT^sB{+n73S)7_XL& z&Hqw;4)cB0!YA*ZkO0CzKc<}VFJH)d@0rWP0)`#DBBcZ%*R{b);Gawx*s4{oO0?GBYgR5D_#l!@_{TaJ|L5qfx#Wwp*3dthNjLQ(_+Hc#f8iKCA zX>F^ZbuSi#VZl|y<8&8S`5yoI^01^25USuD)d5m#hJihNq`{|0i_x*G~fF2D9Z9tl^PZ_`}gG;aK_pf6F|H zI(qbdQVwaF1-%s-A2o>-a*ztbLUT7z#w|C}@H#aO3P8?Wb65&}d7SnWHy;bQz-n8P zXD(5yGghFtsxt=f`XRsugKHF7aWL*M=@&K9vimz!=o(0d)EUL`IEU6E;nSHczlhzp zo=)eu7EPo+uOVy_xZh}TaZts?LkhSNCw4u3Hq{0aBlC5W*Z2|e=?zZG$;hn+e?ZMI zD@Va5Siy=YsFsiC17t{n$`DQ@czJD9UjfvA5rK(jabKKQxOTjdSP$h^7Y zrAPFs83pj4{|-2--1r)l$ybBpEChS~mgFJ@^;awQH>lV5sC@!xrj)QGZ6kk-t1t|V zauL>1pV?RDQ?6dHPzwz;&5g!}ofG*8hS1Pn?#Dc2NZi@<0s;`VcQ9Zb$WVcU*?4OA zF<(kKSU>{-gwW^`Q*6i-@e)hRoOfQLr%qEBVx!P)G6i6#*X=$|zKrsTKceoo8YF4$h1;YdtaeGYD!op#koXN%#97nk1u<2;j28 z*OUKSI4a}w9shggeF`Gz)P=Re#Sno41k`)lixleG0uEtVAcMrm5ktxF&1NeeX3K-G z{WYGK$@hPKUh>%pz1MACip^C1iAw>pO&ChjRMnMZ^m*E68(D4D#C0C9p07E7R#)#V zev?6FKt$vjYuY~<_m~!7kHH$t2*R}Bq?M+gqYA)`;IM0WEuW97(EL+BLSW5^3BEY0 z!)koWEcqPbm!TQ;=~MNK{{%Os##qOlk)6jXmCLfqYQt>M<)$Wt&^aLp{^m z+dAl}TP50D)~A1srKlDE$BwOHM8DwVf46|?dyJl6s%4%g{*dk4fkg6y@9FP#c6fDZ zw*!KI$+WWmt|}+9#BzL};s<7Pgy z=>J5z7XrTOKP{Z&50yQoT6%_fcj}m=0O{3+WOhrp zSq&xOa)eAry3-j=0r}DYxXJ(XB`YM$j!&DPGD6NLSeIXUL5`tPFcPn~UL~Uq=O)4I z$F&W(5Ra>k@&loA+tNclxv!RKqHU8rwlj$k@hKzHNv$Z}& zW)|3PHrkiPKg*bfb&Es6CJQu>6QS5PyyeEFhrB1%I~jRjsDoJ(*t%*pQS2%`lH=o( z3!MjLZczZLk1E-sc5wfDblc|>HosQ77$)EVYre(T`b~Ah!P@*V_mo+98JuBsxQB_Y zOT%2(-?Jimi^;AVAF+m5dC!GRO&783qfj*?A zXl-0d7Way)G;v?q(9KEV^xbo|VL>k9i)Lm!9K3rNv$(4v%)h(c(JqIkJdTHP%=NW0 z$dSgyYIm}`;jcK40X|*a7y7Kb4ZeYA^%PBoLLBkRDd}TNi%zx6-HNa-4i6%WEyRY$ zW0bYb`FqYQjr`pIP#3dAW^;4;B$s;>XrcnL4eRFBYOJ;O_4Ne>p-ouJ`t*3t7$p-c z`<7o_ipYcsj3P`k%g-=r$FpT1+dNj`@)w5*CkC5>;=tFls8zG+@rMWgFMk%ZjI-d>+E9f#Hqm#eROM=CX&{_3uOM=Hc?pA z>R9pJ*}CfLcvjFJX?Hg{8E+VLssqOpL`+FgiG9wCf!;6grNd8wlRL$8TlwDH?3Ly3 z$#rlw{_%E=T0iHAPl`n0yMfS>kW|hVZ+RA1#`y(X1p4Jh&dRvi(Y0hPvBtG%WuA7J zFmKa2$z!hgmul@W4}6Nb)kNIklQ_n4XP~n#dlLzpXk*e@2KfI(f4aR*7DK$VAGhtZ zG}I7umqN_*txo=jE7X9n8N|ppBUFX>1i*^BOU^0%B9!HUKC@h#jQrtCH_kZ=o#JEw zGnHAFdGeuWy!DDcyc7NtK095r^((C&A)9Efsx?!+FxE(?*1&YF$hyp*z84?=ef2AF zgolYd*wTK(4kF=+GaFfrR8|=@4z~s^?fWDahl-O{Y{aa&qiIfl zlgvwiI%2S$6_3i~Qr%b&XTfVEYNeqUu|wU7|A*ou1^ghK%q^-xKv+p&2Ti7n)FjvK z;t+{^+tn)(gQ|DtbG8h^;PHxu~1>!Ew$BlFKhOM#3mE$i)#`(_k-cBQ%M(eEaTJuDWqqFm4#>R zSSWSX0Ax*R?CN~2WPDLwSW%?pB7)8Wkx09WxRSt?hve+@62STxr0mrxS@Y9 z=8r(re?Omax+7^d9J}5lt0_I~)gWz66l2&J`a5|&zz~*>=JUZj_bWOLwhBSTDYEc? z@XG&wZixrK3o8(fC;|1t^^2jVEn?%iE$kG(`hOrbS_jhj>i1Qa z2|K}#l~!@cKWhxI5QI}+0R$z*e2zFcy$&>Am)?L}&4Pr7TTAeq;B~C?2I0~zw$iYS z|CwqhL6rtu(MEu+jwuJZ9D-jX^Mv6RVwQSdXl45sRQ{P#>$;0Jv1UgKhzLUO%s z*bPc=JgX;_X@3=JY3SC)`X8x`Go1DJY{WgWz}8F^BROt{e_8dnvA-ZRR6B0ZNfqcVi5woQ_lk>^ja50}c)2JU z>3V#im2Xu1?+c3-fW)65a##pQ4NDv$|&SNo|j8!+{>rb&_0;}G}p4pWhWO9)k5Yx28>@t|n z>j<#T5FV<)*=MX4H2{iH`J)n1_Q5W)FL)8M>p5sKr}%U>c~J8TV&xE znT$!MKSI*s1AJoM*#P)z53`o2OJ|1=~%u7JI-=^i3=Cw zWN}ggdF%3d2FP+U-X%}$0VE!$*af)=TMIfUs?%!;IK$ISqNqr!a*KQk(U?G%A)(#lf) zNt1a9?>43(&jR;XiV#lozyb&0hcdbghq|uYdl^Gu=@M2~IsJ}Q<8DKyA(Id|r;*Fav6oyeIQmAS)x*E=bSKuUxiEi|GplXYHADWHt?iAm}hTvm?vA z9-z0GQP+sW+Oa1U_=@41R_5G75VvT{ygFzf1n6)|%f^;zH*T4S=C%BYWqupbRxxy> zqU>YJ#L(!~y2~$#jSV0v6WmFgLBEdZFS>SfU~Z1E=pZgZ{u! z)oFGtz8X=TEO;IPxDmtAdvaL;07g58k|(^EKE$<~DcArLz+` zL9W-_DK5`H=4HPssFCHcS?QvJjpZdXUqobj)~49L>|+xI*Dd-M)Sob&ow=Mzng?Gk zR_SgDl|r187j*ef0w6d$Nfk`R?ot)>AsvGFq{@0)yk1uOM&$qdO9ZeOl^}W8b`kPK`I_ULz-JpAmNayAPnxqIY9O}G z7f9Y=UlJ6na>Rg#^_iNr# zGDmRXbDhN#$J(|1H)O7plJxGw+U0^TEdxf`zb8e{xp$eh6kSyCUQ1MSH0*Y-xu+Yv zl$;)L%qe|LGdT!Nx-Qyc?3QGD!zi3S!THnT9%}VTm+->j^dO-fmcTI8xKf3c%gmsv zZ7?OK--=7>8z?JP^i-;+mB-7Oc4Y^6y?TRuFDSm78JathkGLwpnr(9|7LfU3)SGjki37Q;qUp63t*{qwrY;#iQBPy7*6e zI=&Y;w2%<;eoMWE{L8{1r)bzou5O*dQCOTvKUSrS6{@gPv-B4imyiP8^V#}3HZJO~ z`-@g%$<~9ltYk@ZrB@%cEk$y}Np2}s7IkPD%4w50F2BxL0%EUa#FwAQQK#O;rL7k; z8cuMI17GUU#(@p3t2lt7PEWIv`N}Td4L&vb9>;p%V|QiuZcZ_6?uYZ`n}X{u&N>sV z8uRjk@`PD!2zaujYw#1~zD}8g@=5QEt|$CxI$?T5&d3_QL2=Kg!beBjl|Q0E$!j_s4NZ7fpso}g0t1=5^D*oS@1%J1^Lr^?0mAU*&iw7)L=|aRk6AWBc8eo*HHXJMZm)Z@G>>vSe##Hsq^YO;$gn#>DImp%k?%k zxCZ5i?qI9<^%1mTOR&~ts@~EupgF&GC4qZ!i`{)*Cdy+z=sfQs!=doPAH_;nCbJZo zQOWHzI*!qHbNTMsA5O z>{y5PT(}@lylJCToh^e%gX`58xr#~!zw`6`{UoFftai8l6#%U}mR^TZTX5lQCneCA z$KD_w4U~M9O1lUTg4Y(W61^r~CerdeURy12SviaBJK(ZhF6Hu}Fj-g^+fsTm92bj0kQJ>oV*-fxP832oIKjp z33#oJlARc?d$=YCAu|9%MO;U#XoW=X7@UW6ryvQvv-{AuAlNy>%UgTa>fZr`Jii5^ z60R;w`|RPvbzmFaC+nk^nh|%tuau!qzZm-#>~gk-%d%v964p{+WlpZ|#Jhx{HAJyb z%bs^tv@mY?BT$p>&Mnh?D@Hwqx_i}fZgsu34Rg%n(qUn9IJu}{c2b_Co1N$KNnB94 z0^|jrcEO(ah;2^L z_xNJ3On5TgouR~jq$#@!YgJ>C$EDaF)|*qFuQfV{-nZ06#d>`62dVR_Ji7hN88{ZJN0 z!2A$K%DWhAd?g1ANvaVbtwN*2V%D5+*v4%SogRvNaG@@XRK_CQO*D`C=Egl|PQP!YzZ}>sP`S#{AA{5e zPeYc+;UlWEkd@)Vq0l%ZL7HYprF#vVaM#P;$8@5zQx3AEdpTFSL%+NJWiH_o$t`JyFMQdFF=K=-jGau3yG1M6P2LHoBjQhFqlX-bk?{Wa7KWm}r( zNi)hD_#$?Iw)AvKuV}YeR!E}Tig476jU9gcfOfmHw!~_hA6g{fMI|L+_ue0Nof8UW z3+A3@*)!D};ql{UF^A^xy5k`e5B=C#J88wPpLxu1x0RED&1ijGb`SKK4-5Ur!tuV_ZW`4s>(?|tTZ)*cjX08*oFJqJ4h$w(ZL1+5$TGG+R^ z z)Xs*f)ga_@?2fnn?aNa{uVSf+bnG0>hE~$J7BmVp2TqZ@lP{X47bPBds4WM@ysOyV zu#dCP&C^S5>D8?*O3rHV)N=iB-c-H66LwwR1R@6K)SqwP(Zm@RJkp^awcHPTFOQMNigJjeD$Jd20g;}W2Wfep@`90X+!4sXB~ zZd4QvN*lEh*JR{6eus>iY--8YR8R71j^ zm8+fF2k?Dk4cCJt=&W09QwGSL4(_}#)&SY4Fmx5nuMl#{VSG;SbN17&>tQ9;i#>PL z!kJBleq_m_bYZ?#Q9s^6XQi{d2C)fqv!^(5{cPQ?Q6L$5?AXsj_H9~=Tqf?s13PR= z(Yy8P)_E6~dAt(FY+_{oekG|ttKp?BO@9u(f^;siWPXLY-BYcF}^Fw?8?Nw!o^fK{@ zY=iRX_c@*yYcF5}@-6up?GfUbwI-N+Cp4NjkzSUCGC5iWv`QhT= zx2S_mqVoow)}ziWm(2} z&h5*VZAx6d=B&zuPuNSr%@)?zqqX~b?#itgJ`voRGRqK8OK7E_(4CLQ zlvphBU*0@_+Jisy+QSh6sz|h9lX-K-^5%M_KY!*8%h!t_onrG2#GP}gX%1tWRc%2x zdqj|8z^y2|@i^3yQtwBi=i=E#k1m(zY_`kYF!iz$`GzQa#rGzygEn2Ot}@H31W*{1 zI%tztWQ;RFfS7Oy#_CnMJ&`^D61h4W$~Oth8S+iEfo6?e^wnNJ_60leI2^YyxK~m) z%;#Z)U8oo+rx-;FkR*iX4mKd;@V!ybX|omI!$7yn4f{ zg9^5*xJ&M?JG2X)f1BdP5vi$s= z#l45A{>fVVs4yO-LGsR!hJ#0h-$UJ4!j8rTLKa36DmKyRCkVK=iTCu|b1RI?IqPAO zH5Tl2Ott&VAw>)vgy9`H@24Y)g3pB5v@F-4aRVuX%>JMWCeaF$ZNPh+< z^z(F#C{CEU=ad?Sfg@}=zF`u=7w_i;?wZfEBdfm<-^vw=8%pCa*L8X7TV>L_H?@n+ z!sXr-;>OE;0;WW6uk^%U0}I6~4MVp;#-5Yg0;BdE1L+h}#Js0X15}3Cp8S>iQ~nM$ zV`$l#b^fmIdWK8g+g=dJyw^d#zNyTrxq?feL1RQduMg}YYXDOq@-ZQBFKrE-{uNqtz?|&jAgj> zm?c0S{n8_4k~1Girs?X(Ub*Ex znG%4y9O=aK$l_2KsY^avh znia*LDymICatoH}j`L;93hjlWr{0^?FJCA~i2%@yhNeokDXGh)cO3}~&3l4^zlcp$ z-s5GHI#pZXI`o|MJVhSxCVQjJbc7{a)1l+LghHT&qioHnRHq zLV}|T$}4(vBTa=;xD^sR40nqsCq_12_)Ij-T=(l!S%RuWe=_XZIkk*E&Y){N*F?=h z-b|(F+6Xe^>z%oor@iqOIejUXO%l=Via%E~Bu+@&n||O9ibg?AH1*)^QY3Jm0#XzC z+Bj^q!sSFLytZ)lOJMvmH?`NQocZMC4eW$!FMhR?^8_dYwLZ&UI$WC#C}I zGb~SNH4Mes0*f-_kO<4=dVkK;@K9my6gzGhrkDRRb;Y9_enT=o%V}V-n7-#%CbEki zbwa5zo0FTX3Q{s#yMQI3b`;pX$FJ^IbspN0t0#Br+LQxT?J$KGt`u1BdYmu9FZR!- zCQ|%lM;EIP)4G|QRC-(-f+MdQ*KyT823acFlyg4=)z8Fa{2}L|-?gL?&h=#v@zz5P z{>`~O8mzNpWn}cR=)8fdHq04J(-qdlavxUxb(O)jwhnzhfgV}idZ%Lp#kJD$z2P(K zV67?2jr9fckb)_z^lb|SPlae^bAXU&D*g$$rFz9!vo(nx+ev^Sspaz`U94?rmU6BU=j|;4{+O6D35?t&%JU5lbZ9(j)n@uAp&q$mr=x- z4X6fJ1%I1d5~*Z5vOi%wgRxlqkLb#&4)6-o_gKqE%sJrC%1L!=N|(v={{)-UGYR|?Q6BmT$yh-j3ZZ0tOP5T_-5noaUjmR z?HcSSq@%TY7C4hPS6`~$S42R48{P+fTMxC160%-y#J%1V1ruI#!Gnu+j4U<`(F%_l zsLmNp9UmLNEI=ay7uN0mWM!7V-pWo^C+W>4w>xWP-m(?xYP83kf2Ap$_06(;cO}!{ zrSDWy4Rm636IEv~p6YU^Uq@8fW|BeTkOc6{^i5%d=PNARrZ*8o2)(0)>XqxC!W-nI zdBT-_j6D}h>^xj%4;d^Mc2o{?j`JSw_tpP&9=*zuPBC7#a39U8)GKF z_4r9dDBN7D1les6I`vew)b9GJc?3FgcDhEa>LzO(4<9P7;f)8@4F1%5v9k#1_@1s4 zTqku`3}8F4?cyw4kgUA1ICDi=cx2Ol@{7x*0^QT2S?qJm!n1xrZ!C8`-{hVea~2R^ z$b*EO1=Ss8g0>yO#f}XZjnU!yNWptlLN3hAde3G~raZ5CIOFp&cQ0K{T$B5S30{xs z1W$_h#Uk}?jDrf?$b+G^5gBXAZpg<={>Q*j>+I!Q;Ew&KspBtkk z^NW44-POl=Yj&G3&JQ{%11rmiTYm_Amt0~WxoSJRyZKq?OD4{b$L@vI9$mpwG)f)P z?wI^U&wGc~CqVm%5MGK*oDXUz0t_Z2`EfInG1;-+1|Ryqanbh}VlrE@PgJMbz0BOZ zoU5ol&!~%cq@0dL?k0^h9ARcD9gVhBch#TsbGEyeKX0TNUDbWafMTWdJTzk@<2xLc z_TIVeoNyxJBgZbxTknt|fR{v0P`bWOLMY~r9fWaA(j#B=5%o=Ei1`en4Na!AR`s}Q z%trcF0XlIHGlTYqa`r`{p}_ZVb-puKeN(d0I|8-vq6(@N6Z?jNxyp62Etrx+q5q|` z=>V@q058kc>{3ZAteZ_veyI4EC}f;-03A+t?MQphk*MX$2~!1xmB+f+j?yozql2|s zivhQSnkIHsEp`V8rl{)t4hd}ciFGeTzLVsW5& z&UPqC`2|`U2b;8LnYvN&x(d$FiiENwhcfPv3;wmvB2}Ip6@EhQkZbTluI4xrqvw|T zMP&wTw7}0|?2iH3`E~k3=GqO}YvyvUyAJdBEXo{?9zti-6RRGgU$2!e4q9E!oeA?- z)L?Ja$>7W$IT{N=L|?y_yVUP$^(H*qbN@-HdHmrnTA@;dn#0C;w_EvM1#Jlm-~8&C zwM}en1VR>DF*ZMa`HKOjO(Cm*wIHl~_)qS5g{-7RlxRKRL} zJX*EEqC!po4=V68a9m0;#d!r}gq_=MvGh&*qyw=xVkqP?(8wRZnq^L3IcQ6OfH8pD zM3OGLiJvNZDkK*aDa%zWo5d}RG2 zb;SDIv&#WL0!aUd7tnot0(tT)EInP&ArbF zGynAMP9uI5wtN*L)fcA1o_%)C&Q-}D-9lh!uk5?%?$T=_FdWDqm5nnNjaaE#ic<<* z4-7R3Nn8VlC_oZnzr8ffuA9eT0 z17fz23!>~flrC%pRlK3@G3VUAaK`05ASX98sp6f0$GkIn)o>##4r(FYTbp|43%ZL_ zGkWy1`o4>aXw&Fc-#MX!6T$!#ILCOb!-Hqlh1kjKJ1 z;bPK?A!NN(3kO5O4er;D&67?$a|1b*ymzoQd5dTFw`sJ`7qvWWAvm3k8=4JeG#%Tx zlVyHAj12UJ!tUPvzAT})4J&m$5<=&psKhe>Ja28bqQ%70nyq2q!tgNyi4s8k zcDzBt=8Ev(VJ|*?r5EBA{}8&}JjB?>SjE z+M~t@)Ast<_F0teXFfa!zBJGtc{^{hsS#UkTk$=zlsiEJ0f5ik~U^2t= zH+#)cTvnD)MudOE5`2f6x*{P_)JTiHd{<(k-_T60wqmh6xg=ME5-aB{#{^eTQa5nR zJKx&VMqp-M4+0>7L=Txb%K`I2AhJ51H6ly)rbSuxl}YS4!Cnr!+rd&(Bj<*AqnCz@ zSWS$p2*t*AxE-IeDGeE8nH>R*SETs=L3A|1^$lEzS~x}t<~Mqwp;9!KHW*$UC77@{ zc!Miow7NJQxxp2LTpMb$aElr6=-=}*jDTD>Y|BBD4rTmNQ{e24~cO;am{ zUF+0%omeqQ$ImbTgh)AdAmJT)uR2;X!>PiOS`(nNhuV!$EWTic^6&w9mTRyFat}U*0?}n1=t=tAO+aCg6 zO(QauvmR!cqLocaRaNhDAYBG_ypa2;J@^0sFe#KHLnXlg4u^P@%*z`Z`^@67=Y1+L z8ta4F3?Q?@uWoICZb_(6Ncx4)|A>mxUk89?wX4Lvaocr0dXt$R(C}XfrIeF{gK87& z=XG+Bk%E34_P{^9fvv3*90e~l2ii|qon>7_ui(s&hv1RJ{9yER?#d}C3 zYl2=D(oyudH_w<^*2GXiM~ANb^rx3EOzzv^>ufbaDAPg))G+S zHg&R+k#SW)o#gfgvUswp7k|EHaf{rTy^MiyRHPNS=)#zzGa5d%5;&a>Uh9kIW_+^H z3AOxWa=qwg2d*#6<0qaSULoAPSnG!9ibg{3aH4?!k}eQe-|5>FGcQTIAj4J;yPbE^scpX zGHT2BUvkV0LQ}BH)#!QqRyIQ}Et@(%+IXsQgvON(#pO>R+_#x1&UGDPW#d?8dIbI3 zH`YBRevhxL#jd-g(rB4zZ}%1rO4U;Xowq=ySK9ERZ?S)Jg+-gHROtfI)4sAyBU)Fg7|G zzS^Uxt*x!3)DjyVzK`U4O!z$*fRzttDd9J|<*7G#2%6T7fw9E|e~n{8#;XDU)B`jm zUdIHTwNdEAuCKra=-py?+?37aqX*)1P^k!x~Ga*Csoz~V>x&mq35H{p2bY{*|k z?m1Z)suSbQcVluZ~~wD zs3!u!`nNcQzpkN3qbC1G%>Jhz>IVRk6MfdRnf8xS{c0AqQ0UIzJH5UC2GC)0|Mw~k z|IKNhcpZ`a)!$z?=xD&YDBLgMza@%)=UMduH#_NFJHJ{+QA_@>hll5_ z0E9f^v~lD_{Pz<9LXYnMK8OD=#s8P$PYm1tPsQKP_x}mh@8s_P1nO_p_W!)%cb4~m zw*8A?11ua*0R@OH;O6E==FacR1&R~Y0fBnHI22|-KU~cLd3{*5Dhso-v(KiGJKLJY z1BijN{me+^X+J-Ie5ai2`n$A}QijT7Fp$X$2n&iPDNaPaiccvW4mn?;{l?qDm6a7Kq1x8YhI;q&U~*8n>>!Xhe+S3{VDrFf zd(xrR5C5sjK4X>H2-8J6pDDhtpzGlyAvb3a521=s%hTAsL>hXac7TKJkD#B@)~(s0 zntU(Da@9Z0ZvoPI04hWMB2XN8Uk_Lc9~_EFART}iWG&HYw?4^jziV}|JwQg*p~7lL zp#)zb*r~sOP9};RD6G&*_y)+f&(OBy$O6ZhlD@O17NKiruBN(U@QR!%) zES1_F#VV=@RAnbap8jrHrzU^tjyQbz{;jaGjdUPy**Bs*Q>N(#PvO|!@YAm=-O&du z*|Xv{9mqt!K4ote6}?e9URj9NRA(3r__c(RDy-TEDW<;qT&=vbDeL))a{iD@^-n0v zUqa+F404d|XNb7C_y8SV#i;`?cs7uVXzy#w0Z_J;`yNa19EE@*b$mx4Ny{hmv3%_c zD+(BBL0^12$@V9s3K<*p>|iZ50>5S1I{FT)KjHXvXZ`>mkv@`ye%E}{C&rKm9a%8S zz~qz}LU}%K&JXXF>7o=c7V+K(?n|cIb?YF9k~A>Y1klE{0>vDB?~^S4{52WD;4i6? zA*<;~9f%{CcZ-ZS-}6O|Wo0zdai%oeS;GcLH#{3_P z$}4{@_w5ZUt%8xRU3-mUYa{;dMnrtKcW(q+LG;RZz*6G>y4*v@)P7 zFleCCH&az6X)!4X>@Xv?u_s+=fZl8GGOmNW>D0tb;BXi$7j;fo8YZ6`R`P)A6QbJW3`As@6^+9^y^TKD!dN55og9BrN z?j4^;+Oy)=%%@-bhQ_i%7O`{zh;*dp$dZctqy(VX^cc}pnXgJdJ7ngvVi`E%w!h)q zwv;geKm*@Xp3&}|nwVoH_SNb~#lO;_bRwKz8vHPN#rHhdt>k{ymzLNQG`uA=RCaDP zkNacQBcm3gS-m|sQP(6OA+;aY0uha_0z}- z&wbQ~3;i+g_xg^BQzl%s=vSWI!IyuJwVe@fPX&Lb9gN0=gIyBZF;i;B^i9{E3nz&0 zzPj&yIUGyUbo2X|iAM6ao5OU;cv}lwf*g%FBGO1TeIdW%E&ryOdQ&$#ilILu*5M=x zbbahflI~k=U`Hz}c5Z_uXz(_Oy!8p1^XG}e=~wTIe?9;$FL|`W{7%I{OI{SuKj1O_ z0c2?&)g1gFob(+DchReCBwk1;B`}SEzSWgB zDk}n$QMx`$pNoBzMEPr);p;pH)DQ0hl%IPy2i{Ie_0DSaU3!!lk8d7J^V|nOF`-k369e^=Ts`#O;_1v#@g|RdWgXDzT7XPuKau`q7(oT5eoxl@rM3WlP_q3-W zKKtCuc_kCivENIV);!7?AHm=j04mQKQ_1SD+L*SQF%x4m&#{Q|2b5+s?CIjQnmucB z^Z68BYGktY1RDFPSO3wyfY03g05Ok}fe1CVzmxsvSVU06Yx}#nu2_k*c0^rA0$l6-xwXA~&3azo z?=~NAM=_+w^c~>h_U_O5IZy_mI-FKj3y}kcmHK4-(og2wDC56vL4Z2N(&4dcBzieu zloTo=Y37gJA!lYNp-_VTM!|00&Z9fivnMp6x%tihT0sIGpMkhs1rizy7 zwsDj|E$p_%0B4M`Zc!?|`hAYT=J+VuC_QHUCC`WlQ;%(Hl1_`4MR+v5TlIhac2I)4 zUd3T@lt6`##w(uar*Ub;4x|IP{X_aO_R=zQIb%m?ztL%NEZgHXm_Yk~#= zX>yjFdDB+P+58=luK6EEpqsaY(v5~$N?E;1<{)}gC}lT@-U|y8lO>>}AOYfqiaXAg zz!=+GpT+u*`QENG2MYRI26@(nCrlVJc8*EfgPl4_LC#)vD12|rH|B|@%^L@I2FHAu zQ-1|#{>G!y{Jc!x5~e+Wzc^C1@*rmg($|4%4OG*q<6TV4Lw0h^t3LkYukG}o8}A4B zy9oWT{89qRMpy7-2qXC((nBJm1=8|Q1W&X;TyZkG&Dx098ufpIIsg8AXhK;~TG@OxwHC|jSZ-DUQr{5(#L6H6_yfJTl=DyZFPck#z>R`PS_Qd zTJpPA3_Ox2VP1?%y~AA-1Q|MVgW(1kPG9G_@6b@{1q0T68^;whG8|^PL8}SS+d2l> z$#`iEAjJ4@a>(sH(8yPX-eFVI(roq!O)1sP_u7+5oJ}&Z61W`o)0Bd)Ta#r0sD0Mu zEhItrV#wLt&r~R4M{Ai2{@21h`QoK&8;72*!SG5t<}gSo^)sL&nCeGKlA>{BDH3db@Z>1F+T!zo< z9T(VL?{7Ur4s-;Hgu#K#_>9Zkog?P*dJOJ~qjWDxYD!mKre}^+(tlN0_-(%A(Ow%0 znapyw8uWb{A4@2H=n_0tk7#;gW@aWkPGehVDPEd$g4kVX(X6makG;5XbHtQ5v<+RA z%@A<@Ob-8)Rt2+l8>x2tU&Eyq@;==p_}0CY;jt|nOxT#=(FGqX&;`;7)MB5Zah%-f z8U`IWlcFbahe3Yxcabej3hp1rt>mfRhB5ypwzDDIl_ApDqgT(lNdgJzGo9eG+ST@Z zWwU`u8K2L*rJd0VRWHNK{biAWN}h#(0_*>NLaD!^CLEUW4^+$F6*w|Si9s2t*M(%e zAomE?wojE?3b-CVC4J-`uvqq~l;V%azVzFJ1jd9Qm74#x`TH?Z@kVNgX#&FHpBL71N z{(92hV(8DAndJ0rSK02p>?H^I4GR+!6AO!Mn#Rvk5hMn+R3Pc11.6.1.   简单实现

先看一个简单例子。给定一个长度为 \(n\) 的数组 nums ，元素皆为 非负整数。计数排序的整体流程为：

    -
  1. 统计数组的最大数字，记为 \(m\) ，并建立一个长度为 \(m + 1\) 的辅助数组 counter ；
    2. +
  2. 遍历记录数组中的最大数字，记为 \(m\) ，并建立一个长度为 \(m + 1\) 的辅助数组 counter ；
  3. 借助 counter 统计 nums 中各数字的出现次数，其中 counter[num] 对应数字 num 的出现次数。统计方法很简单，只需遍历 nums （设当前数字为 num），每轮将 counter[num] 自增 \(1\) 即可。
  4. 由于 counter 的各个索引是天然有序的，因此相当于所有数字已经被排序好了。接下来，我们遍历 counter ，根据各数字的出现次数，将各数字按从小到大的顺序填入 nums 即可。
-
-
-
-

counting_sort_naive_step1

-
-
-

counting_sort_naive_step2

-
-
-

counting_sort_naive_step3

-
-
-
-

以下是实现代码，计数排序名副其实，确实是通过“统计数量”来实现排序的。

-
+

观察发现，计数排序名副其实，是通过“统计元素数量”来实现排序的。

+

counting_sort_overview

+

Fig. counting_sort_overview

+ +
counting_sort.java
/* 计数排序 */
@@ -1915,7 +1905,7 @@
 
  • 令前缀和 prefix[num] 自减 \(1\) ，从而得到下次放置 num 的索引；
    • 
 
 
    完成遍历后，数组 res 中就是排序好的结果，最后使用 res 覆盖原数组 nums 即可；
    
-
    
+
    
 
    
 
    
 
    counting_sort_step1
    
@@ -1944,7 +1934,7 @@
 
    
 
    
 
    计数排序的实现代码如下所示。
    
-
    
+
    
 
    
 
    
 
    counting_sort.java
    /* 计数排序 */
diff --git a/search/search_index.json b/search/search_index.json
index 2bc6f3316..90a616a24 100644
--- a/search/search_index.json
+++ b/search/search_index.json
@@ -1 +1 @@
-{"config":{"lang":["ja"],"separator":"[\\s\\-\uff0c\u3002]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"
      \u300a Hello \u7b97\u6cd5 \u300b  
     \u52a8\u753b\u56fe\u89e3\u3001\u80fd\u8fd0\u884c\u3001\u53ef\u63d0\u95ee\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5feb\u901f\u5165\u95e8\u6559\u7a0b 
     
     
     @Krahets 
      \u63a8\u8350\u8bed  
    Quote
     
    \u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d
     
    \u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388
     
    Quote
     
    \u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d
     
    \u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6
      \u300c\u5168\u4e66\u52a8\u753b\u56fe\u89e3\u300d  
     \u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u7535\u8111\u3001\u5e73\u677f\u3001\u624b\u673a\u5168\u7ec8\u7aef\u9605\u8bfb 
     
     
     \"A picture is worth a thousand words.\" 
     
     \u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d 
      \u300c\u4ee3\u7801\u4e00\u952e\u8fd0\u884c\u300d  
     \u63d0\u4f9b\u5404\u4e2a\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u7b80\u6d01\u5b9e\u73b0\u4e0e\u6d4b\u8bd5\u6837\u4f8b\uff0c\u7686\u53ef\u76f4\u63a5\u8fd0\u884c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00 
     
     
     \"Talk is cheap. Show me the code.\" 
     
     \u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d 
      \u300c\u53ef\u8ba8\u8bba\u4e0e\u63d0\u95ee\u300d  
     \u9f13\u52b1\u5c0f\u4f19\u4f34\u4eec\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u6210\u957f\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d 
     
     
     \u201c\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d 
     
     \u4e00\u8d77\u52a0\u6cb9\uff01 
      \u5e8f  
    \u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u300a\u5251\u6307 Offer\u300b\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u5f88\u591a\u5c0f\u4f19\u4f34\u7684\u559c\u7231\u4e0e\u652f\u6301\u3002\u5728\u6b64\u671f\u95f4\uff0c\u6211\u4e5f\u56de\u590d\u4e86\u8bb8\u591a\u8bfb\u8005\u7684\u8bc4\u8bba\u95ee\u9898\uff0c\u9047\u5230\u6700\u591a\u7684\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u5b66\u4e60\u7b97\u6cd5\u201d\u3002\u6211\u6e10\u6e10\u4e5f\u5bf9\u8fd9\u4e2a\u95ee\u9898\u597d\u5947\u4e86\u8d77\u6765\u3002
     
    \u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u5e94\u8be5\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u5f0f\uff0c\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\u3002\u7136\u800c\uff0c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5730\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u53d7\u632b\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e66\u7c4d\u4e5f\u662f\u5e38\u7528\u65b9\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u5e94\u4ed8\u7b14\u9762\u8bd5\u5df2\u7ecf\u5360\u7528\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u539a\u91cd\u7684\u4e66\u672c\u4e5f\u56e0\u6b64\u6210\u4e3a\u5de8\u5927\u7684\u6311\u6218\u3002
     
    \u5982\u679c\u4f60\u4e5f\u6709\u4e0a\u8ff0\u70e6\u607c\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u4e8e\u8be5\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u867d\u7136\u4e0d\u4e00\u5b9a\u6b63\u786e\uff0c\u4f46\u81f3\u5c11\u4ee3\u8868\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u8fd9\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u5f97\u5fc3\u5e94\u624b\u5730\u5237\u9898\u4e0e\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u642d\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002
      \u4f5c\u8005\u7b80\u4ecb  
    \u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u5176 LeetBook\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 22 \u4e07\u672c\u3002
      \u81f4\u8c22  
    \u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u7684\u7fa4\u7b56\u7fa4\u529b\u4e0b\u9010\u6b65\u6210\u957f\uff0c\u611f\u8c22\u6bcf\u4e00\u4f4d\u64b0\u7a3f\u4eba\uff0c\u662f\u4ed6\u4eec\u7684\u65e0\u79c1\u5949\u732e\u8ba9\u8fd9\u672c\u4e66\u53d8\u5f97\u66f4\u597d\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\uff09\uff1a
     
     
     
    \u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 justin\u2010tse, krahets, nuomi1, Reanon, sjinzh \u5b8c\u6210\uff0c\u611f\u8c22\u4ed6\u4eec\u7684\u8f9b\u52e4\u4ed8\u51fa\uff01
     justin-tseJS / TS krahetsJava / Python nuomi1Swift ReanonGo / C sjinzhRust / Zig"},{"location":"chapter_appendix/contribution/","title":"12.2. \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"
    \u5f00\u6e90\u7684\u9b45\u529b
     
    \u7eb8\u8d28\u4e66\u7c4d\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u9700\u8981\u6570\u5e74\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002\u4f46\u5728\u672c\u5f00\u6e90 HTML \u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002
     
    \u7531\u4e8e\u4f5c\u8005\u6c34\u5e73\u6709\u9650\uff0c\u4e66\u4e2d\u5185\u5bb9\u96be\u514d\u758f\u6f0f\u8c2c\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u53d1\u73b0\u7b14\u8bef\u3001\u65e0\u6548\u94fe\u63a5\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u3001\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u60a8\u5e2e\u5fd9\u4fee\u6b63\uff0c\u4ee5\u5e2e\u52a9\u5176\u4ed6\u8bfb\u8005\u83b7\u53d6\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u5185\u5bb9\u3002\u6240\u6709\u64b0\u7a3f\u4eba\u5c06\u88ab\u5c55\u793a\u5728\u4ed3\u5e93\u4e0e\u7f51\u7ad9\u4e3b\u9875\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\uff01
    "},{"location":"chapter_appendix/contribution/#1221","title":"12.2.1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"
    \u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u4e00\u4e2a\u300c\u7f16\u8f91\u300d\u56fe\u6807\uff0c\u4f60\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u5b57\u6216\u4ee3\u7801\uff1a
     
       
    1. \u70b9\u51fb\u7f16\u8f91\u6309\u94ae\uff0c\u5982\u679c\u9047\u5230\u63d0\u793a\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\uff0c\u8bf7\u901a\u8fc7\uff1b
      2.  
    2. \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u5e76\u68c0\u67e5\u5185\u5bb9\u6b63\u786e\u6027\uff0c\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7edf\u4e00\uff1b
      3.  
    3. \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u66f4\u6539\u8bf4\u660e\uff0c\u7136\u540e\u5355\u51fb\u201cPropose file change\u201d\u6309\u94ae\uff1b\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u5373\u53ef\u3002
      4.  
     
     
     Fig. \u9875\u9762\u7f16\u8f91\u6309\u952e 
     
    \u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u56fe\u7247\u95ee\u9898\uff0c\u6211\u4f1a\u7b2c\u4e00\u65f6\u95f4\u91cd\u65b0\u753b\u56fe\u5e76\u66ff\u6362\u56fe\u7247\u3002
    "},{"location":"chapter_appendix/contribution/#1222","title":"12.2.2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"
    \u5982\u679c\u60a8\u60f3\u8981\u53c2\u4e0e\u672c\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u7ffb\u8bd1\u4ee3\u7801\u81f3\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u62d3\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd Pull Request \u5de5\u4f5c\u6d41\u7a0b\uff1a
     
       
    1. \u767b\u5f55 GitHub \uff0c\u5e76 Fork \u672c\u4ed3\u5e93 \u81f3\u4e2a\u4eba\u8d26\u53f7\uff1b
      2.  
    2. \u8fdb\u5165 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 git clone \u514b\u9686\u8be5\u4ed3\u5e93\u81f3\u672c\u5730\uff1b
      3.  
    3. \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u901a\u8fc7\u8fd0\u884c\u6d4b\u8bd5\u6765\u9a8c\u8bc1\u4ee3\u7801\u6b63\u786e\u6027\uff1b
      4.  
    4. \u5c06\u672c\u5730\u66f4\u6539 Commit \uff0c\u5e76 Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\uff1b
      5.  
    5. \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u5373\u53ef\uff1b
      6.  
    "},{"location":"chapter_appendix/contribution/#1223-docker","title":"12.2.3. \u00a0 Docker \u90e8\u7f72","text":"
    \u4f60\u53ef\u4ee5\u4f7f\u7528 Docker \u6765\u90e8\u7f72\u672c\u9879\u76ee\u3002\u7a0d\u7b49\u7247\u523b\uff0c\u5373\u53ef\u4f7f\u7528\u6d4f\u89c8\u5668\u6253\u5f00 http://localhost:8000 \u8bbf\u95ee\u672c\u9879\u76ee\u3002
     
    git clone https://github.com/krahets/hello-algo.git\ncd hello-algo\ndocker-compose up -d\n
     
    \u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\u3002
     
    docker-compose down\n
    "},{"location":"chapter_appendix/installation/","title":"12.1. \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1211-vscode","title":"12.1.1. \u00a0 \u5b89\u88c5 VSCode","text":"
    \u672c\u4e66\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730 IDE \uff0c\u4e0b\u8f7d\u5e76\u5b89\u88c5 VSCode \u3002
    "},{"location":"chapter_appendix/installation/#1212-java","title":"12.1.2. \u00a0 Java \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 java \uff0c\u5b89\u88c5 Java Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1213-cc","title":"12.1.3. \u00a0 C/C++ \u73af\u5883","text":"
       
    1. Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff0cMacOS \u81ea\u5e26 Clang \u65e0\u9700\u5b89\u88c5\u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c++ \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1214-python","title":"12.1.4. \u00a0 Python \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 python \uff0c\u5b89\u88c5 Python Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1215-go","title":"12.1.5. \u00a0 Go \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 go \uff0c\u5b89\u88c5 Go \u3002
      3.  
    3. \u5feb\u6377\u952e Ctrl + Shift + P \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 Go: Install/Update Tools \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
      4.  
    "},{"location":"chapter_appendix/installation/#1216-javascript","title":"12.1.6. \u00a0 JavaScript \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 javascript \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1217-c","title":"12.1.7. \u00a0 C# \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 6.0 \uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c# \uff0c\u5b89\u88c5 c# \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1218-swift","title":"12.1.8. \u00a0 Swift \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift\uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 swift\uff0c\u5b89\u88c5 Swift for Visual Studio Code\u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1219-rust","title":"12.1.9. \u00a0 Rust \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust\uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 rust\uff0c\u5b89\u88c5 rust-analyzer\u3002
      3.  
    "},{"location":"chapter_array_and_linkedlist/array/","title":"4.1. \u00a0 \u6570\u7ec4","text":"
    \u300c\u6570\u7ec4 Array\u300d\u662f\u4e00\u79cd\u5c06 \u76f8\u540c\u7c7b\u578b\u5143\u7d20 \u5b58\u50a8\u5728 \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u7684\u6570\u636e\u7ed3\u6784\uff0c\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u5143\u7d20\u7684\u300c\u7d22\u5f15 Index\u300d\u3002
     
     
     Fig. \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f 
     
    Note
     
    \u89c2\u5bdf\u4e0a\u56fe\uff0c\u6211\u4eec\u53d1\u73b0 \u6570\u7ec4\u9996\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \u3002\u4f60\u53ef\u80fd\u4f1a\u60f3\uff0c\u8fd9\u5e76\u4e0d\u7b26\u5408\u65e5\u5e38\u4e60\u60ef\uff0c\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(1\\) \u5462\uff0c\u8fd9\u4e0d\u662f\u66f4\u52a0\u81ea\u7136\u5417\uff1f\u6211\u8ba4\u540c\u4f60\u7684\u60f3\u6cd5\uff0c\u4f46\u8bf7\u5148\u8bb0\u4f4f\u8fd9\u4e2a\u8bbe\u5b9a\uff0c\u540e\u9762\u8bb2\u5185\u5b58\u5730\u5740\u8ba1\u7b97\u65f6\uff0c\u6211\u4f1a\u5c1d\u8bd5\u89e3\u7b54\u8fd9\u4e2a\u95ee\u9898\u3002
     
    \u6570\u7ec4\u521d\u59cb\u5316\u3002\u4e00\u822c\u4f1a\u7528\u5230\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u4e24\u79cd\u5199\u6cd5\uff0c\u53ef\u6839\u636e\u9700\u6c42\u9009\u53d6\u3002\u5728\u4e0d\u7ed9\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u4e00\u822c\u6240\u6709\u5143\u7d20\u4f1a\u88ab\u521d\u59cb\u5316\u4e3a\u9ed8\u8ba4\u503c \\(0\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
     array.cpp
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
     array.py
    \"\"\" \u521d\u59cb\u5316\u6570\u7ec4 \"\"\"\narr: List[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: List[int] = [1, 3, 2, 5, 4]  \n
     array.go
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
     array.js
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
     array.ts
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
     array.c
    int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
     array.cs
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
     array.swift
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
     array.zig
    // \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n
    "},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1. \u00a0 \u6570\u7ec4\u4f18\u70b9","text":"
    \u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u6570\u7ec4\u4e2d\uff0c\u8ba1\u7b97\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u3001\u548c\u4e00\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5229\u7528\u4ee5\u4e0b\u516c\u5f0f\u53ef\u4ee5\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u6b64\u5143\u7d20\u3002
     
     
     Fig. \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 
     
    # \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n
     
    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u5143\u7d20\u7d22\u5f15\u4ece 0 \u5f00\u59cb\u7f16\u53f7\uff1f \u6839\u636e\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u8868\u793a\u7684\u662f\u5185\u5b58\u5730\u5740\u504f\u79fb\u91cf\uff0c\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u90a3\u4e48\u7d22\u5f15\u662f \\(0\\) \u4e5f\u5c31\u5f88\u81ea\u7136\u4e86\u3002
     
    \u8bbf\u95ee\u5143\u7d20\u7684\u9ad8\u6548\u6027\u5e26\u6765\u4e86\u8bb8\u591a\u4fbf\u5229\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u83b7\u53d6\u4e00\u4e2a\u6570\u7ec4\u4e2d\u7684\u5143\u7d20\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = ThreadLocalRandom.current().\nnextInt(0, nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.cpp
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int* nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.py
    def random_access(nums: List[int]) -> int:\n\"\"\" \u968f\u673a\u8bbf\u95ee\u5143\u7d20 \"\"\"\n# \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandom_index = random.randint(0, len(nums) - 1)\n# \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandom_num = nums[random_index]\nreturn random_num\n
     array.go
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandomIndex := rand.Intn(len(nums))\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandomNum = nums[randomIndex]\nreturn\n}\n
     array.js
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
     array.ts
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
     array.c
    [class]{}-[func]{randomAccess}\n
     array.cs
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums)\n{\nRandom random = new();\n// \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = random.Next(nums.Length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.swift
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n// \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet randomIndex = nums.indices.randomElement()!\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet randomNum = nums[randomIndex]\nreturn randomNum\n}\n
     array.zig
    // \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n// \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\nvar randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nvar randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2. \u00a0 \u6570\u7ec4\u7f3a\u70b9","text":"
    \u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7531\u4e8e\u7cfb\u7edf\u65e0\u6cd5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u56e0\u6b64\u6570\u7ec4\u957f\u5ea6\u65e0\u6cd5\u6269\u5c55\u3002\u800c\u82e5\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u65b0\u5efa\u4e00\u4e2a\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u662f\u975e\u5e38\u8017\u65f6\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.cpp
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint* extend(int* nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint* res = new int[size + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] nums;\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.py
    def extend(nums: List[int], enlarge: int) -> List[int]:\n\"\"\" \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 \"\"\"\n# \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres = [0] * (len(nums) + enlarge)\n# \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in range(len(nums)):\nres[i] = nums[i]\n# \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n
     array.go
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres := make([]int, len(nums)+enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i, num := range nums {\nres[i] = num\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
     array.js
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.ts
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.c
    [class]{}-[func]{extend}\n
     array.cs
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge)\n{\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.Length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.swift
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = Array(repeating: 0, count: nums.count + enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in nums.indices {\nres[i] = nums[i]\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
     array.zig
    // \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = try mem_allocator.alloc(i32, nums.len + enlarge);\nstd.mem.set(i32, res, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(i32, res, nums);\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     
    \u6570\u7ec4\u4e2d\u63d2\u5165\u6216\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\u4e0b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u653e\u4efb\u4f55\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4e0d\u5f97\u4e0d\u5c06\u6b64\u7d22\u5f15\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002
     
     
     Fig. \u6570\u7ec4\u63d2\u5165\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#Swift array.java
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.cpp
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int* nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.py
    def insert(nums: List[int], num: int, index: int) -> None:\n\"\"\" \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num \"\"\"\n# \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in range(len(nums) - 1, index, -1):\nnums[i] = nums[i - 1]\n# \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n
     array.go
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i := len(nums) - 1; i > index; i-- {\nnums[i] = nums[i-1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
     array.js
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.ts
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.c
    [class]{}-[func]{insert}\n
     array.cs
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index)\n{\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.Length - 1; i > index; i--)\n{\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.swift
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: nums.count - 1, next: { $0 > index + 1 ? $0 - 1 : nil }) {\nnums[i] = nums[i - 1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
     
    \u5220\u9664\u5143\u7d20\u4e5f\u662f\u7c7b\u4f3c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5220\u9664\u5143\u7d20\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6211\u4eec\u65e0\u9700\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002
     
     
     Fig. \u6570\u7ec4\u5220\u9664\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.cpp
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int* nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.py
    def remove(nums: List[int], index: int) -> None:\n\"\"\" \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 \"\"\"\n# \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in range(index, len(nums) - 1):\nnums[i] = nums[i + 1]\n
     array.go
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i := index; i < len(nums)-1; i++ {\nnums[i] = nums[i+1]\n}\n}\n
     array.js
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums, index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.ts
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.c
    [class]{}-[func]{removeItem}\n
     array.cs
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index)\n{\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.Length - 1; i++)\n{\nnums[i] = nums[i + 1];\n}\n}\n
     array.swift
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\nlet count = nums.count\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: index, next: { $0 < count - 1 - 1 ? $0 + 1 : nil }) {\nnums[i] = nums[i + 1]\n}\n}\n
     array.zig
    // \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar i = index;\nwhile (i < nums.len - 1) : (i += 1) {\nnums[i] = nums[i + 1];\n}\n}\n
     
    \u603b\u7ed3\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\uff1a
     
       
    • \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(N)\\) \uff0c\u5176\u4e2d \\(N\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
      •  
    • \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u88ab\u4e22\u5931\u3002
      •  
    • \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u4e00\u822c\u4f1a\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u6211\u4eec\u4e0d\u5173\u5fc3\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u540c\u65f6\u4e5f\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
      •  
    "},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3. \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":"
    \u6570\u7ec4\u904d\u5386\u3002\u4ee5\u4e0b\u4ecb\u7ecd\u4e24\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int num : nums) {\ncount++;\n}\n}\n
     array.cpp
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int* nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
     array.py
    def traverse(nums: List[int]) -> None:\n\"\"\" \u904d\u5386\u6570\u7ec4 \"\"\"\ncount = 0\n# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor num in nums:\ncount += 1\n
     array.go
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\ncount := 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\ncount++\n}\ncount = 0\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor range nums {\ncount++\n}\n}\n
     array.js
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
     array.ts
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
     array.c
    [class]{}-[func]{traverse}\n
     array.cs
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums)\n{\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nforeach (int num in nums)\n{\ncount++;\n}\n}\n
     array.swift
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\nvar count = 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor _ in nums.indices {\ncount += 1\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor _ in nums {\ncount += 1\n}\n}\n
     array.zig
    // \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\nvar count: i32 = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nvar i: i32 = 0;\nwhile (i < nums.len) : (i += 1) {\ncount += 1;\n}\ncount = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (nums) |_| {\ncount += 1;\n}\n}\n
     
    \u6570\u7ec4\u67e5\u627e\u3002\u901a\u8fc7\u904d\u5386\u6570\u7ec4\uff0c\u67e5\u627e\u6570\u7ec4\u5185\u7684\u6307\u5b9a\u5143\u7d20\uff0c\u5e76\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.cpp
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int* nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.py
    def find(nums: List[int], target: int) -> int:\n\"\"\" \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 \"\"\"\nfor i in range(len(nums)):\nif nums[i] == target:\nreturn i\nreturn -1\n
     array.go
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\nindex = -1\nfor i := 0; i < len(nums); i++ {\nif nums[i] == target {\nindex = i\nbreak\n}\n}\nreturn\n}\n
     array.js
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
     array.ts
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] === target) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     array.c
    [class]{}-[func]{find}\n
     array.cs
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target)\n{\nfor (int i = 0; i < nums.Length; i++)\n{\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.swift
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\nfor i in nums.indices {\nif nums[i] == target {\nreturn i\n}\n}\nreturn -1\n}\n
     array.zig
    // \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\nfor (nums) |num, i| {\nif (num == target) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#414","title":"4.1.4. \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"
    \u968f\u673a\u8bbf\u95ee\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u6837\u672c\u7684\u968f\u673a\u62bd\u53d6\u3002
     
    \u4e8c\u5206\u67e5\u627e\u3002\u4f8b\u5982\u524d\u6587\u67e5\u5b57\u5178\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u4e2d\u7684\u6240\u6709\u5b57\u6309\u7167\u62fc\u97f3\u987a\u5e8f\u5b58\u50a8\u5728\u6570\u7ec4\u4e2d\uff0c\u7136\u540e\u4f7f\u7528\u4e0e\u65e5\u5e38\u67e5\u7eb8\u8d28\u5b57\u5178\u76f8\u540c\u7684\u201c\u7ffb\u5f00\u4e2d\u95f4\uff0c\u6392\u9664\u4e00\u534a\u201d\u7684\u65b9\u5f0f\uff0c\u6765\u5b9e\u73b0\u4e00\u4e2a\u67e5\u7535\u5b50\u5b57\u5178\u7684\u7b97\u6cd5\u3002
     
    \u6df1\u5ea6\u5b66\u4e60\u3002\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002
    "},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2. \u00a0 \u94fe\u8868","text":"
    \u5f15\u8a00
     
    \u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u6392\u9664\u5df2\u5360\u7528\u7684\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u5f80\u5f80\u662f\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u7684\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u5185\u5b58\u7a7a\u95f4\u8fde\u7eed\uff0c\u5f53\u6211\u4eec\u9700\u8981\u7533\u8bf7\u4e00\u4e2a\u5f88\u5927\u7684\u6570\u7ec4\u65f6\uff0c\u7cfb\u7edf\u4e0d\u4e00\u5b9a\u5b58\u5728\u8fd9\u4e48\u5927\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u3002\u800c\u94fe\u8868\u5219\u66f4\u52a0\u7075\u6d3b\uff0c\u4e0d\u9700\u8981\u5185\u5b58\u662f\u8fde\u7eed\u7684\uff0c\u53ea\u8981\u5269\u4f59\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u591f\u7528\u5373\u53ef\u3002
     
    \u300c\u94fe\u8868 Linked List\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u5355\u72ec\u7684\u5bf9\u8c61\uff0c\u5404\u4e2a\u5143\u7d20\uff08\u4e00\u822c\u79f0\u4e3a\u7ed3\u70b9\uff09\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u8fde\u63a5\u3002\u7531\u4e8e\u7ed3\u70b9\u4e2d\u8bb0\u5f55\u4e86\u8fde\u63a5\u5173\u7cfb\uff0c\u56e0\u6b64\u94fe\u8868\u7684\u5b58\u50a8\u65b9\u5f0f\u76f8\u6bd4\u4e8e\u6570\u7ec4\u66f4\u52a0\u7075\u6d3b\uff0c\u7cfb\u7edf\u4e0d\u5fc5\u4fdd\u8bc1\u5185\u5b58\u5730\u5740\u7684\u8fde\u7eed\u6027\u3002
     
    \u94fe\u8868\u7684\u300c\u7ed3\u70b9 Node\u300d\u5305\u542b\u4e24\u9879\u6570\u636e\uff0c\u4e00\u662f\u7ed3\u70b9\u300c\u503c Value\u300d\uff0c\u4e8c\u662f\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u300c\u6307\u9488 Pointer\u300d\uff08\u6216\u79f0\u300c\u5f15\u7528 Reference\u300d\uff09\u3002
     
     
     Fig. \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u7ed3\u70b9\u503c\nListNode *next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
     
    \"\"\" \u94fe\u8868\u7ed3\u70b9\u7c7b \"\"\" \nclass ListNode:\ndef __init__(self, val: int):\nself.val: int = val                  # \u7ed3\u70b9\u503c\nself.next: Optional[ListNode] = None # \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\nVal  int       // \u7ed3\u70b9\u503c\nNext *ListNode // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\nreturn &ListNode{\nVal:  val,\nNext: nil,\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nconstructor(val, next) {\nthis.val = (val === undefined ? 0 : val);       // \u7ed3\u70b9\u503c\nthis.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nconstructor(val?: number, next?: ListNode | null) {\nthis.val = val === undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u7ed3\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\n// typedef \u4f5c\u7528\u662f\u4e3a\u4e00\u79cd\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\u4e00\u4e2a\u65b0\u540d\u5b57\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570\uff0c\u521d\u59cb\u5316\u4e00\u4e2a\u65b0\u7ed3\u70b9 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nreturn node;\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode\n{\nint val;         // \u7ed3\u70b9\u503c\nListNode next;   // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\nListNode(int x) => val = x;  //\u6784\u9020\u51fd\u6570\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
     
    // \u94fe\u8868\u7ed3\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u7ed3\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\n}\n};\n}\n
     
    \u5c3e\u7ed3\u70b9\u6307\u5411\u4ec0\u4e48\uff1f \u6211\u4eec\u4e00\u822c\u5c06\u94fe\u8868\u7684\u6700\u540e\u4e00\u4e2a\u7ed3\u70b9\u79f0\u4e3a\u300c\u5c3e\u7ed3\u70b9\u300d\uff0c\u5176\u6307\u5411\u7684\u662f\u300c\u7a7a\u300d\uff0c\u5728 Java / C++ / Python \u4e2d\u5206\u522b\u8bb0\u4e3a null / nullptr / None \u3002\u5728\u4e0d\u5f15\u8d77\u6b67\u4e49\u4e0b\uff0c\u672c\u4e66\u90fd\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u3002
     
    \u94fe\u8868\u521d\u59cb\u5316\u65b9\u6cd5\u3002\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u5f15\u7528\u6307\u5411\u5173\u7cfb\u3002\u5b8c\u6210\u540e\uff0c\u5373\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u9996\u4e2a\u7ed3\u70b9\uff08\u5373\u5934\u7ed3\u70b9\uff09\u51fa\u53d1\uff0c\u8bbf\u95ee\u5176\u4f59\u6240\u6709\u7684\u7ed3\u70b9\u3002
     
    Tip
     
    \u6211\u4eec\u901a\u5e38\u5c06\u5934\u7ed3\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u4f8b\u5982\u5934\u7ed3\u70b9 head \u548c\u94fe\u8868 head \u5b9e\u9645\u4e0a\u662f\u540c\u4e49\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.cpp
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
     linked_list.py
    \"\"\" \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 \"\"\"\n# \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
     linked_list.go
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
     linked_list.js
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.ts
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.c
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
     linked_list.cs
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.swift
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
     linked_list.zig
    // \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1. \u00a0 \u94fe\u8868\u4f18\u70b9","text":"
    \u5728\u94fe\u8868\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u7684\u64cd\u4f5c\u6548\u7387\u9ad8\u3002\u6bd4\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u5728\u94fe\u8868\u4e2d\u95f4\u7684\u4e24\u4e2a\u7ed3\u70b9 A , B \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u7ed3\u70b9 P \uff0c\u6211\u4eec\u53ea\u9700\u8981\u6539\u53d8\u4e24\u4e2a\u7ed3\u70b9\u6307\u9488\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u76f8\u6bd4\u6570\u7ec4\u7684\u63d2\u5165\u64cd\u4f5c\u9ad8\u6548\u5f88\u591a\u3002
     
     
     Fig. \u94fe\u8868\u63d2\u5165\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.cpp
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode* n0, ListNode* P) {\nListNode* n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
     linked_list.py
    def insert(n0: ListNode, P: ListNode) -> None:\n\"\"\" \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P \"\"\"\nn1 = n0.next\nP.next = n1\nn0.next = P\n
     linked_list.go
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\nn1 := n0.Next\nP.Next = n1\nn0.Next = P\n}\n
     linked_list.js
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunction insert(n0, P) {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.ts
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.c
    [class]{}-[func]{insertNode}\n
     linked_list.cs
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode n0, ListNode P)\n{\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.swift
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\nlet n1 = n0.next\nP.next = n1\nn0.next = P\n}\n
     linked_list.zig
    // \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\nvar n1 = n0.?.next;\nP.?.next = n1;\nn0.?.next = P;\n}\n
     
    \u5728\u94fe\u8868\u4e2d\u5220\u9664\u7ed3\u70b9\u4e5f\u5f88\u65b9\u4fbf\uff0c\u53ea\u9700\u8981\u6539\u53d8\u4e00\u4e2a\u7ed3\u70b9\u6307\u9488\u5373\u53ef\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u867d\u7136\u5728\u5b8c\u6210\u5220\u9664\u540e\u7ed3\u70b9 P \u4ecd\u7136\u6307\u5411 n1 \uff0c\u4f46\u5b9e\u9645\u4e0a P \u5df2\u7ecf\u4e0d\u5c5e\u4e8e\u6b64\u94fe\u8868\u4e86\uff0c\u56e0\u4e3a\u904d\u5386\u6b64\u94fe\u8868\u662f\u65e0\u6cd5\u8bbf\u95ee\u5230 P \u7684\u3002
     
     
     Fig. \u94fe\u8868\u5220\u9664\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.cpp
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode* n0) {\nif (n0->next == nullptr)\nreturn;\n// n0 -> P -> n1\nListNode* P = n0->next;\nListNode* n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\ndelete P;\n}\n
     linked_list.py
    def remove(n0: ListNode) -> None:\n\"\"\" \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 \"\"\"\nif not n0.next:\nreturn\n# n0 -> P -> n1\nP = n0.next\nn1 = P.next\nn0.next = n1\n
     linked_list.go
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc removeNode(n0 *ListNode) {\nif n0.Next == nil {\nreturn\n}\n// n0 -> P -> n1\nP := n0.Next\nn1 := P.Next\nn0.Next = n1\n}\n
     linked_list.js
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction remove(n0) {\nif (!n0.next)\nreturn;\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.ts
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction remove(n0: ListNode): void {\nif (!n0.next) {\nreturn;\n}\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.c
    [class]{}-[func]{removeNode}\n
     linked_list.cs
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode n0)\n{\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.swift
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc remove(n0: ListNode) {\nif n0.next == nil {\nreturn\n}\n// n0 -> P -> n1\nlet P = n0.next\nlet n1 = P?.next\nn0.next = n1\nP?.next = nil\n}\n
     linked_list.zig
    // \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\nif (n0.?.next == null) return;\n// n0 -> P -> n1\nvar P = n0.?.next;\nvar n1 = P.?.next;\nn0.?.next = n1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#422","title":"4.2.2. \u00a0 \u94fe\u8868\u7f3a\u70b9","text":"
    \u94fe\u8868\u8bbf\u95ee\u7ed3\u70b9\u6548\u7387\u4f4e\u3002\u4e0a\u8282\u63d0\u5230\uff0c\u6570\u7ec4\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u4efb\u610f\u5143\u7d20\uff0c\u4f46\u94fe\u8868\u65e0\u6cd5\u76f4\u63a5\u8bbf\u95ee\u4efb\u610f\u7ed3\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u8ba1\u7b97\u673a\u9700\u8981\u4ece\u5934\u7ed3\u70b9\u51fa\u53d1\uff0c\u4e00\u4e2a\u4e00\u4e2a\u5730\u5411\u540e\u904d\u5386\u5230\u76ee\u6807\u7ed3\u70b9\u3002\u4f8b\u5982\uff0c\u5018\u82e5\u60f3\u8981\u8bbf\u95ee\u94fe\u8868\u7d22\u5f15\u4e3a index \uff08\u5373\u7b2c index + 1 \u4e2a\uff09\u7684\u7ed3\u70b9\uff0c\u90a3\u4e48\u9700\u8981 index \u6b21\u8bbf\u95ee\u64cd\u4f5c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.cpp
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode* access(ListNode* head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == nullptr)\nreturn nullptr;\nhead = head->next;\n}\nreturn head;\n}\n
     linked_list.py
    def access(head: ListNode, index: int) -> Optional[ListNode]:\n\"\"\" \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 \"\"\"\nfor _ in range(index):\nif not head:\nreturn None\nhead = head.next\nreturn head\n
     linked_list.go
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\nfor i := 0; i < index; i++ {\nif head == nil {\nreturn nil\n}\nhead = head.Next\n}\nreturn head\n}\n
     linked_list.js
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunction access(head, index) {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.ts
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.c
    [class]{}-[func]{access}\n
     linked_list.cs
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode? access(ListNode head, int index)\n{\nfor (int i = 0; i < index; i++)\n{\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.swift
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\nvar head: ListNode? = head\nfor _ in 0 ..< index {\nif head == nil {\nreturn nil\n}\nhead = head?.next\n}\nreturn head\n}\n
     linked_list.zig
    // \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\nvar head = node;\nvar i: i32 = 0;\nwhile (i < index) : (i += 1) {\nhead = head.?.next;\nif (head == null) return null;\n}\nreturn head;\n}\n
     
    \u94fe\u8868\u7684\u5185\u5b58\u5360\u7528\u591a\u3002\u94fe\u8868\u4ee5\u7ed3\u70b9\u4e3a\u5355\u4f4d\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u9664\u4e86\u4fdd\u5b58\u503c\u5916\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u8fd9\u610f\u5473\u7740\u540c\u6837\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u9700\u8981\u5360\u7528\u66f4\u591a\u5185\u5b58\u7a7a\u95f4\u3002
    "},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3. \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u904d\u5386\u94fe\u8868\u67e5\u627e\u3002\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u94fe\u8868\u5185\u503c\u4e3a target \u7684\u7ed3\u70b9\uff0c\u8f93\u51fa\u7ed3\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.cpp
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode* head, int target) {\nint index = 0;\nwhile (head != nullptr) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.py
    def find(head: ListNode, target: int) -> int:\n\"\"\" \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 \"\"\"\nindex = 0\nwhile head:\nif head.val == target:\nreturn index\nhead = head.next\nindex += 1\nreturn -1\n
     linked_list.go
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc findNode(head *ListNode, target int) int {\nindex := 0\nfor head != nil {\nif head.Val == target {\nreturn index\n}\nhead = head.Next\nindex++\n}\nreturn -1\n}\n
     linked_list.js
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction find(head, target) {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
     linked_list.ts
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
     linked_list.c
    [class]{}-[func]{findNode}\n
     linked_list.cs
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode head, int target)\n{\nint index = 0;\nwhile (head != null)\n{\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.swift
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\nvar head: ListNode? = head\nvar index = 0\nwhile head != nil {\nif head?.val == target {\nreturn index\n}\nhead = head?.next\nindex += 1\n}\nreturn -1\n}\n
     linked_list.zig
    // \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\nvar head = node;\nvar index: i32 = 0;\nwhile (head != null) {\nif (head.?.val == target) return index;\nhead = head.?.next;\nindex += 1;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4. \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"
    \u5355\u5411\u94fe\u8868\u3002\u5373\u4e0a\u8ff0\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u7ed3\u70b9\u6709\u300c\u503c\u300d\u548c\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u300c\u6307\u9488\uff08\u5f15\u7528\uff09\u300d\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u7ed3\u70b9\u79f0\u4e3a\u5934\u7ed3\u70b9\uff0c\u5c3e\u7ed3\u70b9\u6307\u5411 null \u3002
     
    \u73af\u5f62\u94fe\u8868\u3002\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u7ed3\u70b9\u6307\u5411\u5934\u7ed3\u70b9\uff08\u5373\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u7ed3\u70b9\u770b\u4f5c\u662f\u5934\u7ed3\u70b9\u3002
     
    \u53cc\u5411\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u4ec5\u8bb0\u5f55\u4e86\u4e00\u4e2a\u65b9\u5411\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\uff0c\u5728\u53cc\u5411\u94fe\u8868\u7684\u7ed3\u70b9\u5b9a\u4e49\u4e2d\uff0c\u540c\u65f6\u6709\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\uff08\u540e\u7ee7\u7ed3\u70b9\uff09\u548c\u4e0a\u4e00\u7ed3\u70b9\uff08\u524d\u9a71\u7ed3\u70b9\uff09\u7684\u300c\u6307\u9488\uff08\u5f15\u7528\uff09\u300d\u3002\u53cc\u5411\u94fe\u8868\u76f8\u5bf9\u4e8e\u5355\u5411\u94fe\u8868\u66f4\u52a0\u7075\u6d3b\uff0c\u5373\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u7ed3\u70b9\u503c\nListNode *next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode *prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
     
    \"\"\" \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b \"\"\" \nclass ListNode:\ndef __init__(self, val: int):\nself.val: int = val                   # \u7ed3\u70b9\u503c\nself.next: Optional[ListNode] = None  # \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nself.prev: Optional[ListNode] = None  # \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\nVal  int             // \u7ed3\u70b9\u503c\nNext *DoublyListNode // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nPrev *DoublyListNode // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\nreturn &DoublyListNode{\nVal:  val,\nNext: nil,\nPrev: nil,\n}\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nprev;\nconstructor(val, next) {\nthis.val = val  ===  undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nprev: ListNode | null;\nconstructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\nthis.val = val  ===  undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
     
    \n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) => val = x;  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nvar prev: ListNode? // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
     
    // \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u7ed3\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nprev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n
     
     
     Fig. \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b 
    "},{"location":"chapter_array_and_linkedlist/list/","title":"4.3. \u00a0 \u5217\u8868","text":"
    \u7531\u4e8e\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u6570\u7ec4\u7684\u5b9e\u7528\u6027\u5927\u5927\u964d\u4f4e\u3002\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e8b\u5148\u5e76\u4e0d\u77e5\u9053\u4f1a\u8f93\u5165\u591a\u5c11\u6570\u636e\uff0c\u8fd9\u5c31\u4e3a\u6570\u7ec4\u957f\u5ea6\u7684\u9009\u62e9\u5e26\u6765\u4e86\u5f88\u5927\u56f0\u96be\u3002\u957f\u5ea6\u9009\u5c0f\u4e86\uff0c\u9700\u8981\u5728\u6dfb\u52a0\u6570\u636e\u4e2d\u9891\u7e41\u5730\u6269\u5bb9\u6570\u7ec4\uff1b\u957f\u5ea6\u9009\u5927\u4e86\uff0c\u53c8\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
     
    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u88ab\u79f0\u4e3a\u300c\u5217\u8868 List\u300d\u7684\u6570\u636e\u7ed3\u6784\u3002\u5217\u8868\u53ef\u4ee5\u88ab\u7406\u89e3\u4e3a\u957f\u5ea6\u53ef\u53d8\u7684\u6570\u7ec4\uff0c\u56e0\u6b64\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u7ec4 Dynamic Array\u300d\u3002\u5217\u8868\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\uff0c\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u4f18\u70b9\uff0c\u540c\u65f6\u8fd8\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u5b9e\u65f6\u6269\u5bb9\u3002\u5728\u5217\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u4e0d\u7528\u62c5\u5fc3\u8d85\u8fc7\u5bb9\u91cf\u9650\u5236\u3002
    "},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1. \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u521d\u59cb\u5316\u5217\u8868\u3002\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u5230\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> list1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> list = new ArrayList<>(Arrays.asList(numbers));\n
     list.cpp
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 list\n// \u65e0\u521d\u59cb\u503c\nvector<int> list1;\n// \u6709\u521d\u59cb\u503c\nvector<int> list = { 1, 3, 2, 5, 4 };\n
     list.py
    \"\"\" \u521d\u59cb\u5316\u5217\u8868 \"\"\"\n# \u65e0\u521d\u59cb\u503c\nlist1: List[int] = []\n# \u6709\u521d\u59cb\u503c\nlist: List[int] = [1, 3, 2, 5, 4]\n
     list_test.go
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlist1 := []int\n// \u6709\u521d\u59cb\u503c\nlist := []int{1, 3, 2, 5, 4}\n
     list.js
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1 = [];\n// \u6709\u521d\u59cb\u503c\nconst list = [1, 3, 2, 5, 4];\n
     list.ts
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst list: number[] = [1, 3, 2, 5, 4];\n
     list.c
    \n
     list.cs
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = new ();\n// \u6709\u521d\u59cb\u503c\nint[] numbers = new int[] { 1, 3, 2, 5, 4 };\nList<int> list = numbers.ToList();\n
     list.swift
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet list1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar list = [1, 3, 2, 5, 4]\n
     list.zig
    // \u521d\u59cb\u5316\u5217\u8868\nvar list = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list.deinit();\ntry list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n
     
    \u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\u3002\u5217\u8868\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.cpp
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.py
    \"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\nnum: int = list[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\"\"\" \u66f4\u65b0\u5143\u7d20 \"\"\"\nlist[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list_test.go
    /* \u8bbf\u95ee\u5143\u7d20 */\nnum := list[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.js
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.ts
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.c
    \n
     list.cs
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.swift
    /* \u8bbf\u95ee\u5143\u7d20 */\nlet num = list[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.zig
    // \u8bbf\u95ee\u5143\u7d20\nvar num = list.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n// \u66f4\u65b0\u5143\u7d20\nlist.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n
     
    \u5728\u5217\u8868\u4e2d\u6dfb\u52a0\u3001\u63d2\u5165\u3001\u5220\u9664\u5143\u7d20\u3002\u76f8\u5bf9\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u662f\u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u4e00\u6837\u4f4e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.cpp
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push_back(1);\nlist.push_back(3);\nlist.push_back(2);\nlist.push_back(5);\nlist.push_back(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(list.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.erase(list.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.py
    \"\"\" \u6e05\u7a7a\u5217\u8868 \"\"\"\nlist.clear()\n\"\"\" \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 \"\"\"\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n\"\"\" \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 \"\"\"\nlist.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\"\"\" \u5220\u9664\u5143\u7d20 \"\"\"\nlist.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list_test.go
    /* \u6e05\u7a7a\u5217\u8868 */\nlist = nil\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist = append(list, 1)\nlist = append(list, 3)\nlist = append(list, 2)\nlist = append(list, 5)\nlist = append(list, 4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist = append(list[:3], append([]int{6}, list[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist = append(list[:3], list[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.js
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
     list.ts
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
     list.c
    \n
     list.cs
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.Clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.Add(1);\nlist.Add(3);\nlist.Add(2);\nlist.Add(5);\nlist.Add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.Insert(3, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.RemoveAt(3);\n
     list.swift
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.removeAll()\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.zig
    // \u6e05\u7a7a\u5217\u8868\nlist.clearRetainingCapacity();\n// \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry list.append(1);\ntry list.append(3);\ntry list.append(2);\ntry list.append(5);\ntry list.append(4);\n// \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry list.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n// \u5220\u9664\u5143\u7d20\n_ = list.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     
    \u904d\u5386\u5217\u8868\u3002\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u4f7f\u7528\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528 for-each \u76f4\u63a5\u904d\u5386\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
     list.cpp
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
     list.py
    \"\"\" \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 \"\"\"\ncount: int = 0\nfor i in range(len(list)):\ncount += 1\n\"\"\" \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 \"\"\"\ncount: int = 0\nfor n in list:\ncount += 1\n
     list_test.go
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(list); i++ {\ncount++\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor range list {\ncount++\n}\n
     list.js
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
     list.ts
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
     list.c
    \n
     list.cs
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.Count(); i++)\n{\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int n in list)\n{\ncount++;\n}\n
     list.swift
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor _ in list.indices {\ncount += 1\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _ in list {\ncount += 1\n}\n
     list.zig
    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < list.items.len) : (i += 1) {\ncount += 1;\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (list.items) |_| {\ncount += 1;\n}\n
     
    \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\u3002\u518d\u521b\u5efa\u4e00\u4e2a\u65b0\u5217\u8868 list1 \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u4e2d\u4e00\u4e2a\u5217\u8868\u62fc\u63a5\u5230\u53e6\u4e00\u4e2a\u7684\u5c3e\u90e8\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.cpp
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> list1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\nlist.insert(list.end(), list1.begin(), list1.end());\n
     list.py
    \"\"\" \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 \"\"\"\nlist1: List[int] = [6, 8, 7, 10, 9]\nlist += list1  # \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list_test.go
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlist1 := []int{6, 8, 7, 10, 9}\nlist = append(list, list1...)  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.js
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1 = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.ts
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1: number[] = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.c
    \n
     list.cs
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = new() { 6, 8, 7, 10, 9 };\nlist.AddRange(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.swift
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet list1 = [6, 8, 7, 10, 9]\nlist.append(contentsOf: list1) // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.zig
    // \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar list1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list1.deinit();\ntry list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry list.insertSlice(list.items.len, list1.items); // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     
    \u6392\u5e8f\u5217\u8868\u3002\u6392\u5e8f\u4e5f\u662f\u5e38\u7528\u7684\u65b9\u6cd5\u4e4b\u4e00\uff0c\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u5bdf\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u53cc\u6307\u9488\u300d\u7b97\u6cd5\u4e86\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(list);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.cpp
    /* \u6392\u5e8f\u5217\u8868 */\nsort(list.begin(), list.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.py
    \"\"\" \u6392\u5e8f\u5217\u8868 \"\"\"\nlist.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list_test.go
    /* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(list)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.js
    /* \u6392\u5e8f\u5217\u8868 */  list.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.ts
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.c
    \n
     list.cs
    /* \u6392\u5e8f\u5217\u8868 */\nlist.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.swift
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.zig
    // \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32));\n
    "},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2. \u00a0 \u5217\u8868\u7b80\u6613\u5b9e\u73b0 *","text":"
    \u4e3a\u4e86\u5e2e\u52a9\u52a0\u6df1\u5bf9\u5217\u8868\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5728\u6b64\u63d0\u4f9b\u4e00\u4e2a\u5217\u8868\u7684\u7b80\u6613\u7248\u672c\u7684\u5b9e\u73b0\u3002\u9700\u8981\u5173\u6ce8\u4e09\u4e2a\u6838\u5fc3\u70b9\uff1a
     
       
    • \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u7684\u521d\u59cb\u5bb9\u91cf initialCapacity \u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
      •  
    • \u6570\u91cf\u8bb0\u5f55\uff1a\u9700\u8981\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf size \uff0c\u7528\u6765\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u6709\u591a\u5c11\u4e2a\u5143\u7d20\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u4e0e\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u7684\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002
      •  
    • \u6269\u5bb9\u673a\u5236\uff1a\u63d2\u5165\u5143\u7d20\u6709\u53ef\u80fd\u5bfc\u81f4\u8d85\u51fa\u5217\u8868\u5bb9\u91cf\uff0c\u6b64\u65f6\u9700\u8981\u6269\u5bb9\u5217\u8868\uff0c\u65b9\u6cd5\u662f\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\u6765\u66ff\u6362\u5f53\u524d\u6570\u7ec4\u3002\u9700\u8981\u7ed9\u5b9a\u4e00\u4e2a\u6269\u5bb9\u500d\u6570 extendRatio \uff0c\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002
      •  
     
    \u672c\u793a\u4f8b\u662f\u4e3a\u4e86\u5e2e\u52a9\u8bfb\u8005\u5bf9\u5982\u4f55\u5b9e\u73b0\u5217\u8868\u4ea7\u751f\u76f4\u89c2\u7684\u8ba4\u8bc6\u3002\u5b9e\u9645\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u5217\u8868\u7684\u5b9e\u73b0\u8fdc\u6bd4\u4ee5\u4e0b\u4ee3\u7801\u590d\u6742\u4e14\u6807\u51c6\uff0c\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u5b66\u4e60\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_list.java
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int capacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int size = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[capacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size() {\nreturn size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\nnums[size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = Arrays.copyOf(nums, capacity() * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\ncapacity = nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\nint size = size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[size];\nfor (int i = 0; i < size; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
     my_list.cpp
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate:\nint* nums;                // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nint numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint extendRatio = 2;      // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\nnums = new int[numsCapacity];\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~MyList() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\nnums[size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size() - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size() - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nint newCapacity = capacity() * extendRatio;\nint* tmp = nums;\nnums = new int[newCapacity];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size(); i++) {\nnums[i] = tmp[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] tmp;\nnumsCapacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> vec(size());\nfor (int i = 0; i < size(); i++) {\nvec[i] = nums[i];\n}\nreturn vec;\n}\n};\n
     my_list.py
    class MyList:\n\"\"\" \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__capacity: int = 10                       # \u5217\u8868\u5bb9\u91cf\nself.__nums: List[int] = [0] * self.__capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nself.__size: int = 0                            # \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nself.__extend_ratio: int = 2                    # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 \"\"\"\nreturn self.__size\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u5217\u8868\u5bb9\u91cf \"\"\"\nreturn self.__capacity\ndef get(self, index: int) -> int:\n\"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\n# \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nreturn self.__nums[index]\ndef set(self, num: int, index: int) -> None:\n\"\"\" \u66f4\u65b0\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nself.__nums[index] = num\ndef add(self, num: int) -> None:\n\"\"\" \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 \"\"\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size() == self.capacity():\nself.extend_capacity()\nself.__nums[self.__size] = num\nself.__size += 1\ndef insert(self, num: int, index: int) -> None:\n\"\"\" \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.__size == self.capacity():\nself.extend_capacity()\n# \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in range(self.__size - 1, index - 1, -1):\nself.__nums[j + 1] = self.__nums[j]\nself.__nums[index] = num\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size += 1\ndef remove(self, index: int) -> int:\n\"\"\" \u5220\u9664\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nnum = self.__nums[index]\n# \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in range(index, self.__size - 1):\nself.__nums[j] = self.__nums[j + 1]\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size -= 1\n# \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\ndef extend_capacity(self) -> None:\n\"\"\" \u5217\u8868\u6269\u5bb9 \"\"\"\n# \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a self.__size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nself.__nums = self.__nums + [0] * self.capacity() * (self.__extend_ratio - 1)\n# \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.__capacity = len(self.__nums)\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 \"\"\"\nreturn self.__nums[:self.__size]\n
     my_list.go
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\ntype myList struct {\nnumsCapacity int\nnums         []int\nnumsSize     int\nextendRatio  int\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newMyList() *myList {\nreturn &myList{\nnumsCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\nnums:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio:  2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\nreturn l.numsSize\n}\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\nreturn l.numsCapacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn l.nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nl.nums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\nl.nums[l.numsSize] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j := l.numsSize - 1; j >= index; j-- {\nl.nums[j+1] = l.nums[j]\n}\nl.nums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnum := l.nums[index]\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j := index; j < l.numsSize-1; j++ {\nl.nums[j] = l.nums[j+1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize--\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a self.__size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nl.nums = append(l.nums, make([]int, l.numsCapacity*(l.extendRatio-1))...)\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nl.numsCapacity = len(l.nums)\n}\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nreturn l.nums[:l.numsSize]\n}\n
     my_list.js
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\n#nums = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n#capacity = 10; // \u5217\u8868\u5bb9\u91cf\n#size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n#extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#nums = new Array(this.#capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nsize() {\nreturn this.#size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\ncapacity() {\nreturn this.#capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nget(index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.#nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nset(index, num) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.#nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nadd(num) {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.#nums[this.#size] = num;\nthis.#size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\ninsert(index, num) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this.#size - 1; j >= index; j--) {\nthis.#nums[j + 1] = this.#nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#nums[index] = num;\nthis.#size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nremove(index) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.#nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this.#size - 1; j++) {\nthis.#nums[j] = this.#nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nextendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.#nums = this.#nums.concat(\nnew Array(this.capacity() * (this.#extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis.#capacity = this.#nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\ntoArray() {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
     my_list.ts
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate nums: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\nprivate _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.nums = new Array(this._capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic size(): number {\nreturn this._size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic capacity(): number {\nreturn this._capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic get(index: number): number {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic set(index: number, num: number): void {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic add(num: number): void {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this._size === this._capacity)\nthis.extendCapacity();\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.nums[this._size] = num;\nthis._size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic insert(index: number, num: number): void {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this._size === this._capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this._size - 1; j >= index; j--) {\nthis.nums[j + 1] = this.nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.nums[index] = num;\nthis._size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic remove(index: number): number {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this._size - 1; j++) {\nthis.nums[j] = this.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis._size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic extendCapacity(): void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.nums = this.nums.concat(\nnew Array(this.capacity() * (this.extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis._capacity = this.nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic toArray(): number[] {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
     my_list.c
    [class]{myList}-[func]{}\n
     my_list.cs
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList\n{\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList()\n{\nnums = new int[numsCapacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size()\n{\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity()\n{\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index)\n{\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num)\n{\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\nnums[numsSize] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = numsSize - 1; j >= index; j--)\n{\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < numsSize - 1; j++)\n{\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity()\n{\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a numsCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nSystem.Array.Resize(ref nums, numsCapacity * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nnumsCapacity = nums.Length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray()\n{\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[numsSize];\nfor (int i = 0; i < numsSize; i++)\n{\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
     my_list.swift
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate var nums: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\nprivate var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nnums = Array(repeating: 0, count: _capacity)\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nfunc size() -> Int {\n_size\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc capacity() -> Int {\n_capacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc get(index: Int) -> Int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc set(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc add(num: Int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\nnums[_size] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc insert(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\nnums[j + 1] = nums[j]\n}\nnums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5220\u9664\u5143\u7d20 */\n@discardableResult\nfunc remove(index: Int) -> Int {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nlet num = nums[index]\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in index ..< (_size - 1) {\nnums[j] = nums[j + 1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size -= 1\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = nums + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = nums.count\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nfunc toArray() -> [Int] {\nvar nums = Array(repeating: 0, count: _size)\nfor i in 0 ..< _size {\nnums[i] = get(index: i)\n}\nreturn nums\n}\n}\n
     my_list.zig
    // \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\nfn MyList(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnums_capacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\nnum_size: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextend_ratio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.nums = try self.mem_allocator.alloc(T, self.nums_capacity);\nstd.mem.set(T, self.nums, @as(T, 0));\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\npub fn size(self: *Self) usize {\nreturn self.num_size;\n}\n// \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.nums_capacity;\n}\n// \u8bbf\u95ee\u5143\u7d20\npub fn get(self: *Self, index: usize) T {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn self.nums[index];\n}  // \u66f4\u65b0\u5143\u7d20\npub fn set(self: *Self, index: usize, num: T) void {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nself.nums[index] = num;\n}  // \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\npub fn add(self: *Self, num: T) !void {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\nself.nums[self.size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size += 1;\n}  // \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\npub fn insert(self: *Self, index: usize, num: T) !void {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar j = self.size() - 1;\nwhile (j >= index) : (j -= 1) {\nself.nums[j + 1] = self.nums[j];\n}\nself.nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size += 1;\n}\n// \u5220\u9664\u5143\u7d20\npub fn remove(self: *Self, index: usize) T {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nvar num = self.nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar j = index;\nwhile (j < self.size() - 1) : (j += 1) {\nself.nums[j] = self.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n// \u5217\u8868\u6269\u5bb9\npub fn extendCapacity(self: *Self) !void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extend_ratio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nvar newCapacity = self.capacity() * self.extend_ratio;\nvar extend = try self.mem_allocator.alloc(T, newCapacity);\nstd.mem.set(T, extend, @as(T, 0));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(T, extend, self.nums);\nself.nums = extend;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.nums_capacity = newCapacity;\n}\n// \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar nums = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, nums, @as(T, 0));\nfor (nums) |*num, i| {\nnum.* = self.get(i);\n}\nreturn nums;\n}\n};\n}\n
    "},{"location":"chapter_array_and_linkedlist/summary/","title":"4.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff0c\u5373\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u4f18\u70b9\u4e0e\u7f3a\u70b9\u5448\u73b0\u51fa\u6b64\u6d88\u5f7c\u957f\u7684\u5173\u7cfb\u3002
      •  
    • \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5185\u5b58\u7a7a\u95f4\u5360\u7528\u5c0f\uff1b\u4f46\u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
      •  
    • \u94fe\u8868\u53ef\u901a\u8fc7\u66f4\u6539\u6307\u9488\u5b9e\u73b0\u9ad8\u6548\u7684\u7ed3\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u5730\u4fee\u6539\u957f\u5ea6\uff1b\u4f46\u7ed3\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u6709\u5355\u5411\u94fe\u8868\u3001\u5faa\u73af\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002
      •  
    • \u5217\u8868\u53c8\u79f0\u52a8\u6001\u6570\u7ec4\uff0c\u662f\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4fdd\u5b58\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u6539\u53d8\u957f\u5ea6\u3002\u5217\u8868\u7684\u51fa\u73b0\u5927\u5927\u63d0\u5347\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
      •  
    • \u4e0b\u8868\u603b\u7ed3\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u9879\u7279\u6027\u3002
      •  
     \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u79bb\u6563\u5185\u5b58\u7a7a\u95f4 \u6570\u636e\u7ed3\u6784\u957f\u5ea6 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u957f\u5ea6\u53ef\u53d8 \u5185\u5b58\u4f7f\u7528\u7387 \u5360\u7528\u5185\u5b58\u5c11\u3001\u7f13\u5b58\u5c40\u90e8\u6027\u597d \u5360\u7528\u5185\u5b58\u591a \u4f18\u52bf\u64cd\u4f5c \u968f\u673a\u8bbf\u95ee \u63d2\u5165\u3001\u5220\u9664 
    \u7f13\u5b58\u5c40\u90e8\u6027\u7684\u7b80\u5355\u89e3\u91ca
     
    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6570\u636e\u8bfb\u5199\u901f\u5ea6\u6392\u5e8f\u662f\u201c\u786c\u76d8 < \u5185\u5b58 < CPU \u7f13\u5b58\u201d\u3002\u5f53\u6211\u4eec\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u5b83\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u8ba1\u7b97\u673a\u53ea\u80fd\u6328\u4e2a\u5730\u7f13\u5b58\u5404\u4e2a\u7ed3\u70b9\uff0c\u8fd9\u6837\u7684\u591a\u6b21\u201c\u642c\u8fd0\u201d\u964d\u4f4e\u4e86\u6574\u4f53\u6548\u7387\u3002
     
       
    • \u4e0b\u8868\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u79cd\u64cd\u4f5c\u6548\u7387\u3002
      •  
     \u64cd\u4f5c \u6570\u7ec4 \u94fe\u8868 \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(N)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(N)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(N)\\) \\(O(1)\\)"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1. \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1. \u00a0 \u7b97\u6cd5\u8bc4\u4ef7\u7ef4\u5ea6","text":"
    \u5728\u5f00\u59cb\u5b66\u4e60\u7b97\u6cd5\u4e4b\u524d\uff0c\u6211\u4eec\u9996\u5148\u8981\u60f3\u6e05\u695a\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4ec0\u4e48\uff0c\u6216\u8005\u8bf4\uff0c\u5982\u4f55\u6765\u8bc4\u5224\u7b97\u6cd5\u7684\u597d\u4e0e\u574f\u3002\u6574\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u8bbe\u8ba1\u7b97\u6cd5\u65f6\u8ffd\u6c42\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002
     
       
    1. \u627e\u5230\u95ee\u9898\u89e3\u6cd5\u3002\u7b97\u6cd5\u9700\u8981\u80fd\u591f\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u4e0b\uff0c\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
      2.  
    2. \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\u3002\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u800c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u6548\u7387\u5c3d\u53ef\u80fd\u7684\u9ad8\u3002
      3.  
     
    \u6362\u8a00\u4e4b\uff0c\u5728\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5219\u662f\u4e3b\u8981\u8bc4\u4ef7\u7ef4\u5ea6\uff0c\u5305\u62ec\uff1a
     
       
    • \u65f6\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u7684\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
      •  
    • \u7a7a\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u3002
      •  
     
    \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8ffd\u6c42\u201c\u8fd0\u884c\u901f\u5ea6\u5feb\u3001\u5360\u7528\u5185\u5b58\u5c11\u201d\uff0c\u800c\u5982\u4f55\u53bb\u8bc4\u4ef7\u7b97\u6cd5\u6548\u7387\u5219\u662f\u975e\u5e38\u91cd\u8981\u7684\u95ee\u9898\uff0c\u56e0\u4e3a\u53ea\u6709\u77e5\u9053\u5982\u4f55\u8bc4\u4ef7\u7b97\u6cd5\uff0c\u624d\u80fd\u53bb\u505a\u7b97\u6cd5\u4e4b\u95f4\u7684\u5bf9\u6bd4\u5206\u6790\uff0c\u4ee5\u53ca\u4f18\u5316\u7b97\u6cd5\u8bbe\u8ba1\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2. \u00a0 \u6548\u7387\u8bc4\u4f30\u65b9\u6cd5","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#_1","title":"\u5b9e\u9645\u6d4b\u8bd5","text":"
    \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 A \u548c \u7b97\u6cd5 B \uff0c\u90fd\u80fd\u591f\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u4e24\u4e2a\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002\u6211\u4eec\u80fd\u591f\u60f3\u5230\u7684\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\uff0c\u5c31\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u628a\u4e24\u4e2a\u7b97\u6cd5\u90fd\u5b8c\u6574\u8dd1\u4e00\u904d\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u662f\u4e5f\u5b58\u5728\u5f88\u5927\u7684\u786c\u4f24\u3002
     
    \u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u5230\u7b97\u6cd5\u7684\u6027\u80fd\u8868\u73b0\u3002\u4f8b\u5982\uff0c\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 A \u6bd4\u7b97\u6cd5 B \u8fd0\u884c\u65f6\u95f4\u66f4\u77ed\uff1b\u4f46\u6362\u5230\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u4f1a\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u5c55\u5f00\u6d4b\u8bd5\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002
     
    \u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u5927\u5c0f\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u5448\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u8868\u73b0\u3002\u6bd4\u5982\uff0c\u6709\u53ef\u80fd\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 A \u8fd0\u884c\u65f6\u95f4\u77ed\u4e8e\u7b97\u6cd5 B \uff0c\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u622a\u7136\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u82e5\u60f3\u8981\u8fbe\u5230\u5177\u6709\u8bf4\u670d\u529b\u7684\u5bf9\u6bd4\u7ed3\u679c\uff0c\u90a3\u4e48\u9700\u8981\u8f93\u5165\u5404\u79cd\u4f53\u91cf\u6570\u636e\uff0c\u8fd9\u6837\u7684\u6d4b\u8bd5\u9700\u8981\u5360\u7528\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#_2","title":"\u7406\u8bba\u4f30\u7b97","text":"
    \u65e2\u7136\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u5f88\u5927\u7684\u5c40\u9650\u6027\uff0c\u90a3\u4e48\u6211\u4eec\u662f\u5426\u53ef\u4ee5\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\uff0c\u5c31\u83b7\u77e5\u7b97\u6cd5\u7684\u6548\u7387\u6c34\u5e73\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\uff0c\u6211\u4eec\u5c06\u6b64\u4f30\u7b97\u65b9\u6cd5\u79f0\u4e3a\u300c\u590d\u6742\u5ea6\u5206\u6790 Complexity Analysis\u300d\u6216\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 Asymptotic Complexity Analysis\u300d\u3002
     
    \u590d\u6742\u5ea6\u5206\u6790\u8bc4\u4f30\u7684\u662f\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u53e5\u8bdd\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\uff1a
     
       
    • \u201c\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u201d\u53ef\u5206\u4e3a\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u548c\u201c\u5360\u7528\u7a7a\u95f4\u201d\uff0c\u8fdb\u800c\u53ef\u5c06\u590d\u6742\u5ea6\u5206\u4e3a\u300c\u65f6\u95f4\u590d\u6742\u5ea6 Time Complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u3002
      •  
    • \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u201d\u4ee3\u8868\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u6709\u5173\uff0c\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff1b
      •  
    • \u201c\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u4e0d\u5173\u5fc3\u7b97\u6cd5\u5177\u4f53\u4f7f\u7528\u4e86\u591a\u5c11\u65f6\u95f4\u6216\u5360\u7528\u4e86\u591a\u5c11\u7a7a\u95f4\uff0c\u800c\u662f\u7ed9\u51fa\u4e00\u79cd\u201c\u8d8b\u52bf\u6027\u5206\u6790\u201d\uff1b
      •  
     
    \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\u3002\u4e00\u662f\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002\u4e8c\u662f\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u53ef\u4ee5\u53cd\u6620\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002
     
    \u5982\u679c\u611f\u89c9\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u7684\u6982\u5ff5\u4e00\u77e5\u534a\u89e3\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u540e\u7eed\u7ae0\u8282\u4f1a\u5c55\u5f00\u4ecb\u7ecd\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#213","title":"2.1.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790\u91cd\u8981\u6027","text":"
    \u590d\u6742\u5ea6\u5206\u6790\u7ed9\u51fa\u4e00\u628a\u8bc4\u4ef7\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u544a\u8bc9\u6211\u4eec\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u9700\u8981\u591a\u5c11\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u4e5f\u8ba9\u6211\u4eec\u53ef\u4ee5\u5f00\u5c55\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u5bf9\u6bd4\u3002
     
    \u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u51fa\u53d1\uff0c\u5176\u5e76\u4e0d\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u7ae0\u5185\u5bb9\u3002\u4f46\u662f\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u8005\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u9700\u8981\u5206\u6790\u5b83\u7684\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002\u56e0\u6b64\uff0c\u5728\u5c55\u5f00\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5efa\u8bae\u8bfb\u8005\u5148\u5bf9\u590d\u6742\u5ea6\u5efa\u7acb\u8d77\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u5e76\u4e14\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u6848\u4f8b\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/","title":"2.3. \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"
    \u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u7edf\u8ba1 \u7b97\u6cd5\u4f7f\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5f88\u7c7b\u4f3c\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/#231","title":"2.3.1. \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"
    \u7b97\u6cd5\u8fd0\u884c\u4e2d\uff0c\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u6709\u4ee5\u4e0b\u51e0\u79cd\uff1a
     
       
    • \u300c\u8f93\u5165\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\uff1b
      •  
    • \u300c\u6682\u5b58\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\uff1b
      •  
    • \u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\uff1b
      •  
     
    Tip
     
    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u8303\u56f4\u662f\u300c\u6682\u5b58\u7a7a\u95f4\u300d+\u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u3002
     
    \u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a
     
       
    • \u300c\u6682\u5b58\u6570\u636e\u300d\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5404\u79cd \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61 \u7b49\u3002
      •  
    • \u300c\u6808\u5e27\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u90fd\u4f1a\u5728\u6808\u7684\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u65f6\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
      •  
    • \u300c\u6307\u4ee4\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u4e00\u822c\u5ffd\u7565\u4e0d\u8ba1\u3002
      •  
     
     
     Fig. \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nfinal int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7ed3\u6784\u4f53 */\nstruct Node {\nint val;\nNode *next;\nNode(int x) : val(x), next(nullptr) {}\n};\n/* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    \"\"\" \u7c7b \"\"\"\nclass Node:\ndef __init__(self, x: int):\nself.val: int = x                 # \u7ed3\u70b9\u503c\nself.next: Optional[Node] = None  # \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n\"\"\" \u51fd\u6570 \"\"\"\ndef function() -> int:\n# do something...\nreturn 0\ndef algorithm(n) -> int:  # \u8f93\u5165\u6570\u636e\nA: int = 0            # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\nb: int = 0            # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnode = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc: int = function()   # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn A + b + c      # \u8f93\u51fa\u6570\u636e\n
     
    /* \u7ed3\u6784\u4f53 */\ntype node struct {\nval  int\nnext *node\n}\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\nreturn &node{val: val}\n}\n/* \u51fd\u6570 */\nfunc function() int {\n// do something...\nreturn 0\n}\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\nconst a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nb := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnewNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nval;\nnext;\nconstructor(val) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc() {\n// do something\nreturn 0;\n}\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\nconst a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nval: number;\nnext: Node | null;\nconstructor(val?: number) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n// do something\nreturn 0;\n}\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\nconst a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n
     
    \n
     
    /* \u7c7b */\nclass Node\n{\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function()\n{\n// do something...\nreturn 0;\n}\nint algorithm(int n)          // \u8f93\u5165\u6570\u636e\n{\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nvar val: Int\nvar next: Node?\ninit(x: Int) {\nval = x\n}\n}\n/* \u51fd\u6570 */\nfunc function() -> Int {\n// do something...\nreturn 0\n}\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\nlet a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nvar b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/space_complexity/#232","title":"2.3.2. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"
    \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u548c\u65f6\u95f4\u590d\u6742\u5ea6\u603b\u4f53\u7c7b\u4f3c\uff0c\u53ea\u662f\u4ece\u7edf\u8ba1\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u201d\u53d8\u4e3a\u7edf\u8ba1\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u4e00\u822c\u53ea\u5173\u6ce8\u300c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u300d\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u4e2a\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u4fdd\u8bc1\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002
     
    \u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\uff0c\u5206\u522b\u4e3a\u8f93\u5165\u6570\u636e\u7684\u6700\u5dee\u5206\u5e03\u3001\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u6700\u5dee\u65f6\u95f4\u70b9\u3002
     
       
    • \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\u3002\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u662f\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 nums \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
      •  
    • \u4ee5\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\u3002\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 nums \u65f6\uff0c\u7a0b\u5e8f\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\nint[] nums = new int[n]; // O(n)\n}\n
     
    void algorithm(int n) {\nint a = 0;               // O(1)\nvector<int> b(10000);    // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
     
    def algorithm(n: int) -> None:\na: int = 0                     # O(1)\nb: List[int] = [0] * 10000     # O(1)\nif n > 10:\nnums: List[int] = [0] * n  # O(n)\n
     
    func algorithm(n int) {\na := 0                      // O(1)\nb := make([]int, 10000)     // O(1)\nvar nums []int\nif n > 10 {\nnums := make([]int, n)  // O(n)\n}\nfmt.Println(a, b, nums)\n}\n
     
    function algorithm(n) {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
     
    function algorithm(n: number): void {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
     
    \n
     
    void algorithm(int n)\n{\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\n{\nint[] nums = new int[n]; // O(n)\n}\n}\n
     
    func algorithm(n: Int) {\nlet a = 0 // O(1)\nlet b = Array(repeating: 0, count: 10000) // O(1)\nif n > 10 {\nlet nums = Array(repeating: 0, count: n) // O(n)\n}\n}\n
     
    \n
     
    \u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u4f8b\u5982\u51fd\u6570 loop()\uff0c\u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 function() \uff0c\u6bcf\u8f6e\u4e2d\u7684 function() \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002\u800c\u9012\u5f52\u51fd\u6570 recur() \u5728\u8fd0\u884c\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 recur() \uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
     
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
     
    def function() -> int:\n# do something\nreturn 0\n\"\"\" \u5faa\u73af O(1) \"\"\"\ndef loop(n: int) -> None:\nfor _ in range(n):\nfunction()\n\"\"\" \u9012\u5f52 O(n) \"\"\"\ndef recur(n: int) -> int:\nif n == 1: return\nreturn recur(n - 1)\n
     
    func function() int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\nfor i := 0; i < n; i++ {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\nif n == 1 {\nreturn\n}\nrecur(n - 1)\n}\n
     
    function constFunc() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
     
    function constFunc(): number {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
     
    \n
     
    int function()\n{\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nint recur(int n)\n{\nif (n == 1) return 1;\nreturn recur(n - 1);\n}\n
     
    @discardableResult\nfunc function() -> Int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\nfor _ in 0 ..< n {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\nif n == 1 {\nreturn\n}\nrecur(n: n - 1)\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/space_complexity/#233","title":"2.3.3. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09
     \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\] 
     
     Fig. \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b 
     
    Tip
     
    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u5148\u805a\u7126\u5728\u7406\u89e3\u7a7a\u95f4\u590d\u6742\u5ea6\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u4e0a\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"
    \u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002
     
    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u5373\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
     space_complexity.cpp
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nvector<int> nums(10000);\nListNode node(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
     space_complexity.py
    def constant(n: int) -> None:\n\"\"\" \u5e38\u6570\u9636 \"\"\"\n# \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\na: int = 0\nnums: List[int] = [0] * 10000\nnode = ListNode(0)\n# \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nc: int = 0\n# \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nfunction()\n
     space_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0\nb := 0\nnums := make([]int, 10000)\nListNode := newNode(0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nvar c int\nfor i := 0; i < n; i++ {\nc = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i := 0; i < n; i++ {\nfunction()\n}\nfmt.Println(a, b, nums, c, ListNode)\n}\n
     space_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
     space_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceConstant}\n
     space_complexity.cs
    /* \u5e38\u6570\u9636 */\nvoid constant(int n)\n{\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nint a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++)\n{\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n
     space_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nlet a = 0\nvar b = 0\nlet nums = Array(repeating: 0, count: 10000)\nlet node = ListNode(x: 0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nlet c = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nfunction()\n}\n}\n
     space_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a: i32 = 0;\nvar b: i32 = 0;\nvar nums = [_]i32{0}**10000;\nvar node = inc.ListNode(i32){.val = 0};\nvar i: i32 = 0;\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nwhile (i < n) : (i += 1) {\nvar c: i32 = 0;\n_ = c;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\ni = 0;\nwhile (i < n) : (i += 1) {\n_ = function();\n}\n_ = a;\n_ = b;\n_ = nums;\n_ = node;\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"
    \u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nnodes.add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<Integer, String> map = new HashMap<>();\nfor (int i = 0; i < n; i++) {\nmap.put(i, String.valueOf(i));\n}\n}\n
     space_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvector<int> nums(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvector<ListNode> nodes;\nfor (int i = 0; i < n; i++) {\nnodes.push_back(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nunordered_map<int, string> map;\nfor (int i = 0; i < n; i++) {\nmap[i] = to_string(i);\n}\n}\n
     space_complexity.py
    def linear(n: int) -> None:\n\"\"\" \u7ebf\u6027\u9636 \"\"\"\n# \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nnums: List[int] = [0] * n\n# \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nmapp: Dict = {}\nfor i in range(n):\nmapp[i] = str(i)\n
     space_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n_ = make([]int, n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes []*node\nfor i := 0; i < n; i++ {\nnodes = append(nodes, newNode(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nm := make(map[int]string, n)\nfor i := 0; i < n; i++ {\nm[i] = strconv.Itoa(i)\n}\n}\n
     space_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
     space_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes: ListNode[] = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceLinear}\n
     space_complexity.cs
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n)\n{\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new();\nfor (int i = 0; i < n; i++)\n{\nnodes.Add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nDictionary<int, String> map = new();\nfor (int i = 0; i < n; i++)\n{\nmap.Add(i, i.ToString());\n}\n}\n
     space_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet nums = Array(repeating: 0, count: n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet nodes = (0 ..< n).map { ListNode(x: $0) }\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
     space_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvar nums = [_]i32{0}**n;\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ntry nodes.append(i);\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\ndefer map.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\nconst string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\ndefer std.heap.page_allocator.free(string);\ntry map.put(i, string);\n}\n_ = nums;\n}\n
     
    \u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nSystem.out.println(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\ncout << \"\u9012\u5f52 n = \" << n << endl;\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.py
    def linear_recur(n: int) -> None:\n\"\"\" \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 \"\"\"\nprint(\"\u9012\u5f52 n =\", n)\nif n == 1: return\nlinear_recur(n - 1)\n
     space_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\nfmt.Println(\"\u9012\u5f52 n =\", n)\nif n == 1 {\nreturn\n}\nspaceLinearRecur(n - 1)\n}\n
     space_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.c
    [class]{}-[func]{spaceLinearRecur}\n
     space_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n)\n{\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\nprint(\"\u9012\u5f52 n = \\(n)\")\nif n == 1 {\nreturn\n}\nlinearRecur(n: n - 1)\n}\n
     space_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\nstd.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     
     
     Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"
    \u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\u7684\u77e9\u9635\u3001\u56fe\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[][] numMatrix = new int[n][n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<Integer>> numList = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<Integer> tmp = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n
     space_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvector<vector<int>> numMatrix;\nfor (int i = 0; i < n; i++) {\nvector<int> tmp;\nfor (int j = 0; j < n; j++) {\ntmp.push_back(0);\n}\nnumMatrix.push_back(tmp);\n}\n}\n
     space_complexity.py
    def quadratic(n: int) -> None:\n\"\"\" \u5e73\u65b9\u9636 \"\"\"\n# \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nnum_matrix: List[List[int]] = [[0] * n for _ in range(n)]\n
     space_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nnumMatrix := make([][]int, n)\nfor i := 0; i < n; i++ {\nnumMatrix[i] = make([]int, n)\n}\n}\n
     space_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n).fill(null).map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
     space_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceQuadratic}\n
     space_complexity.cs
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n)\n{\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[,] numMatrix = new int[n, n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = new();\nfor (int i = 0; i < n; i++)\n{\nList<int> tmp = new();\nfor (int j = 0; j < n; j++)\n{\ntmp.Add(0);\n}\nnumList.Add(tmp);\n}\n}\n
     space_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
     space_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvar nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\nvar tmp = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer tmp.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ntry tmp.append(0);\n}\ntry nodes.append(tmp);\n}\n}\n
     
    \u5728\u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \uff0c\u5e76\u4e14\u6bcf\u4e2a\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n, n-1, n-2, ..., 2, 1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nint[] nums = new int[n];\nSystem.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nvector<int> nums(n);\ncout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.py
    def quadratic_recur(n: int) -> int:\n\"\"\" \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 \"\"\"\nif n <= 0: return 0\n# \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nnums: List[int] = [0] * n\nreturn quadratic_recur(n - 1)\n
     space_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\nif n <= 0 {\nreturn 0\n}\nnums := make([]int, n)\nfmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\nreturn spaceQuadraticRecur(n - 1)\n}\n
     space_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.c
    [class]{}-[func]{spaceQuadraticRecur}\n
     space_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n)\n{\nif (n <= 0) return 0;\nint[] nums = new int[n];\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\nif n <= 0 {\nreturn 0\n}\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = Array(repeating: 0, count: n)\nprint(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\nreturn quadraticRecur(n: n - 1)\n}\n
     space_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\nif (n <= 0) return 0;\nvar nums = [_]i32{0}**n;\nstd.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\nreturn quadraticRecur(n - 1);\n}\n
     
     
     Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"
    \u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u7684\u7ed3\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u4f7f\u7528 \\(O(2^n)\\) \u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode* buildTree(int n) {\nif (n == 0) return nullptr;\nTreeNode* root = new TreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.py
    def build_tree(n: int) -> Optional[TreeNode]:\n\"\"\" \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 \"\"\"\nif n == 0: return None\nroot = TreeNode(0)\nroot.left = build_tree(n - 1)\nroot.right = build_tree(n - 1)\nreturn root\n
     space_complexity.go
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\nif n == 0 {\nreturn nil\n}\nroot := newTreeNode(0)\nroot.left = buildTree(n - 1)\nroot.right = buildTree(n - 1)\nreturn root\n}\n
     space_complexity.js
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.c
    [class]{}-[func]{buildTree}\n
     space_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n)\n{\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\nif n == 0 {\nreturn nil\n}\nlet root = TreeNode(x: 0)\nroot.left = buildTree(n: n - 1)\nroot.right = buildTree(n: n - 1)\nreturn root\n}\n
     space_complexity.zig
    // \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\nif (n == 0) return null;\nconst root = try mem_allocator.create(inc.TreeNode(i32));\nroot.init(0);\nroot.left = try buildTree(mem_allocator, n - 1);\nroot.right = try buildTree(mem_allocator, n - 1);\nreturn root;\n}\n
     
     
     Fig. \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"
    \u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3001\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u7b49\u3002
     
    \u4f8b\u5982\u300c\u5f52\u5e76\u6392\u5e8f\u300d\uff0c\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u53ef\u4ee5\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     
    \u518d\u4f8b\u5982\u300c\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\u300d\uff0c\u8f93\u5165\u4efb\u610f\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n) = O(\\log n)\\) \u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/","title":"2.4. \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u591f\u8fbe\u5230\u6700\u4f18\uff0c\u800c\u5b9e\u9645\u4e0a\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u662f\u975e\u5e38\u56f0\u96be\u7684\u3002
     
    \u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5f80\u5f80\u662f\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\u7684\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u628a\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u300c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u300d\uff1b\u53cd\u4e4b\uff0c\u79f0\u4e4b\u4e3a\u300c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u300d\u3002\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002
     
    \u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u90fd\u662f\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\u7684\uff0c\u53ea\u8981\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u8981\u592a\u79bb\u8c31\u3001\u80fd\u63a5\u53d7\u5c31\u884c\uff0c\u56e0\u6b64\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u6700\u4e3a\u5e38\u7528\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#241","title":"2.4.1. \u00a0 \u793a\u4f8b\u9898\u76ee *","text":"
    \u4ee5 LeetCode \u5168\u7ad9\u7b2c\u4e00\u9898 \u4e24\u6570\u4e4b\u548c \u4e3a\u4f8b\u3002
     
    \u4e24\u6570\u4e4b\u548c
     
    \u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u6574\u6570\u76ee\u6807\u503c target \uff0c\u8bf7\u4f60\u5728\u8be5\u6570\u7ec4\u4e2d\u627e\u51fa\u201c\u548c\u201d\u4e3a\u76ee\u6807\u503c target \u7684\u90a3\u4e24\u4e2a\u6574\u6570\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u4e0b\u6807\u3002
     
    \u4f60\u53ef\u4ee5\u5047\u8bbe\u6bcf\u79cd\u8f93\u5165\u53ea\u4f1a\u5bf9\u5e94\u4e00\u4e2a\u7b54\u6848\u3002\u4f46\u662f\uff0c\u6570\u7ec4\u4e2d\u540c\u4e00\u4e2a\u5143\u7d20\u5728\u7b54\u6848\u91cc\u4e0d\u80fd\u91cd\u590d\u51fa\u73b0\u3002
     
    \u4f60\u53ef\u4ee5\u6309\u4efb\u610f\u987a\u5e8f\u8fd4\u56de\u7b54\u6848\u3002
     
    \u300c\u66b4\u529b\u679a\u4e3e\u300d\u548c\u300c\u8f85\u52a9\u54c8\u5e0c\u8868\u300d\u5206\u522b\u5bf9\u5e94 \u7a7a\u95f4\u6700\u4f18 \u548c \u65f6\u95f4\u6700\u4f18 \u7684\u4e24\u79cd\u89e3\u6cd5\u3002\u672c\u7740\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\u7684\u539f\u5219\uff0c\u540e\u8005\u662f\u672c\u9898\u7684\u6700\u4f73\u89e3\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#_1","title":"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e","text":"
    \u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u6240\u6709\u53ef\u80fd\u6027\u3002\u901a\u8fc7\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a target \uff0c\u82e5\u662f\u5219\u8fd4\u56de\u5b83\u4fe9\u7684\u7d22\u5f15\uff08\u5373\u4e0b\u6807\uff09\u5373\u53ef\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig leetcode_two_sum.java
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.cpp
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int>& nums, int target) {\nint size = nums.size();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn { i, j };\n}\n}\nreturn {};\n}\n
     leetcode_two_sum.py
    def two_sum_brute_force(nums: List[int], target: int) -> List[int]:\n\"\"\" \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e \"\"\"\n# \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in range(len(nums) - 1):\nfor j in range(i + 1, len(nums)):\nif nums[i] + nums[j] == target:\nreturn [i, j]\nreturn []\n
     leetcode_two_sum.go
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\nsize := len(nums)\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i := 0; i < size-1; i++ {\nfor j := i + 1; i < size; j++ {\nif nums[i]+nums[j] == target {\nreturn []int{i, j}\n}\n}\n}\nreturn nil\n}\n
     leetcode_two_sum.js
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
     leetcode_two_sum.ts
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n};\n
     leetcode_two_sum.c
    [class]{}-[func]{twoSumBruteForce}\n
     leetcode_two_sum.cs
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target)\n{\nint size = nums.Length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++)\n{\nfor (int j = i + 1; j < size; j++)\n{\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.swift
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in nums.indices.dropLast() {\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[i] + nums[j] == target {\nreturn [i, j]\n}\n}\n}\nreturn [0]\n}\n
     leetcode_two_sum.zig
    // \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\nvar size: usize = nums.len;\nvar i: usize = 0;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nwhile (i < size - 1) : (i += 1) {\nvar j = i + 1;\nwhile (j < size) : (j += 1) {\nif (nums[i] + nums[j] == target) {\nreturn [_]i32{@intCast(i32, i), @intCast(i32, j)};\n}\n}\n}\nreturn null;\n}\n
     
    \u8be5\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5c5e\u4e8e\u65f6\u95f4\u6362\u7a7a\u95f4\u3002\u672c\u65b9\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u8f83\u9ad8\uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#_2","title":"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868","text":"
    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0ckey \u4e3a\u6570\u7ec4\u5143\u7d20\u3001value \u4e3a\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5e76\u6267\u884c\uff1a
     
       
    1. \u5224\u65ad\u6570\u5b57 target - num \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\u5219\u76f4\u63a5\u8fd4\u56de\u8be5\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1b
      2.  
    2. \u5c06\u5143\u7d20 num \u548c\u5176\u7d22\u5f15\u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\uff1b
      3.  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig leetcode_two_sum.java
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nMap<Integer, Integer> dic = new HashMap<>();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn new int[] { dic.get(target - nums[i]), i };\n}\ndic.put(nums[i], i);\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.cpp
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int>& nums, int target) {\nint size = nums.size();\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nunordered_map<int, int> dic;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.find(target - nums[i]) != dic.end()) {\nreturn { dic[target - nums[i]], i };\n}\ndic.emplace(nums[i], i);\n}\nreturn {};\n}\n
     leetcode_two_sum.py
    def two_sum_hash_table(nums: List[int], target: int) -> List[int]:\n\"\"\" \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 \"\"\"\n# \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\ndic = {}\n# \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in range(len(nums)):\nif target - nums[i] in dic:\nreturn [dic[target - nums[i]], i]\ndic[nums[i]] = i\nreturn []\n
     leetcode_two_sum.go
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nhashTable := map[int]int{}\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor idx, val := range nums {\nif preIdx, ok := hashTable[target-val]; ok {\nreturn []int{preIdx, idx}\n}\nhashTable[val] = idx\n}\nreturn nil\n}\n
     leetcode_two_sum.js
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m = {};\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nif (m[nums[i]] !== undefined) {\nreturn [m[nums[i]], i];\n} else {\nm[target - nums[i]] = i;\n}\n}\nreturn [];\n}\n
     leetcode_two_sum.ts
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m: Map<number, number> = new Map();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nlet index = m.get(nums[i]);\nif (index !== undefined) {\nreturn [index, i];\n} else {\nm.set(target - nums[i], i);\n}\n}\nreturn [];\n};\n
     leetcode_two_sum.c
    [class]{}-[func]{twoSumHashTable}\n
     leetcode_two_sum.cs
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target)\n{\nint size = nums.Length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nDictionary<int, int> dic = new();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++)\n{\nif (dic.ContainsKey(target - nums[i]))\n{\nreturn new int[] { dic[target - nums[i]], i };\n}\ndic.Add(nums[i], i);\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.swift
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic: [Int: Int] = [:]\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in nums.indices {\nif let j = dic[target - nums[i]] {\nreturn [j, i]\n}\ndic[nums[i]] = i\n}\nreturn [0]\n}\n
     leetcode_two_sum.zig
    // \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\nvar size: usize = nums.len;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\ndefer dic.deinit();\nvar i: usize = 0;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nwhile (i < size) : (i += 1) {\nif (dic.contains(target - nums[i])) {\nreturn [_]i32{dic.get(target - nums[i]).?, @intCast(i32, i)};\n}\ntry dic.put(nums[i], @intCast(i32, i));\n}\nreturn null;\n}\n
     
    \u8be5\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff0c\u4f53\u73b0\u7a7a\u95f4\u6362\u65f6\u95f4\u3002\u672c\u65b9\u6cd5\u867d\u7136\u5f15\u5165\u4e86\u989d\u5916\u7a7a\u95f4\u4f7f\u7528\uff0c\u4f46\u65f6\u95f4\u548c\u7a7a\u95f4\u4f7f\u7528\u6574\u4f53\u66f4\u52a0\u5747\u8861\uff0c\u56e0\u6b64\u4e3a\u672c\u9898\u6700\u4f18\u89e3\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/summary/","title":"2.5. \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#_1","title":"\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"
       
    • \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u7b97\u6cd5\u6027\u80fd\u7684\u4e24\u4e2a\u91cd\u8981\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3002
      •  
    • \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\uff0c\u5e76\u4e14\u975e\u5e38\u8017\u8d39\u8ba1\u7b97\u8d44\u6e90\u3002
      •  
    • \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u5927\u5c0f\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\u3002
      •  
    "},{"location":"chapter_computational_complexity/summary/#_2","title":"\u65f6\u95f4\u590d\u6742\u5ea6","text":"
       
    • \u65f6\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u6bd4\u5982\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u6027\u3002
      •  
    • \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5373\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u5176\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u4e8e\u6b63\u65e0\u7a77\u65f6\uff0c\\(T(n)\\) \u5904\u4e8e\u4f55\u79cd\u589e\u957f\u7ea7\u522b\u3002
      •  
    • \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\uff0c\u518d\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
      •  
    • \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n \\log n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) , \\(O(n!)\\) \u3002
      •  
    • \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u540e\u8005\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u9700\u8981\u6ee1\u8db3\u82db\u523b\u7684\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
      •  
    • \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u53cd\u6620\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u6700\u8d34\u5408\u5b9e\u9645\u4f7f\u7528\u60c5\u51b5\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\uff0c\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002
      •  
    "},{"location":"chapter_computational_complexity/summary/#_3","title":"\u7a7a\u95f4\u590d\u6742\u5ea6","text":"
       
    • \u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5b9a\u4e49\u7c7b\u4f3c\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u7b97\u6cd5\u5360\u7528\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002
      •  
    • \u7b97\u6cd5\u8fd0\u884c\u4e2d\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u8ba1\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6307\u4ee4\u7a7a\u95f4\u3001\u6570\u636e\u7a7a\u95f4\u3001\u6808\u5e27\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u4e00\u822c\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u624d\u4f1a\u5f71\u54cd\u5230\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
      •  
    • \u6211\u4eec\u4e00\u822c\u53ea\u5173\u5fc3\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u95f4\u70b9\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
      •  
    • \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) \u3002
      •  
    "},{"location":"chapter_computational_complexity/time_complexity/","title":"2.2. \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/time_complexity/#221","title":"2.2.1. \u00a0 \u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4","text":"
    \u8fd0\u884c\u65f6\u95f4\u80fd\u591f\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u4f53\u73b0\u51fa\u7b97\u6cd5\u7684\u6548\u7387\u6c34\u5e73\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981 \u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4 \uff0c\u8be5\u5982\u4f55\u505a\u5462\uff1f
     
       
    1. \u9996\u5148\u9700\u8981 \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0 \uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u90fd\u4f1a\u5f71\u54cd\u5230\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
      2.  
    2. \u8bc4\u4f30 \u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u7684\u6240\u9700\u8fd0\u884c\u65f6\u95f4 \uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c + \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c * \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c\u9700\u8981 5 ns \u7b49\u3002
      3.  
    3. \u6839\u636e\u4ee3\u7801 \u7edf\u8ba1\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf \uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u5373\u53ef\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002
      4.  
     
    \u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e3a \\(6n + 12\\) ns \u3002
     \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nSystem.out.println(0);     // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\ncout << 0 << endl;         // 5 ns\n}\n}\n
     
    # \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int) -> None:\na = 2      # 1 ns\na = a + 1  # 1 ns\na = a * 2  # 10 ns\n# \u5faa\u73af n \u6b21\nfor _ in range(n):  # 1 ns\nprint(0)        # 5 ns\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\na := 2      // 1 ns\na = a + 1   // 1 ns\na = a * 2   // 10 ns\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {    // 1 ns\nfmt.Println(a)          // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\nvar a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\nvar a: number = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintf(\"%d\", 0);            // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n)\n{\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++)\n{  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nConsole.WriteLine(0);     // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\nvar a = 2 // 1 ns\na = a + 1 // 1 ns\na = a * 2 // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // 1 ns\nprint(0) // 5 ns\n}\n}\n
     
    \n
     
    \u4f46\u5b9e\u9645\u4e0a\uff0c \u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u6bd5\u7adf\u7b97\u6cd5\u9700\u8981\u8dd1\u5728\u5404\u5f0f\u5404\u6837\u7684\u5e73\u53f0\u4e4b\u4e0a\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u4e00\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u4e3a\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#222","title":"2.2.2. \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"
    \u300c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u300d\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u505a\u6cd5\uff0c\u5176\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f \u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf \u3002
     
    \u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u4f8b\u5b50\u6765\u7406\u89e3\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 A , B , C \u3002
     
       
    • \u7b97\u6cd5 A \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
      •  
    • \u7b97\u6cd5 B \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u6210\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u300c\u7ebf\u6027\u9636\u300d\u3002
      •  
    • \u7b97\u6cd5 C \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u4ecd\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c A \u76f8\u540c\uff0c\u4ecd\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nSystem.out.println(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nSystem.out.println(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nSystem.out.println(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\ncout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\ncout << 0 << endl;\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\ncout << 0 << endl;\n}\n}\n
     
    # \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int) -> None:\nprint(0)\n# \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int) -> None:\nfor _ in range(n):\nprint(0)\n# \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int) -> None:\nfor _ in range(1000000):\nprint(0)\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\nfmt.Println(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\nfor i := 0; i < n; i++ {\nfmt.Println(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\nfor i := 0; i < 1000000; i++ {\nfmt.Println(0)\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nprintf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nprintf(\"%d\", 0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprintf(\"%d\", 0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n)\n{\nConsole.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nConsole.WriteLine(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n)\n{\nfor (int i = 0; i < 1000000; i++)\n{\nConsole.WriteLine(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\nprint(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\nfor _ in 0 ..< n {\nprint(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\nfor _ in 0 ..< 1000000 {\nprint(0)\n}\n}\n
     
    \n
     
     
     Fig. \u7b97\u6cd5 A, B, C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf 
     
    \u76f8\u6bd4\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7684\u505a\u6cd5\u6709\u4ec0\u4e48\u597d\u5904\u5462\uff1f\u4ee5\u53ca\u6709\u4ec0\u4e48\u4e0d\u8db3\uff1f
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u7b97\u6cd5 B \u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u662f\u7ebf\u6027\u7684\uff0c\u5728 \\(n > 1\\) \u65f6\u6162\u4e8e\u7b97\u6cd5 A \uff0c\u5728 \\(n > 1000000\\) \u65f6\u6162\u4e8e\u7b97\u6cd5 C \u3002\u5b9e\u8d28\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u300c\u7ebf\u6027\u9636\u300d\u7684\u7b97\u6cd5\uff0c\u8fd9\u4e5f\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u52a0\u7b80\u4fbf\u3002\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7edf\u8ba1\u300c\u8ba1\u7b97\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\u300d\u7b80\u5316\u4e3a\u7edf\u8ba1\u300c\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf\u300d\uff0c\u8fd9\u662f\u56e0\u4e3a\uff0c\u65e0\u8bba\u662f\u8fd0\u884c\u5e73\u53f0\u8fd8\u662f\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\uff0c\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u7edf\u4e00\u770b\u4f5c\u662f\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u8fd9\u6837\u7684\u7b80\u5316\u505a\u6cd5\u5927\u5927\u964d\u4f4e\u4e86\u4f30\u7b97\u96be\u5ea6\u3002
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u6bd4\u5982\uff0c\u867d\u7136\u7b97\u6cd5 A \u548c C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u662f\u5b9e\u9645\u7684\u8fd0\u884c\u65f6\u95f4\u6709\u975e\u5e38\u5927\u7684\u5dee\u522b\u3002\u518d\u6bd4\u5982\uff0c\u867d\u7136\u7b97\u6cd5 B \u6bd4 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8981\u66f4\u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6bd4\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 B \u662f\u8981\u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 C \u7684\u3002\u5bf9\u4e8e\u4ee5\u4e0a\u60c5\u51b5\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u5224\u5b9a\u7b97\u6cd5\u6548\u7387\u9ad8\u4f4e\u3002\u7136\u800c\uff0c\u5373\u4f7f\u5b58\u5728\u8fd9\u4e9b\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u7684\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#223","title":"2.2.3. \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"
    \u8bbe\u7b97\u6cd5\u300c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u300d\u4e3a \\(T(n)\\) \uff0c\u5176\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u4ee5\u4e0b\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a
     \\[ T(n) = 3 + 2n \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nSystem.out.println(0);    // +1\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\ncout << 0 << endl;    // +1\n}\n}\n
     
    def algorithm(n: int) -> None:\na: int = 1  # +1\na = a + 1  # +1\na = a * 2  # +1\n# \u5faa\u73af n \u6b21\nfor i in range(n):  # +1\nprint(0)        # +1\n
     
    func algorithm(n int) {\na := 1      // +1\na = a + 1   // +1\na = a * 2   // +1\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {   // +1\nfmt.Println(a)         // +1\n}\n}\n
     
    function algorithm(n) {\nvar a = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
     
    function algorithm(n: number): void{\nvar a: number = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintf(\"%d\", 0);            // +1\n}\n}  
     
    void algorithm(int n)\n{\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n{\nConsole.WriteLine(0);   // +1\n}\n}\n
     
    func algorithm(n: Int) {\nvar a = 1 // +1\na = a + 1 // +1\na = a * 2 // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // +1\nprint(0) // +1\n}\n}\n
     
    \n
     
    \\(T(n)\\) \u662f\u4e2a\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u6613\u5f97\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002
     
    \u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u88ab\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 Big-\\(O\\) Notation\u300d\uff0c\u4ee3\u8868\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002
     
    \u6211\u4eec\u8981\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u672c\u8d28\u4e0a\u662f\u5728\u8ba1\u7b97\u300c\u64cd\u4f5c\u6570\u91cf\u51fd\u6570 \\(T(n)\\) \u300d\u7684\u6e10\u8fd1\u4e0a\u754c\u3002\u4e0b\u9762\u6211\u4eec\u5148\u6765\u770b\u770b\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5b9a\u4e49\u3002
     
    \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c
     
    \u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 $$ T(n) \\leq c \\cdot f(n) $$ \u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a $$ T(n) = O(f(n)) $$
     
     
     Fig. \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c 
     
    \u672c\u8d28\u4e0a\u770b\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5728\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5728 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff08\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\uff09\u3002
     
    Tip
     
    \u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u56e0\u4e3a\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\u6211\u4eec\u53ea\u9700\u8981\u4f1a\u63a8\u7b97\u5373\u53ef\uff0c\u6570\u5b66\u610f\u4e49\u53ef\u4ee5\u6162\u6162\u9886\u609f\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#224","title":"2.2.4. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"
    \u63a8\u7b97\u51fa \\(f(n)\\) \u540e\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\uff0c\u5982\u4f55\u6765\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u300c\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\u300d\uff0c\u7136\u540e\u300c\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u300d\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1) \u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"
    \u5bf9\u7740\u4ee3\u7801\uff0c\u4ece\u4e0a\u5230\u4e0b\u4e00\u884c\u4e00\u884c\u5730\u8ba1\u6570\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u5077\u61d2\u6280\u5de7\uff1a
     
       
    1. \u8df3\u8fc7\u6570\u91cf\u4e0e \\(n\\) \u65e0\u5173\u7684\u64cd\u4f5c\u3002\u56e0\u4e3a\u4ed6\u4eec\u90fd\u662f \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\uff0c\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
      2.  
    2. \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u3001\u2026\u2026\uff0c\u90fd\u53ef\u4ee5\u5316\u7b80\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
      3.  
    3. \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u4e0a\u8ff0 1. \u548c 2. \u6280\u5de7\u3002
      4.  
     
    \u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u3001\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\u3002
     \\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\] 
    \u6700\u7ec8\uff0c\u4e24\u8005\u90fd\u80fd\u63a8\u51fa\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7ed3\u679c\uff0c\u5373 \\(O(n^2)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nSystem.out.println(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nSystem.out.println(0);\n}\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\ncout << 0 << endl;\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\ncout << 0 << endl;\n}\n}\n}\n
     
    def algorithm(n: int) -> None:\na: int = 1  # +0\uff08\u6280\u5de7 1\uff09\na = a + n   # +0\uff08\u6280\u5de7 1\uff09\n# +n\uff08\u6280\u5de7 2\uff09\nfor i in range(5 * n + 1):\nprint(0)\n# +n*n\uff08\u6280\u5de7 3\uff09\nfor i in range(2 * n):\nfor j in range(n + 1):\nprint(0)\n
     
    func algorithm(n int) {\na := 1      // +0\uff08\u6280\u5de7 1\uff09\na = a + n  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i := 0; i < 5 * n + 1; i++ {\nfmt.Println(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i := 0; i < 2 * n; i++ {\nfor j := 0; j < n + 1; j++ {\nfmt.Println(0)\n}\n}\n}\n
     
    function algorithm(n) {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
     
    function algorithm(n: number): void {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprintf(\"%d\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprintf(\"%d\", 0);\n}\n}\n}\n
     
    void algorithm(int n)\n{\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++)\n{\nConsole.WriteLine(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++)\n{\nfor (int j = 0; j < n + 1; j++)\n{\nConsole.WriteLine(0);\n}\n}\n}\n
     
    func algorithm(n: Int) {\nvar a = 1 // +0\uff08\u6280\u5de7 1\uff09\na = a + n // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor _ in 0 ..< (5 * n + 1) {\nprint(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor _ in 0 ..< (2 * n) {\nfor _ in 0 ..< (n + 1) {\nprint(0)\n}\n}\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2) \u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\u7531\u591a\u9879\u5f0f \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u5904\u4e8e\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u5b83\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002
     
    \u4ee5\u4e0b\u8868\u683c\u7ed9\u51fa\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u6709\u4e00\u4e9b\u5938\u5f20\u7684\u503c\uff0c\u662f\u60f3\u8981\u5411\u5927\u5bb6\u5f3a\u8c03 \u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570 \u8fd9\u4e00\u7ed3\u8bba\u3002\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u90fd\u662f\u201c\u6d6e\u4e91\u201d\u3002
     \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#225","title":"2.2.5. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09
     \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\] 
     
     Fig. \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b 
     
    Tip
     
    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u5148\u805a\u7126\u5728\u7406\u89e3\u65f6\u95f4\u590d\u6742\u5ea6\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u4e0a\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"
    \u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002
     
    \u5bf9\u4e8e\u4ee5\u4e0b\u7b97\u6cd5\uff0c\u65e0\u8bba\u64cd\u4f5c\u6570\u91cf size \u6709\u591a\u5927\uff0c\u53ea\u8981\u4e0e\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4ecd\u4e3a \\(O(1)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.py
    def constant(n: int) -> int:\n\"\"\" \u5e38\u6570\u9636 \"\"\"\ncount: int = 0\nsize: int = 100000\nfor _ in range(size):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\ncount := 0\nsize := 100000\nfor i := 0; i < size; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{constant}\n
     time_complexity.cs
    /* \u5e38\u6570\u9636 */\nint constant(int n)\n{\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\nvar count = 0\nlet size = 100_000\nfor _ in 0 ..< size {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n_ = n;\nvar count: i32 = 0;\nconst size: i32 = 100_000;\nvar i: i32 = 0;\nwhile(i<size) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"
    \u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u8f93\u5165\u6570\u636e\u5927\u5c0f\u6210\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u5e38\u51fa\u73b0\u4e8e\u5355\u5c42\u5faa\u73af\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.py
    def linear(n: int) -> int:\n\"\"\" \u7ebf\u6027\u9636 \"\"\"\ncount: int = 0\nfor _ in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\ncount := 0\nfor i := 0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{linear}\n
     time_complexity.cs
    /* \u7ebf\u6027\u9636 */\nint linear(int n)\n{\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\nvar count = 0\nfor _ in 0 ..< n {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
     
    \u300c\u904d\u5386\u6570\u7ec4\u300d\u548c\u300c\u904d\u5386\u94fe\u8868\u300d\u7b49\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\u3002
     
    Tip
     
    \u6570\u636e\u5927\u5c0f \\(n\\) \u662f\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u786e\u5b9a\u7684\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u76f4\u63a5\u5c06 \\(n\\) \u770b\u4f5c\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u4ee5\u4e0b\u904d\u5386\u6570\u7ec4\u793a\u4f8b\u4e2d\uff0c\u6570\u636e\u5927\u5c0f \\(n\\) \u4e3a\u6570\u7ec4\u7684\u957f\u5ea6\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int>& nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def array_traversal(nums: List[int]) -> int:\n\"\"\" \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\ncount: int = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor num in nums:\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor range nums {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{arrayTraversal}\n
     time_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums)\n{\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nforeach (int num in nums)\n{\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\nvar count: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (nums) |_| {\ncount += 1;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"
    \u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u8f93\u5165\u6570\u636e\u5927\u5c0f\u6210\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u90fd\u4e3a \\(O(n)\\) \uff0c\u603b\u4f53\u4e3a \\(O(n^2)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.py
    def quadratic(n: int) -> int:\n\"\"\" \u5e73\u65b9\u9636 \"\"\"\ncount: int = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i in range(n):\nfor j in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i := 0; i < n; i++ {\nfor j := 0; j < n; j++ {\ncount++\n}\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{quadratic}\n
     time_complexity.cs
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n)\n{\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++)\n{\nfor (int j = 0; j < n; j++)\n{\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0 ..< n {\nfor _ in 0 ..< n {\ncount += 1\n}\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ncount += 1;\n}\n}\nreturn count;\n}\n
     
     
     Fig. \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u3001\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u4ee5\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af \\(n-1, n-2, \\cdots, 2, 1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(\\frac{n}{2}\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
     \\[ O((n - 1) \\frac{n}{2}) = O(n^2) \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int>& nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.py
    def bubble_sort(nums: List[int]) -> int:\n\"\"\" \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\ncount: int = 0  # \u8ba1\u6570\u5668\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(len(nums) - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp: int = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\nreturn count\n
     time_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\ncount := 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp := nums[j]\nnums[j] = nums[j+1]\nnums[j+1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{bubbleSort}\n
     time_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums)\n{\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\nvar count = 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in 0 ..< i {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\nvar count: i32 = 0;  // \u8ba1\u6570\u5668 \n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: i32 = @intCast(i32, nums.len ) - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"
    Note
     
    \u751f\u7269\u5b66\u79d1\u4e2d\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u5373\u662f\u6307\u6570\u9636\u589e\u957f\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u4e3a \\(4\\) \u4e2a\uff0c\u2026\u2026\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002
     
    \u6307\u6570\u9636\u589e\u957f\u5f97\u975e\u5e38\u5feb\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u4e00\u822c\u662f\u4e0d\u80fd\u88ab\u63a5\u53d7\u7684\u3002\u82e5\u4e00\u4e2a\u95ee\u9898\u4f7f\u7528\u300c\u66b4\u529b\u679a\u4e3e\u300d\u6c42\u89e3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(2^n)\\) \uff0c\u90a3\u4e48\u4e00\u822c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u52a8\u6001\u89c4\u5212\u300d\u6216\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u7b49\u7b97\u6cd5\u6765\u6c42\u89e3\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.py
    def exponential(n: int) -> int:\n\"\"\" \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount: int = 0\nbase: int = 1\n# cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in range(n):\nfor _ in range(base):\ncount += 1\nbase *= 2\n# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n
     time_complexity.go
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\ncount, base := 0, 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor i := 0; i < n; i++ {\nfor j := 0; j < base; j++ {\ncount++\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
     time_complexity.js
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{exponential}\n
     time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n)\n{\nint count = 0, bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++)\n{\nfor (int j = 0; j < bas; j++)\n{\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\nvar count = 0\nvar base = 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0 ..< n {\nfor _ in 0 ..< base {\ncount += 1\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
     time_complexity.zig
    // \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\nvar count: i32 = 0;\nvar bas: i32 = 1;\nvar i: i32 = 0;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < bas) : (j += 1) {\ncount += 1;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     
     
     Fig. \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u4e0d\u65ad\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u5206\u88c2 \\(n\\) \u6b21\u540e\u505c\u6b62\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.py
    def exp_recur(n: int) -> int:\n\"\"\" \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 1: return 1\nreturn exp_recur(n - 1) + exp_recur(n - 1) + 1\n
     time_complexity.go
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n-1) + expRecur(n-1) + 1\n}\n
     time_complexity.js
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.c
    [class]{}-[func]{expRecur}\n
     time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n)\n{\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
     time_complexity.zig
    // \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"
    \u5bf9\u6570\u9636\u4e0e\u6307\u6570\u9636\u6b63\u597d\u76f8\u53cd\uff0c\u540e\u8005\u53cd\u6620\u201c\u6bcf\u8f6e\u589e\u52a0\u5230\u4e24\u500d\u7684\u60c5\u51b5\u201d\uff0c\u800c\u524d\u8005\u53cd\u6620\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u7684\u60c5\u51b5\u201d\u3002\u5bf9\u6570\u9636\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\uff0c\u65f6\u95f4\u589e\u957f\u5f97\u5f88\u6162\uff0c\u662f\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
     
    \u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u5206\u6cbb\u7b97\u6cd5\u300d\u4e2d\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u591a\u201d\u3001\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002
     
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def logarithmic(n: float) -> int:\n\"\"\" \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount: int = 0\nwhile n > 1:\nn = n / 2\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\ncount := 0\nfor n > 1 {\nn = n / 2\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{logarithmic}\n
     time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n)\n{\nint count = 0;\nwhile (n > 1)\n{\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\nvar count = 0\nvar n = n\nwhile n > 1 {\nn = n / 2\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\nvar count: i32 = 0;\nvar n_var = n;\nwhile (n_var > 1)\n{\nn_var = n_var / 2;\ncount +=1;\n}\nreturn count;\n}\n
     
     
     Fig. \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.py
    def log_recur(n: float) -> int:\n\"\"\" \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 1: return 0\nreturn log_recur(n / 2) + 1\n
     time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n/2) + 1\n}\n
     time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.c
    [class]{}-[func]{logRecur}\n
     time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n)\n{\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n: n / 2) + 1\n}\n
     time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on-log-n","title":"\u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"
    \u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002
     
    \u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(n \\log n )\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def linear_log_recur(n: float) -> int:\n\"\"\" \u7ebf\u6027\u5bf9\u6570\u9636 \"\"\"\nif n <= 1: return 1\ncount: int = linear_log_recur(n // 2) + \\\n                 linear_log_recur(n // 2)\nfor _ in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\nif n <= 1 {\nreturn 1\n}\ncount := linearLogRecur(n/2) +\nlinearLogRecur(n/2)\nfor i := 0.0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{linearLogRecur}\n
     time_complexity.cs
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n)\n{\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++)\n{\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 1\n}\nvar count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\nfor _ in stride(from: 0, to: n, by: 1) {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\nif (n <= 1) return 1;\nvar count: i32 = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nvar i: f32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
     
     
     Fig. \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/time_complexity/#on_1","title":"\u9636\u4e58\u9636 \\(O(n!)\\)","text":"
    \u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u300c\u5168\u6392\u5217\u300d\u3002\u5373\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u5219\u65b9\u6848\u6570\u91cf\u4e3a
     \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\cdots \\times 2 \\times 1 \\] 
    \u9636\u4e58\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u2026\u2026 \uff0c\u76f4\u81f3\u5230\u7b2c \\(n\\) \u5c42\u65f6\u7ec8\u6b62\u5206\u88c2\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.py
    def factorial_recur(n: int) -> int:\n\"\"\" \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 0: return 1\ncount: int = 0\n# \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in range(n):\ncount += factorial_recur(n - 1)\nreturn count\n
     time_complexity.go
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\nif n == 0 {\nreturn 1\n}\ncount := 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor i := 0; i < n; i++ {\ncount += factorialRecur(n - 1)\n}\nreturn count\n}\n
     time_complexity.js
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{factorialRecur}\n
     time_complexity.cs
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n)\n{\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++)\n{\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\nif n == 0 {\nreturn 1\n}\nvar count = 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0 ..< n {\ncount += factorialRecur(n: n - 1)\n}\nreturn count\n}\n
     time_complexity.zig
    // \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\nif (n == 0) return 1;\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nwhile (i < n) : (i += 1) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     
     
     Fig. \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/time_complexity/#226","title":"2.2.6. \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"
    \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u4e3e\u4e00\u4e2a\u4f8b\u5b50\uff0c\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u6570\u7ec4 nums \uff0c\u5176\u4e2d nums \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff1b\u7b97\u6cd5\u7684\u4efb\u52a1\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a
     
       
    • \u5f53 nums = [?, ?, ..., 1]\uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u5219\u9700\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1b
      •  
    • \u5f53 nums = [1, ?, ?, ...] \uff0c\u5373\u5f53\u9996\u4e2a\u6570\u5b57\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \uff1b
      •  
     
    \u300c\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u300d\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002\u4e0e\u4e4b\u5bf9\u5e94\uff0c\u300c\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\u300d\u7528 \\(\\Omega\\) \u8bb0\u53f7\uff08Omega Notation\uff09\u6765\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig worst_best_time_complexity.java
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nInteger[] nums = new Integer[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nCollections.shuffle(Arrays.asList(nums));\n// Integer[] -> int[]\nint[] res = new int[n];\nfor (int i = 0; i < n; i++) {\nres[i] = nums[i];\n}\nreturn res;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.cpp
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\nvector<int> nums(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\nunsigned seed = chrono::system_clock::now().time_since_epoch().count();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nshuffle(nums.begin(), nums.end(), default_random_engine(seed));\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int>& nums) {\nfor (int i = 0; i < nums.size(); i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.py
    def random_numbers(n: int) -> List[int]:\n\"\"\" \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71 \"\"\"\n# \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\nnums: List[int] = [i for i in range(1, n + 1)]\n# \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrandom.shuffle(nums)\nreturn nums\ndef find_one(nums: List[int]) -> int:\n\"\"\" \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 \"\"\"\nfor i in range(len(nums)):\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1:\nreturn i\nreturn -1\n
     worst_best_time_complexity.go
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\nnums := make([]int, n)\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor i := 0; i < n; i++ {\nnums[i] = i + 1\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrand.Shuffle(len(nums), func(i, j int) {\nnums[i], nums[j] = nums[j], nums[i]\n})\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\nfor i := 0; i < len(nums); i++ {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
     worst_best_time_complexity.js
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     worst_best_time_complexity.ts
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     worst_best_time_complexity.c
    [class]{}-[func]{randomNumbers}\n[class]{}-[func]{findOne}\n
     worst_best_time_complexity.cs
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n)\n{\nint[] nums = new int[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++)\n{\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = 0; i < nums.Length; i++)\n{\nvar index = new Random().Next(i, nums.Length);\nvar tmp = nums[i];\nvar ran = nums[index];\nnums[i] = ran;\nnums[index] = tmp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums)\n{\nfor (int i = 0; i < nums.Length; i++)\n{\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.swift
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nvar nums = Array(1 ... n)\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle()\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\nfor i in nums.indices {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
     worst_best_time_complexity.zig
    // \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\npub fn randomNumbers(comptime n: usize) [n]i32 {\nvar nums: [n]i32 = undefined;\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (nums) |*num, i| {\nnum.* = @intCast(i32, i) + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nconst rand = std.crypto.random;\nrand.shuffle(i32, &nums);\nreturn nums;\n}\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\npub fn findOne(nums: []i32) i32 {\nfor (nums) |num, i| {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (num == 1) return @intCast(i32, i);\n}\nreturn -1;\n}\n
     
    Tip
     
    \u6211\u4eec\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5f88\u5c11\u4f7f\u7528\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u56e0\u4e3a\u5f80\u5f80\u53ea\u6709\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u53cd\u4e4b\uff0c\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u6700\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u201c\u6548\u7387\u5b89\u5168\u503c\u201d\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002
     
    \u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6216\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u5728\u201c\u7279\u6b8a\u5206\u5e03\u7684\u6570\u636e\u201d\u4e2d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u5f80\u5f80\u5f88\u5c0f\uff0c\u56e0\u6b64\u5e76\u4e0d\u80fd\u6700\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u5bf9\u5730\uff0c\u300c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\uff08Theta Notation\uff09\u6765\u8868\u793a\u3002
     
    \u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5219\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(\\frac{n}{2}\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(\\frac{n}{2}) = \\Theta(n)\\) \u3002
     
    \u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5c24\u5176\u662f\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u7b80\u4fbf\u5730\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e00\u822c\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002
     
    \u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f
     
    \u5b9e\u9645\u4e2d\u6211\u4eec\u7ecf\u5e38\u4f7f\u7528\u300c\u5927 \\(O\\) \u7b26\u53f7\u300d\u6765\u8868\u793a\u300c\u5e73\u5747\u590d\u6742\u5ea6\u300d\uff0c\u8fd9\u6837\u4e25\u683c\u610f\u4e49\u4e0a\u6765\u8bf4\u662f\u4e0d\u89c4\u8303\u7684\u3002\u8fd9\u53ef\u80fd\u662f\u56e0\u4e3a \\(O\\) \u7b26\u53f7\u5b9e\u5728\u662f\u592a\u6717\u6717\u4e0a\u53e3\u4e86\u3002\u5982\u679c\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\u770b\u5230\u7c7b\u4f3c \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u7684\u8868\u8ff0\uff0c\u8bf7\u4f60\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u5373\u53ef\u3002
    "},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"
    \u6570\u636e\u7ed3\u6784\u4e3b\u8981\u53ef\u6839\u636e\u300c\u903b\u8f91\u7ed3\u6784\u300d\u548c\u300c\u7269\u7406\u7ed3\u6784\u300d\u4e24\u79cd\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002
    "},{"location":"chapter_data_structure/classification_of_data_structure/#321","title":"3.2.1. \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"
    \u300c\u903b\u8f91\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u6570\u7ec4\u548c\u94fe\u8868\u7684\u6570\u636e\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u6392\u5217\uff0c\u53cd\u6620\u4e86\u6570\u636e\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u6811\u4ece\u9876\u81f3\u5e95\u6309\u5c42\u7ea7\u6392\u5217\uff0c\u53cd\u6620\u4e86\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u7531\u7ed3\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7f51\u7edc\u5173\u7cfb\u3002
     
    \u6211\u4eec\u4e00\u822c\u5c06\u903b\u8f91\u7ed3\u6784\u5206\u4e3a\u300c\u7ebf\u6027\u300d\u548c\u300c\u975e\u7ebf\u6027\u300d\u4e24\u79cd\u3002\u201c\u7ebf\u6027\u201d\u8fd9\u4e2a\u6982\u5ff5\u5f88\u76f4\u89c2\uff0c\u5373\u8868\u660e\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u662f\u6392\u6210\u4e00\u6761\u7ebf\u7684\uff1b\u800c\u5982\u679c\u6570\u636e\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u662f\u975e\u7ebf\u6027\u7684\uff08\u4f8b\u5982\u662f\u7f51\u72b6\u6216\u6811\u72b6\u7684\uff09\uff0c\u90a3\u4e48\u5c31\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
     
       
    • \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff1b
      •  
    • \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u56fe\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff1b
      •  
     
     
     Fig. \u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 
    "},{"location":"chapter_data_structure/classification_of_data_structure/#322","title":"3.2.2. \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u79bb\u6563","text":"
    Note
     
    \u82e5\u611f\u5230\u9605\u8bfb\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u770b\u5b8c\u4e0b\u4e2a\u7ae0\u8282\u300c\u6570\u7ec4\u4e0e\u94fe\u8868\u300d\uff0c\u518d\u56de\u8fc7\u5934\u6765\u7406\u89e3\u7269\u7406\u7ed3\u6784\u7684\u542b\u4e49\u3002
     
    \u300c\u7269\u7406\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u5206\u522b\u662f \u6570\u7ec4\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8 \u548c \u94fe\u8868\u7684\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u4e0a\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u6b64\u6d88\u5f7c\u957f\u7684\u7279\u6027\u3002
     
     
     Fig. \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8 
     
    \u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u6216\u94fe\u8868\u3001\u6216\u4e24\u8005\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\u6808\u548c\u961f\u5217\uff0c\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u3001\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u800c\u4f8b\u5982\u54c8\u5e0c\u8868\uff0c\u5176\u5b9e\u73b0\u540c\u65f6\u5305\u542b\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u3002
     
       
    • \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\uff1b
      •  
    • \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\uff1b
      •  
     
    \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u300c\u9759\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u610f\u5473\u7740\u8be5\u6570\u636e\u7ed3\u6784\u5728\u5728\u88ab\u521d\u59cb\u5316\u540e\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u53cd\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8be5\u6570\u636e\u7ed3\u6784\u5728\u88ab\u521d\u59cb\u5316\u540e\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u4fee\u6539\u5176\u957f\u5ea6\u3002
     
    Tip
     
    \u6570\u7ec4\u4e0e\u94fe\u8868\u662f\u5176\u4ed6\u6240\u6709\u6570\u636e\u7ed3\u6784\u7684\u201c\u5e95\u5c42\u79ef\u6728\u201d\uff0c\u5efa\u8bae\u8bfb\u8005\u4e00\u5b9a\u8981\u591a\u82b1\u4e9b\u65f6\u95f4\u4e86\u89e3\u3002
    "},{"location":"chapter_data_structure/data_and_memory/","title":"3.1. \u00a0 \u6570\u636e\u4e0e\u5185\u5b58","text":""},{"location":"chapter_data_structure/data_and_memory/#311","title":"3.1.1. \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"
    \u8c08\u5230\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\uff0c\u6211\u4eec\u80fd\u591f\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u7b49\uff0c\u8fd9\u4e9b\u6570\u636e\u867d\u7136\u7ec4\u7ec7\u5f62\u5f0f\u4e0d\u540c\uff0c\u4f46\u90fd\u662f\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u7684\u3002
     
    \u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\u3002
     
       
    • \u300c\u6574\u6570\u300d\u6839\u636e\u4e0d\u540c\u7684\u957f\u5ea6\u5206\u4e3a byte, short, int, long \uff0c\u6839\u636e\u7b97\u6cd5\u9700\u6c42\u9009\u7528\uff0c\u5373\u5728\u6ee1\u8db3\u53d6\u503c\u8303\u56f4\u7684\u60c5\u51b5\u4e0b\u5c3d\u91cf\u51cf\u5c0f\u5185\u5b58\u7a7a\u95f4\u5360\u7528\uff1b
      •  
    • \u300c\u6d6e\u70b9\u6570\u300d\u4ee3\u8868\u5c0f\u6570\uff0c\u6839\u636e\u957f\u5ea6\u5206\u4e3a float, double \uff0c\u540c\u6837\u6839\u636e\u7b97\u6cd5\u7684\u5b9e\u9645\u9700\u6c42\u9009\u7528\uff1b
      •  
    • \u300c\u5b57\u7b26\u300d\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u5b57\u7b26\u96c6\u7684\u5f62\u5f0f\u4fdd\u5b58\u7684\uff0cchar \u7684\u503c\u5b9e\u9645\u4e0a\u662f\u6570\u5b57\uff0c\u4ee3\u8868\u5b57\u7b26\u96c6\u4e2d\u7684\u7f16\u53f7\uff0c\u8ba1\u7b97\u673a\u901a\u8fc7\u5b57\u7b26\u96c6\u67e5\u8868\u6765\u5b8c\u6210\u7f16\u53f7\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002\u5360\u7528\u7a7a\u95f4\u901a\u5e38\u4e3a 2 bytes \u6216 1 byte \uff1b
      •  
    • \u300c\u5e03\u5c14\u300d\u4ee3\u8868\u903b\u8f91\u4e2d\u7684\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\uff0c\u5176\u5360\u7528\u7a7a\u95f4\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u786e\u5b9a\uff0c\u901a\u5e38\u4e3a 1 byte \u6216 1 bit \uff1b
      •  
     \u7c7b\u522b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u53d6\u503c\u8303\u56f4 \u9ed8\u8ba4\u503c \u6574\u6570 byte 1 byte \\(-2^7\\) ~ \\(2^7 - 1\\) ( \\(-128\\) ~ \\(127\\) ) \\(0\\) short 2 bytes \\(-2^{15}\\) ~ \\(2^{15} - 1\\) \\(0\\) int 4 bytes \\(-2^{31}\\) ~ \\(2^{31} - 1\\) \\(0\\) long 8 bytes \\(-2^{63}\\) ~ \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 float 4 bytes \\(-3.4 \\times 10^{38}\\) ~ \\(3.4 \\times 10^{38}\\) \\(0.0\\) f double 8 bytes \\(-1.7 \\times 10^{308}\\) ~ \\(1.7 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 char 2 bytes / 1 byte \\(0\\) ~ \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 bool 1 byte / 1 bit \\(\\text{true}\\) \u6216 \\(\\text{false}\\) \\(\\text{false}\\) 
    Tip
     
    \u4ee5\u4e0a\u8868\u683c\u4e2d\uff0c\u52a0\u7c97\u9879\u5728\u300c\u7b97\u6cd5\u9898\u300d\u4e2d\u6700\u4e3a\u5e38\u7528\u3002\u6b64\u8868\u683c\u65e0\u9700\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_1","title":"\u6574\u6570\u8868\u793a\u65b9\u5f0f","text":"
    \u6574\u6570\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u53d8\u91cf\u4f7f\u7528\u7684\u5185\u5b58\u957f\u5ea6\uff0c\u5373\u5b57\u8282\uff08\u6216\u6bd4\u7279\uff09\u6570\u3002\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c 1 \u5b57\u8282 (byte) = 8 \u6bd4\u7279 (bit) \uff0c 1 \u6bd4\u7279\u5373 1 \u4e2a\u4e8c\u8fdb\u5236\u4f4d\u3002\u4ee5 int \u7c7b\u578b\u4e3a\u4f8b\uff1a
     
       
    1. \u6574\u6570\u7c7b\u578b int \u5360\u7528 4 bytes = 32 bits \uff0c\u56e0\u6b64\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u4e0d\u540c\u7684\u6570\u5b57\uff1b
      2.  
    2. \u5c06\u6700\u9ad8\u4f4d\u770b\u4f5c\u7b26\u53f7\u4f4d\uff0c\\(0\\) \u4ee3\u8868\u6b63\u6570\uff0c\\(1\\) \u4ee3\u8868\u8d1f\u6570\uff0c\u4ece\u800c\u53ef\u4ee5\u8868\u793a \\(2^{31}\\) \u4e2a\u6b63\u6570\u548c \\(2^{31}\\) \u4e2a\u8d1f\u6570\uff1b
      3.  
    3. \u5f53\u6240\u6709 bits \u4e3a 0 \u65f6\u4ee3\u8868\u6570\u5b57 \\(0\\) \uff0c\u4ece\u96f6\u5f00\u59cb\u589e\u5927\uff0c\u53ef\u5f97\u6700\u5927\u6b63\u6570\u4e3a \\(2^{31} - 1\\) \uff1b
      4.  
    4. \u5269\u4f59 \\(2^{31}\\) \u4e2a\u6570\u5b57\u5168\u90e8\u7528\u6765\u8868\u793a\u8d1f\u6570\uff0c\u56e0\u6b64\u6700\u5c0f\u8d1f\u6570\u4e3a \\(-2^{31}\\) \uff1b\u5177\u4f53\u7ec6\u8282\u6d89\u53ca\u5230\u5230\u201c\u6e90\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u201d\u77e5\u8bc6\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u67e5\u9605\u5b66\u4e60\uff1b  
      5.  
     
    \u5176\u5b83\u6574\u6570\u7c7b\u578b byte, short, long \u53d6\u503c\u8303\u56f4\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e0e int \u7c7b\u4f3c\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_2","title":"\u6d6e\u70b9\u6570\u8868\u793a\u65b9\u5f0f *","text":"
    Note
     
    \u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u540e\u7684 * \u7b26\u53f7\u4ee3\u8868\u9009\u8bfb\u7ae0\u8282\uff0c\u5982\u679c\u4f60\u89c9\u5f97\u7406\u89e3\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u7eed\u518d\u5355\u72ec\u653b\u514b\u3002 
     
    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u7591\u60d1\uff1a int \u548c float \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48 float \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \uff1f\u6309\u8bf4 float \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002
     
    \u5176\u5b9e\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 float \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002IEEE 754 \u6807\u51c6\u89c4\u5b9a\uff0c32-bit \u957f\u5ea6\u7684 float \u7531\u4ee5\u4e0b\u90e8\u5206\u6784\u6210\uff1a
     
       
    • \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff1b
      •  
    • \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff1b
      •  
    • \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 24 bits \uff0c\u5176\u4e2d 23 \u4f4d\u663e\u5f0f\u5b58\u50a8\uff1b
      •  
     
    \u8bbe 32-bit \u4e8c\u8fdb\u5236\u6570\u7684\u7b2c \\(i\\) \u4f4d\u4e3a \\(b_i\\) \uff0c\u5219 float \u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u5b9a\u4e49\u4e3a
     \\[ \\text { val } = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] 
    \u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a
     \\[ \\text { val }=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] 
    \u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a
     \\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\} , \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] 
     
     Fig. IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u8868\u793a\u65b9\u5f0f 
     
    \u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u6613\u5f97
     \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] 
    \u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u5f00\u59cb\u7684\u95ee\u9898\uff1afloat \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c float \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002
     
    \u6d6e\u70b9\u6570 float \u867d\u7136\u62d3\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b int \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 float \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002
     
    \u8fdb\u4e00\u6b65\u5730\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002
     \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) 
    \u7279\u522b\u5730\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u5c0f\u6570\u7cbe\u5ea6\uff1a
     
       
    • \u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\approx 1.18 \\times 10^{-38}\\) \uff1b
      •  
    • \u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23} \\approx 1.4 \\times 10^{-45}\\) \uff1b
      •  
     
    \u53cc\u7cbe\u5ea6 double \u4e5f\u91c7\u7528\u7c7b\u4f3c float \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_3","title":"\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb","text":"
    \u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5b83\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\uff0c\u800c\u4e0d\u662f\u201c\u6570\u636e\u201d\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u81ea\u7136\u60f3\u5230\u4f7f\u7528\u300c\u6570\u7ec4\u300d\u6570\u636e\u7ed3\u6784\u3002\u6570\u7ec4\u7684\u5b58\u50a8\u65b9\u5f0f\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u3001\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5176\u4e2d\u5b58\u50a8\u7684\u662f\u6574\u6570 int \uff0c\u8fd8\u662f\u5c0f\u6570 float \uff0c\u6216\u662f\u5b57\u7b26 char \uff0c\u5219\u4e0e\u6240\u8c13\u7684\u6570\u636e\u7684\u7ed3\u6784\u65e0\u5173\u4e86\u3002
     
    \u6362\u8a00\u4e4b\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] booleans = new boolean[5];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool booleans[5];\n
     
    \"\"\" Python \u7684 list \u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 \"\"\"\nlist = [0, 0.0, 'a', False]\n
     
    // \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar booleans = [5]bool{}\n
     
    /* JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 */\nconst array = [0, 0.0, 'a', false];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst booleans: boolean[] = [];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool booleans[10];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] booleans = new bool[5];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet booleans = Array(repeating: Bool(), count: 5)\n
     
    \n
    "},{"location":"chapter_data_structure/data_and_memory/#312","title":"3.1.2. \u00a0 \u8ba1\u7b97\u673a\u5185\u5b58","text":"
    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u5185\u5b58\u548c\u786c\u76d8\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u5b58\u50a8\u786c\u4ef6\u8bbe\u5907\u3002\u300c\u786c\u76d8\u300d\u4e3b\u8981\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5bb9\u91cf\u8f83\u5927\uff08\u901a\u5e38\u53ef\u8fbe\u5230 TB \u7ea7\u522b\uff09\u3001\u901f\u5ea6\u8f83\u6162\u3002\u300c\u5185\u5b58\u300d\u7528\u4e8e\u8fd0\u884c\u7a0b\u5e8f\u65f6\u6682\u5b58\u6570\u636e\uff0c\u901f\u5ea6\u8f83\u5feb\uff0c\u4f46\u5bb9\u91cf\u8f83\u5c0f\uff08\u901a\u5e38\u4e3a GB \u7ea7\u522b\uff09\u3002
     
    \u7b97\u6cd5\u8fd0\u884c\u4e2d\uff0c\u76f8\u5173\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8 1 byte \u7684\u6570\u636e\uff0c\u5728\u7b97\u6cd5\u8fd0\u884c\u65f6\uff0c\u6240\u6709\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u8fd9\u4e9b\u5355\u5143\u683c\u4e2d\u3002
     
    \u7cfb\u7edf\u901a\u8fc7\u300c\u5185\u5b58\u5730\u5740 Memory Location\u300d\u6765\u8bbf\u95ee\u76ee\u6807\u5185\u5b58\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u7ed9\u8868\u683c\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u7f16\u53f7\uff0c\u4fdd\u8bc1\u6bcf\u5757\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u72ec\u7acb\u7684\u5185\u5b58\u5730\u5740\u3002\u81ea\u6b64\uff0c\u7a0b\u5e8f\u4fbf\u901a\u8fc7\u8fd9\u4e9b\u5730\u5740\uff0c\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002
     
     
     Fig. \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 
     
    \u5185\u5b58\u8d44\u6e90\u662f\u8bbe\u8ba1\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u8003\u8651\u56e0\u7d20\u3002\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5f53\u5185\u5b58\u88ab\u67d0\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u4e0d\u80fd\u88ab\u5176\u5b83\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u3002\u6211\u4eec\u9700\u8981\u6839\u636e\u5269\u4f59\u5185\u5b58\u8d44\u6e90\u7684\u60c5\u51b5\u6765\u8bbe\u8ba1\u7b97\u6cd5\u3002\u4f8b\u5982\uff0c\u82e5\u5269\u4f59\u5185\u5b58\u7a7a\u95f4\u6709\u9650\uff0c\u5219\u8981\u6c42\u7b97\u6cd5\u5360\u7528\u7684\u5cf0\u503c\u5185\u5b58\u4e0d\u80fd\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u5185\u5b58\uff1b\u82e5\u8fd0\u884c\u7684\u7a0b\u5e8f\u5f88\u591a\u3001\u7f3a\u5c11\u5927\u5757\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u5219\u8981\u6c42\u9009\u53d6\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u79bb\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002
    "},{"location":"chapter_data_structure/summary/","title":"3.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6574\u6570 byte, short, int, long \u3001\u6d6e\u70b9\u6570 float, double \u3001\u5b57\u7b26 char \u3001\u5e03\u5c14 boolean \u662f\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u51b3\u5b9a\u4e86\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u3002
      •  
    • \u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u7684\u5185\u5b58\u4e2d\u3002\u5185\u5b58\u4e2d\u6bcf\u5757\u7a7a\u95f4\u90fd\u6709\u72ec\u7acb\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u662f\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u6570\u636e\u7684\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\u4e3b\u8981\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u4e2d\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u5f62\u5f0f\u3002
      •  
    • \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u6709\u7ebf\u6027\u3001\u6811\u72b6\u3001\u7f51\u72b6\u7b49\u3002\u6211\u4eec\u4e00\u822c\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u6839\u636e\u5b9e\u73b0\u65b9\u5f0f\u7684\u4e0d\u540c\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u662f\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u3002
      •  
    • \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u6709\u4e24\u79cd\uff0c\u5206\u522b\u662f\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\uff0c\u6240\u6709\u7684\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u6216\u94fe\u8868\u3001\u6216\u4e24\u8005\u7ec4\u5408\u5b9e\u73b0\u7684\u3002
      •  
    "},{"location":"chapter_graph/graph/","title":"9.1. \u00a0 \u56fe","text":"
    \u300c\u56fe Graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 Vertex\u300d\u548c\u300c\u8fb9 Edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4f8b\u5982\uff0c\u4ee5\u4e0b\u8868\u793a\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe
     \\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\] 
     
     Fig. \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb 
     
    \u90a3\u4e48\uff0c\u56fe\u4e0e\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb\u662f\u4ec0\u4e48\uff1f\u5982\u679c\u6211\u4eec\u628a\u300c\u9876\u70b9\u300d\u770b\u4f5c\u7ed3\u70b9\uff0c\u628a\u300c\u8fb9\u300d\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u7ed3\u70b9\u7684\u6307\u9488\uff0c\u5219\u53ef\u5c06\u300c\u56fe\u300d\u770b\u6210\u4e00\u79cd\u4ece\u300c\u94fe\u8868\u300d\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u76f8\u6bd4\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4e5f\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002
    "},{"location":"chapter_graph/graph/#911","title":"9.1.1. \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b","text":"
    \u6839\u636e\u8fb9\u662f\u5426\u6709\u65b9\u5411\uff0c\u5206\u4e3a\u300c\u65e0\u5411\u56fe Undirected Graph\u300d\u548c\u300c\u6709\u5411\u56fe Directed Graph\u300d\u3002
     
       
    • \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u201c\u53cc\u5411\u201d\u7684\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\uff1b
      •  
    • \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u662f\u6709\u65b9\u5411\u7684\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\uff1b
      •  
     
     
     Fig. \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe 
     
    \u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u5206\u4e3a\u300c\u8fde\u901a\u56fe Connected Graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe Disconnected Graph\u300d\u3002
     
       
    • \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\uff1b
      •  
    • \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\uff1b
      •  
     
     
     Fig. \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe 
     
    \u6211\u4eec\u53ef\u4ee5\u7ed9\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u5f97\u5230\u300c\u6709\u6743\u56fe Weighted Graph\u300d\u3002\u4f8b\u5982\uff0c\u5728\u738b\u8005\u8363\u8000\u7b49\u6e38\u620f\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u4f7f\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002
     
     
     Fig. \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe 
    "},{"location":"chapter_graph/graph/#912","title":"9.1.2. \u00a0 \u56fe\u5e38\u7528\u672f\u8bed","text":"
       
    • \u300c\u90bb\u63a5 Adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u6b64\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u4f8b\u5982\uff0c\u4e0a\u56fe\u4e2d\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2, 3, 5 \u3002
      •  
    • \u300c\u8def\u5f84 Path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u8d70\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\uff0c\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u4f8b\u5982\uff0c\u4e0a\u56fe\u4e2d\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u4e2a\u8def\u5f84\u3002
      •  
    • \u300c\u5ea6 Degree\u300d\u8868\u793a\u4e00\u4e2a\u9876\u70b9\u5177\u6709\u591a\u5c11\u6761\u8fb9\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 In-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 Out-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
      •  
    "},{"location":"chapter_graph/graph/#913","title":"9.1.3. \u00a0 \u56fe\u7684\u8868\u793a","text":"
    \u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u6cd5\u6709\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u3002\u4ee5\u4e0b\u4f7f\u7528\u300c\u65e0\u5411\u56fe\u300d\u6765\u4e3e\u4f8b\u3002
    "},{"location":"chapter_graph/graph/#_1","title":"\u90bb\u63a5\u77e9\u9635","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 Adjacency Matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u4f7f\u7528 \\(1\\) \u6216 \\(0\\) \u6765\u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8bb0\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\) \u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u5219\u77e9\u9635\u5143\u7d20 \\(M[i][j] = 1\\) \u4ee3\u8868\u7740\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u6709\u8fb9\uff0c\u76f8\u53cd\u5730 \\(M[i][j] = 0\\) \u4ee3\u8868\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002
     
     
     Fig. \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a 
     
    \u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a
     
       
    • \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u800c\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
      •  
    • \u300c\u65e0\u5411\u56fe\u300d\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
      •  
    • \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) , \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u80fd\u591f\u8868\u793a\u300c\u6709\u6743\u56fe\u300d\u3002
      •  
     
    \u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u6765\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u5927\u3002
    "},{"location":"chapter_graph/graph/#_2","title":"\u90bb\u63a5\u8868","text":"
    \u300c\u90bb\u63a5\u8868 Adjacency List\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u7ed3\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u5373\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002
     
     
     Fig. \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a 
     
    \u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u5f80\u5f80\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u4f46\u662f\uff0c\u56e0\u4e3a\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u6240\u4ee5\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002
     
    \u89c2\u5bdf\u4e0a\u56fe\u53d1\u73b0\uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u300c\u94fe\u5730\u5740\u6cd5\u300d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u7c7b\u4f3c\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\uff0c\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u4e2d\u5e8f\u904d\u5386\u83b7\u53d6\u6709\u5e8f\u5e8f\u5217\uff1b\u8fd8\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a\u54c8\u5e0c\u8868\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u4f4e\u81f3 \\(O(1)\\) \u3002
    "},{"location":"chapter_graph/graph/#914","title":"9.1.4. \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"
    \u73b0\u5b9e\u4e2d\u7684\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u4f7f\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u5bf9\u5e94\u7684\u5f85\u6c42\u89e3\u95ee\u9898\u4e5f\u53ef\u4ee5\u88ab\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002
     \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2. \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"
    \u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u5206\u4e3a\u5bf9\u300c\u8fb9\u300d\u7684\u64cd\u4f5c\u548c\u5bf9\u300c\u9876\u70b9\u300d\u7684\u64cd\u4f5c\uff0c\u5728\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u8fd9\u4e24\u79cd\u8868\u793a\u4e0b\u7684\u5b9e\u73b0\u65b9\u5f0f\u4e0d\u540c\u3002
    "},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6709\uff1a
     
       
    • \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u8fb9\u7684\u5bf9\u5e94\u5143\u7d20\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
      •  
    • \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
      •  
    • \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 vertices \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 adjMat \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
      •  
     \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_adjacency_matrix.java
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<Integer> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new ArrayList<>();\nthis.adjMat = new ArrayList<>();\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (int[] e : edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<Integer> newRow = new ArrayList<>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.add(0);\n}\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<Integer> row : adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfBoundsException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<Integer> row : adjMat) {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat.get(i).set(j, 1);\nadjMat.get(j).set(i, 1);\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\nadjMat.get(i).set(j, 0);\nadjMat.get(j).set(i, 0);\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nSystem.out.print(\"\u9876\u70b9\u5217\u8868 = \");\nSystem.out.println(vertices);\nSystem.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
     graph_adjacency_matrix.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvector<int> vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(const vector<int>& vertices, const vector<vector<int>>& edges) {\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const vector<int>& edge : edges) {\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() const {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.push_back(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nadjMat.emplace_back(n, 0);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (vector<int>& row : adjMat) {\nrow.push_back(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.erase(vertices.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.erase(adjMat.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (vector<int>& row : adjMat) {\nrow.erase(row.begin() + index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid print() {\ncout << \"\u9876\u70b9\u5217\u8868 = \";\nPrintUtil::printVector(vertices);\ncout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\nPrintUtil::printVectorMatrix(adjMat);\n}\n};\n
     graph_adjacency_matrix.py
    class GraphAdjMat:\n\"\"\" \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b \"\"\"\n# \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices: List[int] = []\n# \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadj_mat: List[List[int]] = []\ndef __init__(self, vertices: List[int], edges: List[List[int]]) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.vertices: List[int] = []\nself.adj_mat: List[List[int]] = []\n# \u6dfb\u52a0\u9876\u70b9\nfor val in vertices:\nself.add_vertex(val)\n# \u6dfb\u52a0\u8fb9\n# \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges:\nself.add_edge(e[0], e[1])\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u9876\u70b9\u6570\u91cf \"\"\"\nreturn len(self.vertices)\ndef add_vertex(self, val: int) -> None:\n\"\"\" \u6dfb\u52a0\u9876\u70b9 \"\"\"\nn = self.size()\n# \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.append(val)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnew_row = [0] * n\nself.adj_mat.append(new_row)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in self.adj_mat:\nrow.append(0)\ndef remove_vertex(self, index: int) -> None:\n\"\"\" \u5220\u9664\u9876\u70b9 \"\"\"\nif index >= self.size():\nraise IndexError()\n# \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in self.adj_mat:\nrow.pop(index)\ndef add_edge(self, i: int, j: int) -> None:\n\"\"\" \u6dfb\u52a0\u8fb9 \"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\n# \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1\nself.adj_mat[j][i] = 1\ndef remove_edge(self, i: int, j: int) -> None:\n\"\"\" \u5220\u9664\u8fb9 \"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\nself.adj_mat[i][j] = 0\nself.adj_mat[j][i] = 0\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u90bb\u63a5\u77e9\u9635 \"\"\"\nprint(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\nprint(\"\u90bb\u63a5\u77e9\u9635 =\")\nprint_matrix(self.adj_mat)\n
     graph_adjacency_matrix.go
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices []int\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat [][]int\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n// \u6dfb\u52a0\u9876\u70b9\nn := len(vertices)\nadjMat := make([][]int, n)\nfor i := range adjMat {\nadjMat[i] = make([]int, n)\n}\n// \u521d\u59cb\u5316\u56fe\ng := &graphAdjMat{\nvertices: vertices,\nadjMat:   adjMat,\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor i := range edges {\ng.addEdge(edges[i][0], edges[i][1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\nreturn len(g.vertices)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\nn := g.size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\ng.vertices = append(g.vertices, val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnewRow := make([]int, n)\ng.adjMat = append(g.adjMat, newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i], 0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\nif index >= g.size() {\nreturn\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\ng.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\ng.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\ng.adjMat[i][j] = 1\ng.adjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\ng.adjMat[i][j] = 0\ng.adjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\nfmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\nfmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\nfor i := range g.adjMat {\nfmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n}\n}\n
     graph_adjacency_matrix.js
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices;   // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat;     // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices, edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val) {\nconst n = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow = [];\nfor (let j = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index) {\nif (index >= this.size()) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint() {\nconsole.log(\"\u9876\u70b9\u5217\u8868 = \", this.vertices);\nconsole.log(\"\u90bb\u63a5\u77e9\u9635 =\", this.adjMat);\n}\n}\n
     graph_adjacency_matrix.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices: number[];     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat: number[][];     // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices: number[], edges: number[][]) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val: number): void {\nconst n: number = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow: number[] = [];\nfor (let j: number = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index: number): void {\nif (index >= this.size()) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint(): void {\nconsole.log(\"\u9876\u70b9\u5217\u8868 = \", this.vertices);\nconsole.log(\"\u90bb\u63a5\u77e9\u9635 =\", this.adjMat);\n}\n}\n
     graph_adjacency_matrix.c
    [class]{graphAdjMat}-[func]{}\n
     graph_adjacency_matrix.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat\n{\nList<int> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjMat(int[] vertices, int[][] edges)\n{\nthis.vertices = new List<int>();\nthis.adjMat = new List<List<int>>();\n// \u6dfb\u52a0\u9876\u70b9\nforeach (int val in vertices)\n{\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nforeach (int[] e in edges)\n{\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size()\n{\nreturn vertices.Count;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val)\n{\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.Add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = new List<int>(n);\nfor (int j = 0; j < n; j++)\n{\nnewRow.Add(0);\n}\nadjMat.Add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nforeach (List<int> row in adjMat)\n{\nrow.Add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index)\n{\nif (index >= size())\nthrow new IndexOutOfRangeException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nforeach (List<int> row in adjMat)\n{\nrow.RemoveAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j)\n{\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j)\n{\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print()\n{\nConsole.Write(\"\u9876\u70b9\u5217\u8868 = \");\nPrintUtil.PrintList(vertices);\nConsole.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
     graph_adjacency_matrix.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nprivate var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nprivate var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(vertices: [Int], edges: [[Int]]) {\nself.vertices = []\nadjMat = []\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\naddVertex(val: val)\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges {\naddEdge(i: e[0], j: e[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc size() -> Int {\nvertices.count\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc addVertex(val: Int) {\nlet n = size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.append(val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nlet newRow = Array(repeating: 0, count: n)\nadjMat.append(newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i in adjMat.indices {\nadjMat[i].append(0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc removeVertex(index: Int) {\nif index >= size() {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i in adjMat.indices {\nadjMat[i].remove(at: index)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc addEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1\nadjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc removeEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\nadjMat[i][j] = 0\nadjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc print() {\nSwift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\nSwift.print(vertices)\nSwift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\nPrintUtil.printMatrix(matrix: adjMat)\n}\n}\n
     graph_adjacency_matrix.zig
    \n
    "},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u6709\uff1a
     
       
    • \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u5c3e\u90e8\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u8be2\u4e0e\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u4e0e\u6dfb\u52a0\u8fb9\u4e00\u6837\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\u5373\u53ef\uff0c\u5e76\u4ee5\u65b0\u589e\u9876\u70b9\u4e3a\u94fe\u8868\u5934\u7ed3\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
      •  
    • \u5220\u9664\u9876\u70b9\uff1a\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
      •  
    • \u521d\u59cb\u5316\uff1a\u9700\u8981\u5728\u90bb\u63a5\u8868\u4e2d\u5efa\u7acb \\(n\\) \u4e2a\u7ed3\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
      •  
     \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 
     
     
     
     
     
    \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u56fe\u7684\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u6ce8\u610f\u5230\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 Vertex \u7ed3\u70b9\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a
     
       
    • \u5982\u679c\u6211\u4eec\u9009\u62e9\u901a\u8fc7\u9876\u70b9\u503c\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u503c\u91cd\u590d\u7684\u9876\u70b9\u5c06\u65e0\u6cd5\u88ab\u533a\u5206\u3002
      •  
    • \u5982\u679c\u7c7b\u4f3c\u90bb\u63a5\u77e9\u9635\u90a3\u6837\uff0c\u4f7f\u7528\u9876\u70b9\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\u3002\u90a3\u4e48\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u5176\u4e2d \\(> i\\) \u7684\u7d22\u5f15\u5168\u90e8\u6267\u884c \\(-1\\) \uff0c\u8fd9\u6837\u64cd\u4f5c\u6548\u7387\u592a\u4f4e\u3002
      •  
    • \u56e0\u6b64\u6211\u4eec\u8003\u8651\u5f15\u5165\u9876\u70b9\u7c7b Vertex \uff0c\u4f7f\u5f97\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684\u5bf9\u8c61\uff0c\u6b64\u65f6\u5220\u9664\u9876\u70b9\u65f6\u5c31\u65e0\u9700\u6539\u52a8\u5176\u4f59\u9876\u70b9\u4e86\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_adjacency_list.java
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new HashMap<>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (Vertex[] edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList.get(vet1).add(vet2);\nadjList.get(vet2).add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList.get(vet1).remove(vet2);\nadjList.get(vet2).remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.containsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.put(vet, new ArrayList<>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet))\nthrow new IllegalArgumentException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (List<Vertex> list : adjList.values()) {\nlist.remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nSystem.out.println(\"\u90bb\u63a5\u8868 =\");\nfor (Map.Entry<Vertex, List<Vertex>> entry : adjList.entrySet()) {\nList<Integer> tmp = new ArrayList<>();\nfor (Vertex vertex : entry.getValue())\ntmp.add(vertex.val);\nSystem.out.println(entry.getKey().val + \": \" + tmp + \",\");\n}\n}\n}\n
     graph_adjacency_list.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\npublic:\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nunordered_map<Vertex*, vector<Vertex*>> adjList;\n/* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u7ed3\u70b9 */\nvoid remove(vector<Vertex*> &vec, Vertex *vet) {\nfor (int i = 0; i < vec.size(); i++) {\nif (vec[i] == vet) {\nvec.erase(vec.begin() + i);\nbreak;\n}\n}\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(const vector<vector<Vertex*>>& edges) {\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const vector<Vertex*>& edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() { return adjList.size(); }\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex* vet1, Vertex* vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].push_back(vet2);\nadjList[vet2].push_back(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex* vet1, Vertex* vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5220\u9664\u8fb9 vet1 - vet2\nremove(adjList[vet1], vet2);\nremove(adjList[vet2], vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex* vet) {\nif (adjList.count(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = vector<Vertex*>();\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex* vet) {\nif (!adjList.count(vet))\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.erase(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (auto& [key, vec] : adjList) {\nremove(vec, vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid print() {\ncout << \"\u90bb\u63a5\u8868 =\" << endl;\nfor (auto& adj  : adjList) {\nconst auto& key= adj.first;\nconst auto& vec = adj.second;\ncout << key->val << \": \";\nPrintUtil::printVector(vetsToVals(vec));\n}\n}\n};\n
     graph_adjacency_list.py
    class GraphAdjList:\n\"\"\" \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b \"\"\"\ndef __init__(self, edges: List[List[Vertex]]) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nself.adj_list: Dict = {}\n# \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges:\nself.add_vertex(edge[0])\nself.add_vertex(edge[1])\nself.add_edge(edge[0], edge[1])\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u9876\u70b9\u6570\u91cf \"\"\"\nreturn len(self.adj_list)\ndef add_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\" \u6dfb\u52a0\u8fb9 \"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError\n# \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list[vet1].append(vet2)\nself.adj_list[vet2].append(vet1)\ndef remove_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\" \u5220\u9664\u8fb9 \"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError\n# \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list[vet1].remove(vet2)\nself.adj_list[vet2].remove(vet1)\ndef add_vertex(self, vet: Vertex) -> None:\n\"\"\" \u6dfb\u52a0\u9876\u70b9 \"\"\"\nif vet in self.adj_list:\nreturn\n# \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list[vet] = []\ndef remove_vertex(self, vet: Vertex) -> None:\n\"\"\" \u5220\u9664\u9876\u70b9 \"\"\"\nif vet not in self.adj_list:\nraise ValueError\n# \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.pop(vet)\n# \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor vertex in self.adj_list:\nif vet in self.adj_list[vertex]:\nself.adj_list[vertex].remove(vet)\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u90bb\u63a5\u8868 \"\"\"\nprint(\"\u90bb\u63a5\u8868 =\")\nfor vertex in self.adj_list:\ntmp = [v.val for v in self.adj_list[vertex]]\nprint(f\"{vertex.val}: {tmp},\")\n
     graph_adjacency_list.go
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList map[Vertex][]Vertex\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\ng := &graphAdjList{\nadjList: make(map[Vertex][]Vertex),\n}\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor _, edge := range edges {\ng.addVertex(edge[0])\ng.addVertex(edge[1])\ng.addEdge(edge[0], edge[1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\nreturn len(g.adjList)\n}\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\ng.adjList[vet1] = append(g.adjList[vet1], vet2)\ng.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nDeleteSliceElms(g.adjList[vet1], vet2)\nDeleteSliceElms(g.adjList[vet2], vet1)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif ok {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ng.adjList[vet] = make([]Vertex, 0)\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif !ok {\npanic(\"error\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\ndelete(g.adjList, vet)\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor _, list := range g.adjList {\nDeleteSliceElms(list, vet)\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\nvar builder strings.Builder\nfmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\nfor k, v := range g.adjList {\nbuilder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\nfor _, vet := range v {\nbuilder.WriteString(strconv.Itoa(vet.Val) + \" \")\n}\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
     graph_adjacency_list.js
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1, vet2) {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1, vet2) {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet) {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet) {\nif (!this.adjList.has(vet)) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint() {\nconsole.log(\"\u90bb\u63a5\u8868 =\");\nfor (const [key, value] of this.adjList) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + \": \" + tmp.join());\n}\n}\n}\n
     graph_adjacency_list.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList: Map<Vertex, Vertex[]>;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges: Vertex[][]) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1: Vertex, vet2: Vertex): void {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1: Vertex, vet2: Vertex): void {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet: Vertex): void {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet: Vertex): void {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index: number = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint(): void {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList.entries()) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
     graph_adjacency_list.c
    [class]{graphAdjList}-[func]{}\n
     graph_adjacency_list.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList\n{\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic Dictionary<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjList(Vertex[][] edges)\n{\nthis.adjList = new Dictionary<Vertex, List<Vertex>>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nforeach (Vertex[] edge in edges)\n{\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size()\n{\nreturn adjList.Count;\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2)\n{\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].Add(vet2);\nadjList[vet2].Add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2)\n{\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1].Remove(vet2);\nadjList[vet2].Remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet)\n{\nif (adjList.ContainsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.Add(vet, new List<Vertex>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet)\n{\nif (!adjList.ContainsKey(vet))\nthrow new InvalidOperationException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.Remove(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nforeach (List<Vertex> list in adjList.Values)\n{\nlist.Remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print()\n{\nConsole.WriteLine(\"\u90bb\u63a5\u8868 =\");\nforeach (KeyValuePair<Vertex, List<Vertex>> entry in adjList)\n{\nList<int> tmp = new List<int>();\nforeach (Vertex vertex in entry.Value)\ntmp.Add(vertex.Val);\nConsole.WriteLine(entry.Key.Val + \": [\" + string.Join(\", \", tmp) + \"],\");\n}\n}\n}\n
     graph_adjacency_list.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic private(set) var adjList: [Vertex: [Vertex]]\n/* \u6784\u9020\u65b9\u6cd5 */\npublic init(edges: [[Vertex]]) {\nadjList = [:]\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\naddVertex(vet: edge[0])\naddVertex(vet: edge[1])\naddEdge(vet1: edge[0], vet2: edge[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic func size() -> Int {\nadjList.count\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic func addEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]?.append(vet2)\nadjList[vet2]?.append(vet1)\n}\n/* \u5220\u9664\u8fb9 */\npublic func removeEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]?.removeAll(where: { $0 == vet2 })\nadjList[vet2]?.removeAll(where: { $0 == vet1 })\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic func addVertex(vet: Vertex) {\nif adjList[vet] != nil {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = []\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic func removeVertex(vet: Vertex) {\nif adjList[vet] == nil {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.removeValue(forKey: vet)\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor key in adjList.keys {\nadjList[key]?.removeAll(where: { $0 == vet })\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic func print() {\nSwift.print(\"\u90bb\u63a5\u8868 =\")\nfor entry in adjList {\nvar tmp: [Int] = []\nfor vertex in entry.value {\ntmp.append(vertex.val)\n}\nSwift.print(\"\\(entry.key.val): \\(tmp),\")\n}\n}\n}\n
     graph_adjacency_list.zig
    [class]{GraphAdjList}-[func]{}\n
    "},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3. \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"
    \u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u4e0b\u8868\u4e3a\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\u3002
     \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) 
    \u89c2\u5bdf\u4e0a\u8868\uff0c\u8c8c\u4f3c\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u603b\u7ed3\u4ee5\u4e0a\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
    "},{"location":"chapter_graph/graph_traversal/","title":"9.3. \u00a0 \u56fe\u7684\u904d\u5386","text":"
    \u56fe\u4e0e\u6811\u7684\u5173\u7cfb
     
    \u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u53ef\u4ee5\u4ee3\u8868\u4efb\u610f\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u672c\u8d28\u4e0a\uff0c\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u662f\u56fe\u7684\u4e00\u7c7b\u7279\u4f8b\u3002\u90a3\u4e48\u663e\u7136\uff0c\u6811\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u4e24\u8005\u7684\u65b9\u6cd5\u662f\u975e\u5e38\u7c7b\u4f3c\u7684\uff0c\u5efa\u8bae\u4f60\u5728\u5b66\u4e60\u672c\u7ae0\u8282\u7684\u8fc7\u7a0b\u4e2d\u5c06\u4e24\u8005\u878d\u4f1a\u8d2f\u901a\u3002
     
    \u300c\u56fe\u300d\u4e0e\u300c\u6811\u300d\u90fd\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u641c\u7d22\u7b97\u6cd5\u300d\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002
     
    \u7c7b\u4f3c\u5730\uff0c\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u5206\u4e3a\u4e24\u79cd\uff0c\u5373\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 Breadth-First Traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Travsersal\u300d\uff0c\u4e5f\u79f0\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 Depth-First Search\u300d\uff0c\u7b80\u79f0\u4e3a BFS \u548c DFS \u3002
    "},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1. \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u4f18\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\u5f00\u59cb\u8bbf\u95ee\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u968f\u540e\u904d\u5386\u4e0b\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026
     
     
     Fig. \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 
    "},{"location":"chapter_graph/graph_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    BFS \u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u662f\u5f02\u66f2\u540c\u5de5\u7684\u3002
     
       
    1. \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 startVet \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\uff1b
      2.  
    2. \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5f39\u51fa\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u5e76\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\uff1b
      3.  
    3. \u5faa\u73af 2. \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6210\u540e\u7ed3\u675f\uff1b
      4.  
     
    \u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u54ea\u4e9b\u7ed3\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_bfs.java
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>() {{ add(startVet); }};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new LinkedList<>() {{ offer(startVet); }};\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.isEmpty()) {\nVertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.cpp
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex*> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex*> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex*> visited = { startVet };\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nqueue<Vertex*> que;\nque.push(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.empty()) {\nVertex *vet = que.front();\nque.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (auto adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;           // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.push(adjVet);       // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.py
    def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:\n\"\"\" \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS \"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set([start_vet])\n# \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nque = collections.deque([start_vet])\n# \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile len(que) > 0:\nvet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet)      # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adj_vet in graph.adj_list[vet]:\nif adj_vet in visited:\ncontinue         # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adj_vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n
     graph_bfs.go
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\nvisited[startVet] = struct{}{}\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\nqueue := make([]Vertex, 0)\nqueue = append(queue, startVet)\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nfor len(queue) > 0 {\n// \u961f\u9996\u9876\u70b9\u51fa\u961f\nvet := queue[0]\nqueue = queue[1:]\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres = append(res, vet)\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nif !isExist {\nqueue = append(queue, adjVet)\nvisited[adjVet] = struct{}{}\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_bfs.js
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift();    // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet);              // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue;           // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet);       // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet);    // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.ts
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.c
    [class]{}-[func]{graphBFS}\n
     graph_bfs.cs
    [class]{graph_bfs}-[func]{graphBFS}\n
     graph_bfs.swift
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = [startVet]\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nvar que: [Vertex] = [startVet]\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.isEmpty {\nlet vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_bfs.zig
    [class]{}-[func]{graphBFS}\n
     
    \u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u4ee5\u4e0b\u52a8\u753b\u56fe\u793a\u6765\u52a0\u6df1\u7406\u89e3\u3002
     <1><2><3><4><5><6><7><8><9><10><11> 
     
     
     
     
     
     
     
     
     
     
     
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
     
    \u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\) , \\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u3001\u9876\u70b9 \\(2\\) , \\(4\\) , \\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3001\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026
    "},{"location":"chapter_graph/graph_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u3001\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \uff0c\u961f\u5217 que \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2. \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u4e0d\u65ad\u5730\u8bbf\u95ee\u5f53\u524d\u7ed3\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u56de\u6eaf\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5e95 + \u56de\u6eaf\uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u65f6\u7ed3\u675f\u3002
     
     
     Fig. \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 
    "},{"location":"chapter_graph/graph_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    \u8fd9\u79cd\u201c\u8d70\u5230\u5934 + \u56de\u6eaf\u201d\u7684\u7b97\u6cd5\u5f62\u5f0f\u4e00\u822c\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e BFS \u7c7b\u4f3c\uff0c\u5728 DFS \u4e2d\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_dfs.java
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.cpp
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList& graph, unordered_set<Vertex*>& visited, vector<Vertex*>& res, Vertex* vet) {\nres.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex* adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;     // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex*> graphDFS(GraphAdjList& graph, Vertex* startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex*> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex*> visited;\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.py
    def dfs(graph: GraphAdjList, visited: Set[Vertex], res: List[Vertex], vet: Vertex):\n\"\"\" \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 \"\"\"\nres.append(vet)     # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adj_list[vet]:\nif adjVet in visited:\ncontinue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n# \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet)\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:\n\"\"\" \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \"\"\"\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set()\ndfs(graph, visited, res, start_vet)\nreturn res\n
     graph_dfs.go
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n// append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n*res = append(*res, vet)\nvisited[vet] = struct{}{}\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nif !isExist {\ndfs(g, visited, res, adjVet)\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\ndfs(g, visited, &res, startVet)\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_dfs.js
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\nres.push(vet);      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.ts
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(graph: GraphAdjList, visited: Set<Vertex>, res: Vertex[], vet: Vertex): void {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
     graph_dfs.cs
    [class]{graph_dfs}-[func]{dfs}\n[class]{graph_dfs}-[func]{graphDFS}\n
     graph_dfs.swift
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = []\ndfs(graph: graph, visited: &visited, res: &res, vet: startVet)\nreturn res\n}\n
     graph_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
     
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5176\u4e2d
     
       
    • \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u4ee3\u8868\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\uff1b
      •  
    • \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u4ee3\u8868\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u9012\u5f52\u65b9\u6cd5\u7684\u4f4d\u7f6e\uff1b
      •  
     
    \u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6a21\u62df\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002
     <1><2><3><4><5><6><7><8><9><10><11> 
     
     
     
     
     
     
     
     
     
     
     
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
     
    \u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u884c\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
     
    \u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u4f53\u73b0\u4e09\u79cd\u4e0d\u540c\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u800c\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
    "},{"location":"chapter_graph/graph_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u88ab\u8bbf\u95ee\u4e00\u6b21\uff1b\u6240\u6709\u8fb9\u90fd\u88ab\u8bbf\u95ee\u4e86 \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_graph/summary/","title":"9.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
      •  
    • \u76f8\u6bd4\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4e5f\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002
      •  
    • \u6709\u5411\u56fe\u7684\u8fb9\u5b58\u5728\u65b9\u5411\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u90fd\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
      •  
    • \u90bb\u63a5\u77e9\u9635\u4f7f\u7528\u65b9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u4f7f\u7528 \\(1\\) \u6216 \\(0\\) \u6765\u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u7684\u589e\u5220\u67e5\u64cd\u4f5c\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u5360\u7528\u7a7a\u95f4\u5927\u3002
      •  
    • \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
      •  
    • \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
      •  
    • \u4ece\u7b97\u6cd5\u601d\u60f3\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d
      •  
    • \u56fe\u53ef\u4ee5\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u4f8b\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
      •  
    • \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
      •  
    • \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
      •  
    • \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_hashing/hash_collision/","title":"6.2. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570\u5e94\u8be5\u4e3a\u6bcf\u4e2a\u8f93\u5165\u4ea7\u751f\u552f\u4e00\u7684\u8f93\u51fa\uff0c\u4f7f\u5f97 key \u548c value \u4e00\u4e00\u5bf9\u5e94\u3002\u800c\u5b9e\u9645\u4e0a\uff0c\u5f80\u5f80\u5b58\u5728\u5411\u54c8\u5e0c\u51fd\u6570\u8f93\u5165\u4e0d\u540c\u7684 key \u800c\u4ea7\u751f\u76f8\u540c\u8f93\u51fa\u7684\u60c5\u51b5\uff0c\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4ece\u800c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002
     
    \u90a3\u4e48\uff0c\u4e3a\u4ec0\u4e48\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f\u672c\u8d28\u4e0a\u770b\uff0c\u7531\u4e8e\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u53ef\u907f\u514d\u5730\u4f1a\u51fa\u73b0\u591a\u4e2a\u8f93\u5165\u4ea7\u751f\u76f8\u540c\u8f93\u51fa\u7684\u60c5\u51b5\uff0c\u5373\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u4e00\u4e2a\u56fa\u5b9a\u5927\u5c0f\u7684\u6570\u7ec4\uff0c\u90a3\u4e48\u5fc5\u5b9a\u4f1a\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u5230\u540c\u4e00\u4e2a\u6570\u7ec4\u7d22\u5f15\u3002
     
    \u4e3a\u4e86\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff0c\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u8868\u6269\u5bb9\u6765\u51cf\u5c0f\u51b2\u7a81\u6982\u7387\u3002\u6781\u7aef\u60c5\u51b5\u4e0b\uff0c\u5f53\u8f93\u5165\u7a7a\u95f4\u548c\u8f93\u51fa\u7a7a\u95f4\u5927\u5c0f\u76f8\u7b49\u65f6\uff0c\u54c8\u5e0c\u8868\u5c31\u7b49\u4ef7\u4e8e\u6570\u7ec4\u4e86\uff0c\u6bcf\u4e2a key \u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u53ef\u8c13\u201c\u5927\u529b\u51fa\u5947\u8ff9\u201d\u3002
     
    \u53e6\u4e00\u65b9\u9762\uff0c\u8003\u8651\u901a\u8fc7\u4f18\u5316\u54c8\u5e0c\u8868\u7684\u8868\u793a\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e38\u89c1\u7684\u65b9\u6cd5\u6709\u300c\u94fe\u5f0f\u5730\u5740\u300d\u548c\u300c\u5f00\u653e\u5bfb\u5740\u300d\u3002
    "},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1. \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9","text":"
    \u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6876\u6570\u91cf \\(n\\) \u53d6\u4f59\uff0c\u4ee5\u5c06\u54c8\u5e0c\u503c\u6620\u5c04\u5230\u6876\u7684\u7d22\u5f15\u8303\u56f4\uff0c\u4ece\u800c\u5c06 key \u653e\u5165\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff08\u5373 \\(n\\) \u8d8a\u5927\uff09\u65f6\uff0c\u591a\u4e2a key \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002
     
    \u56e0\u6b64\uff0c\u5728\u54c8\u5e0c\u8868\u5185\u7684\u51b2\u7a81\u6574\u4f53\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u7f16\u7a0b\u8bed\u8a00\u4e00\u822c\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u79fb\u52a8\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u5f00\u9500\u5f88\u5927\u3002
     
    \u7f16\u7a0b\u8bed\u8a00\u4e00\u822c\u4f7f\u7528\u300c\u8d1f\u8f7d\u56e0\u5b50 Load Factor\u300d\u6765\u8bc4\u4f30\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u5e38\u7528\u4f5c\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u6bd4\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50 \\(> 0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06 HashMap \u5bb9\u91cf\u6269\u5145\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
    "},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2. \u00a0 \u94fe\u5f0f\u5730\u5740","text":"
    \u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u53ea\u80fd\u5b58\u50a8\u4e00\u4e2a\u5143\u7d20\uff08\u5373\u952e\u503c\u5bf9\uff09\u3002\u8003\u8651\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u6210\u4e00\u4e2a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u90fd\u5b58\u50a8\u5728\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002
     
     
     Fig. \u94fe\u5f0f\u5730\u5740 
     
    \u94fe\u5f0f\u5730\u5740\u4e0b\uff0c\u54c8\u5e0c\u8868\u64cd\u4f5c\u65b9\u6cd5\u4e3a\uff1a
     
       
    • \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 key \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6570\u7ec4\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u7ed3\u70b9\uff0c\u518d\u901a\u8fc7\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 key \u6765\u67e5\u627e\u952e\u503c\u5bf9\u3002
      •  
    • \u6dfb\u52a0\u5143\u7d20\uff1a\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u518d\u5c06\u7ed3\u70b9\uff08\u5373\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u5934\u90e8\u5373\u53ef\u3002
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u540c\u6837\u5148\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u518d\u904d\u5386\u94fe\u8868\u67e5\u627e\u5bf9\u5e94\u7ed3\u70b9\uff0c\u5220\u9664\u4e4b\u5373\u53ef\u3002
      •  
     
    \u94fe\u5f0f\u5730\u5740\u867d\u7136\u89e3\u51b3\u4e86\u54c8\u5e0c\u51b2\u7a81\u95ee\u9898\uff0c\u4f46\u4ecd\u5b58\u5728\u5c40\u9650\u6027\uff0c\u5305\u62ec\uff1a
     
       
    • \u5360\u7528\u7a7a\u95f4\u53d8\u5927\uff0c\u56e0\u4e3a\u94fe\u8868\u6216\u4e8c\u53c9\u6811\u5305\u542b\u7ed3\u70b9\u6307\u9488\uff0c\u76f8\u6bd4\u4e8e\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\uff1b
      •  
    • \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\uff1b
      •  
     
    \u4e3a\u4e86\u63d0\u5347\u64cd\u4f5c\u6548\u7387\uff0c\u53ef\u4ee5\u628a\u300c\u94fe\u8868\u300d\u8f6c\u5316\u4e3a\u300cAVL \u6811\u300d\u6216\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
    "},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3. \u00a0 \u5f00\u653e\u5bfb\u5740","text":"
    \u300c\u5f00\u653e\u5bfb\u5740\u300d\u4e0d\u5f15\u5165\u989d\u5916\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u6839\u636e\u63a2\u6d4b\u65b9\u6cd5\u7684\u4e0d\u540c\uff0c\u4e3b\u8981\u5206\u4e3a \u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u3002
    "},{"location":"chapter_hashing/hash_collision/#_1","title":"\u7ebf\u6027\u63a2\u6d4b","text":"
    \u300c\u7ebf\u6027\u63a2\u6d4b\u300d\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u67e5\u627e\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002
     
    \u63d2\u5165\u5143\u7d20\uff1a\u5982\u679c\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u4e00\u822c\u53d6 1 \uff09\uff0c\u76f4\u5230\u627e\u5230\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u5219\u5c06\u5143\u7d20\u63d2\u5165\u5230\u8be5\u7a7a\u4f4d\u4e2d\u3002
     
    \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u6267\u884c\u7ebf\u6027\u67e5\u627e\uff0c\u4f1a\u9047\u5230\u4e24\u79cd\u60c5\u51b5\uff1a
     
       
    1. \u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de value \u5373\u53ef\uff1b
      2.  
    2. \u82e5\u9047\u5230\u7a7a\u4f4d\uff0c\u5219\u8bf4\u660e\u67e5\u627e\u952e\u503c\u5bf9\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff1b
      3.  
     
     
     Fig. \u7ebf\u6027\u63a2\u6d4b 
     
    \u7ebf\u6027\u63a2\u6d4b\u5b58\u5728\u4ee5\u4e0b\u7f3a\u9677\uff1a
     
       
    • \u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u5220\u9664\u5143\u7d20\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5185\u51fa\u73b0\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u5728\u67e5\u627e\u5176\u4ed6\u5143\u7d20\u65f6\uff0c\u8be5\u7a7a\u4f4d\u6709\u53ef\u80fd\u5bfc\u81f4\u7a0b\u5e8f\u8ba4\u4e3a\u5143\u7d20\u4e0d\u5b58\u5728\uff08\u5373\u4e0a\u8ff0\u7b2c 2. \u79cd\u60c5\u51b5\uff09\u3002\u56e0\u6b64\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u6807\u8bb0\u5220\u9664\u5143\u7d20\u3002
      •  
    • \u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u6570\u7ec4\u5185\u88ab\u5360\u7528\u7684\u8fde\u7eed\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u8fdb\u8fd9\u4e00\u4f4d\u7f6e\u7684\u201c\u805a\u5806\u751f\u957f\u201d\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u7684\u52a3\u5316\u3002
      •  
    "},{"location":"chapter_hashing/hash_collision/#_2","title":"\u591a\u6b21\u54c8\u5e0c","text":"
    \u987e\u540d\u601d\u4e49\uff0c\u300c\u591a\u6b21\u54c8\u5e0c\u300d\u7684\u601d\u8def\u662f\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) , \\(f_2(x)\\) , \\(f_3(x)\\) , \\(\\cdots\\) \u8fdb\u884c\u63a2\u6d4b\u3002
     
    \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002
     
    \u67e5\u627e\u5143\u7d20\uff1a\u4ee5\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u67e5\u627e\uff0c\u5b58\u5728\u4e24\u79cd\u60c5\u51b5\uff1a
     
       
    1. \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de\u4e4b\uff1b
      2.  
    2. \u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64\u5143\u7d20\uff1b
      3.  
     
    \u76f8\u6bd4\u4e8e\u300c\u7ebf\u6027\u63a2\u6d4b\u300d\uff0c\u300c\u591a\u6b21\u54c8\u5e0c\u300d\u65b9\u6cd5\u66f4\u4e0d\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4ee3\u4ef7\u662f\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u989d\u5916\u8ba1\u7b97\u91cf\u3002
     
    \u5de5\u4e1a\u754c\u65b9\u6848
     
    Java \u91c7\u7528\u300c\u94fe\u5f0f\u5730\u5740\u300d\u3002\u5728 JDK 1.8 \u4e4b\u540e\uff0cHashMap \u5185\u6570\u7ec4\u957f\u5ea6\u5927\u4e8e 64 \u65f6\uff0c\u957f\u5ea6\u5927\u4e8e 8 \u7684\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002
     
    Python \u91c7\u7528\u300c\u5f00\u653e\u5bfb\u5740\u300d\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002 
     
    Golang \u91c7\u7528\u300c\u94fe\u5f0f\u5730\u5740\u300d\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u4fdd\u8bc1\u6027\u80fd\u3002
    "},{"location":"chapter_hashing/hash_map/","title":"6.1. \u00a0 \u54c8\u5e0c\u8868","text":"
    \u54c8\u5e0c\u8868\u901a\u8fc7\u5efa\u7acb\u300c\u952e key\u300d\u548c\u300c\u503c value\u300d\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u3002\u5177\u4f53\u5730\uff0c\u8f93\u5165\u4e00\u4e2a key \uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5e76\u83b7\u53d6 value \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
     
    \u4f8b\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5b66\u751f\u7684\u6570\u636e\u5e93\uff0c\u6bcf\u4e2a\u5b66\u751f\u6709\u201c\u59d3\u540d name \u201d\u548c\u201c\u5b66\u53f7 id \u201d\u4e24\u9879\u6570\u636e\uff0c\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u67e5\u8be2\u529f\u80fd\uff1a\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\uff0c\u5219\u53ef\u4ee5\u4f7f\u7528\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002
     
     
     Fig. \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a 
    "},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1. \u00a0 \u54c8\u5e0c\u8868\u6548\u7387","text":"
    \u9664\u4e86\u54c8\u5e0c\u8868\u4e4b\u5916\uff0c\u8fd8\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u6570\u636e\u7ed3\u6784\u6765\u5b9e\u73b0\u4e0a\u8ff0\u67e5\u8be2\u529f\u80fd\uff1a
     
       
    1. \u65e0\u5e8f\u6570\u7ec4\uff1a\u6bcf\u4e2a\u5143\u7d20\u4e3a  [\u5b66\u53f7, \u59d3\u540d] \uff1b
      2.  
    2. \u6709\u5e8f\u6570\u7ec4\uff1a\u5c06 1. \u4e2d\u7684\u6570\u7ec4\u6309\u7167\u5b66\u53f7\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff1b
      3.  
    3. \u94fe\u8868\uff1a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u503c\u4e3a [\u5b66\u53f7, \u59d3\u540d] \uff1b
      4.  
    4. \u4e8c\u53c9\u641c\u7d22\u6811\uff1a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u503c\u4e3a [\u5b66\u53f7, \u59d3\u540d] \uff0c\u6839\u636e\u5b66\u53f7\u5927\u5c0f\u6765\u6784\u5efa\u6811\uff1b
      5.  
     
    \u4f7f\u7528\u4e0a\u8ff0\u65b9\u6cd5\uff0c\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5982\u4e0b\u8868\u6240\u793a\uff08\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\uff0c\u8be6\u89e3\u53ef\u89c1 \u4e8c\u53c9\u641c\u7d22\u6811\u7ae0\u8282\uff09\u3002\u65e0\u8bba\u662f\u67e5\u627e\u5143\u7d20\u3001\u8fd8\u662f\u589e\u5220\u5143\u7d20\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u5168\u9762\u80dc\u51fa\uff01
     \u65e0\u5e8f\u6570\u7ec4 \u6709\u5e8f\u6570\u7ec4 \u94fe\u8868 \u4e8c\u53c9\u641c\u7d22\u6811 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\)"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2. \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u54c8\u5e0c\u8868\u7684\u57fa\u672c\u64cd\u4f5c\u5305\u62ec \u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u4e0e\u5220\u9664\u952e\u503c\u5bf9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hash_map.java
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   map.put(15937, \"\u5c0f\u5570\");   map.put(16750, \"\u5c0f\u7b97\");   map.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
     hash_map.cpp
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
     hash_map.py
    \"\"\" \u521d\u59cb\u5316\u54c8\u5e0c\u8868 \"\"\"\nmapp: Dict = {}\n\"\"\" \u6dfb\u52a0\u64cd\u4f5c \"\"\"\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n\"\"\" \u67e5\u8be2\u64cd\u4f5c \"\"\"\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = mapp[15937]\n\"\"\" \u5220\u9664\u64cd\u4f5c \"\"\"\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmapp.pop(10583)\n
     hash_map.go
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nmapp := make(map[int]string)\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := mapp[15937]\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(mapp, 10583)\n
     hash_map.js
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new ArrayHashMap();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
     hash_map.ts
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
     hash_map.c
    \n
     hash_map.cs
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, String> map = new ();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.Add(12836, \"\u5c0f\u54c8\");\nmap.Add(15937, \"\u5c0f\u5570\");\nmap.Add(16750, \"\u5c0f\u7b97\");\nmap.Add(13276, \"\u5c0f\u6cd5\");\nmap.Add(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
     hash_map.swift
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
     hash_map.zig
    \n
     
    \u904d\u5386\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u65b9\u5f0f\uff0c\u5373 \u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u3001\u904d\u5386\u503c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hash_map.java
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\nSystem.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\nSystem.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\nSystem.out.println(val);\n}\n
     hash_map.cpp
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\ncout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (auto key: map) {\ncout << key.first << endl;\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (auto val: map) {\ncout << val.second << endl;\n}\n
     hash_map.py
    \"\"\" \u904d\u5386\u54c8\u5e0c\u8868 \"\"\"\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in mapp.items():\nprint(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in mapp.keys():\nprint(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in mapp.values():\nprint(value)\n
     hash_map_test.go
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range mapp {\nfmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range mapp {\nfmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range mapp {\nfmt.Println(value)\n}\n
     hash_map.js
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
     hash_map.ts
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
     hash_map.c
    \n
     hash_map.cs
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map)\n{\nConsole.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys)\n{\nConsole.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (String val in map.Values)\n{\nConsole.WriteLine(val);\n}\n
     hash_map.swift
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\nprint(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\nprint(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\nprint(value)\n}\n
     hash_map.zig
    \n
    "},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3. \u00a0 \u54c8\u5e0c\u51fd\u6570","text":"
    \u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\uff0c\u5e76\u4e14\u53ef\u80fd\u5305\u542b\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff08\u7ea2\u9ed1\u6811\uff09\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6027\u80fd\uff08\u4e0b\u8282\u4f1a\u8ba8\u8bba\uff09\u3002
     
    \u9996\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u300c\u6570\u7ec4\u300d\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u6839\u636e\u4e60\u60ef\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 Bucket\u300d\uff0c\u7528\u4e8e\u5b58\u50a8\u952e\u503c\u5bf9\u3002
     
    \u6211\u4eec\u5c06\u952e\u503c\u5bf9 key, value \u5305\u88c5\u6210\u4e00\u4e2a\u7c7b Entry \uff0c\u5e76\u5c06\u6240\u6709 Entry \u90fd\u653e\u5165\u6570\u7ec4\u4e2d\uff0c\u90a3\u4e48\u6bcf\u4e2a Entry \u5728\u6570\u7ec4\u4e2d\u90fd\u6709\u552f\u4e00\u7684\u7d22\u5f15\u3002\u800c\u4e3a\u4e86\u5efa\u7acb key \u548c\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\uff0c\u6211\u4eec\u9700\u8981\u4f7f\u7528\u300c\u54c8\u5e0c\u51fd\u6570 Hash Function\u300d\u3002
     
    \u8bbe\u54c8\u5e0c\u8868\u7684\u6570\u7ec4\u4e3a buckets \uff0c\u54c8\u5e0c\u51fd\u6570\u4e3a f(x) \uff0c\u90a3\u4e48\u67e5\u8be2\u64cd\u4f5c\u7684\u6b65\u9aa4\u4e3a\uff1a
     
       
    1. \u8f93\u5165 key \uff0c\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u51fa\u7d22\u5f15 index \uff0c\u5373 index = f(key) \uff1b
      2.  
    2. \u901a\u8fc7\u7d22\u5f15\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5230\u952e\u503c\u5bf9 entry \uff0c\u5373 entry = buckets[index] \uff0c\u5e76\u5728 entry \u4e2d\u83b7\u53d6\u5230 value \u5373\u53ef\uff1b
      3.  
     
    \u4ee5\u4e0a\u8ff0\u5b66\u751f\u6570\u636e key \u5b66\u53f7 -> value \u59d3\u540d \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u54c8\u5e0c\u51fd\u6570\u300d\u8bbe\u8ba1\u4e3a
     \\[ f(x) = x \\% 100 \\] 
     
     Fig. \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_hash_map.java
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry {\npublic int key;\npublic String val;\npublic Entry(int key, String val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Entry> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new ArrayList<>();\nfor (int i = 0; i < 100; i++) {\nbuckets.add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\nEntry pair = buckets.get(index);\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\nEntry pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets.set(index, pair);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets.set(index, null);\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Entry> entrySet() {\nList<Entry> entrySet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nentrySet.add(pair);\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<Integer> keySet() {\nList<Integer> keySet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nkeySet.add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet() {\nList<String> valueSet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nvalueSet.add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Entry kv: entrySet()) {\nSystem.out.println(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
     array_hash_map.cpp
    /* \u952e\u503c\u5bf9 int->String */\nstruct Entry {\npublic:\nint key;\nstring val;\nEntry(int key, string val) {\nthis->key = key;\nthis->val = val;\n}\n};\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate:\nvector<Entry*> buckets;\npublic:\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = vector<Entry*>(100);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\nEntry* pair = buckets[index];\nif (pair == nullptr)\nreturn nullptr;\nreturn pair->val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\nEntry* pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a nullptr \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nullptr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvector<Entry*> entrySet() {\nvector<Entry*> entrySet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr) {\nentrySet.push_back(pair);\n}\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvector<int> keySet() {\nvector<int> keySet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr) {\nkeySet.push_back(pair->key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvector<string> valueSet() {\nvector<string> valueSet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr){\nvalueSet.push_back(pair->val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Entry* kv: entrySet()) {\ncout << kv->key << \" -> \" << kv->val << endl;\n}\n}\n};\n
     array_hash_map.py
    class Entry:\n\"\"\" \u952e\u503c\u5bf9 int->String \"\"\"\ndef __init__(self, key: int, val: str):\nself.key = key\nself.val = val\nclass ArrayHashMap:\n\"\"\" \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets: List[Optional[Entry]] = [None] * 100\ndef hash_func(self, key: int) -> int:\n\"\"\" \u54c8\u5e0c\u51fd\u6570 \"\"\"\nindex: int = key % 100\nreturn index\ndef get(self, key: int) -> str:\n\"\"\" \u67e5\u8be2\u64cd\u4f5c \"\"\"\nindex: int = self.hash_func(key)\npair: Entry = self.buckets[index]\nif pair is None:\nreturn None\nreturn pair.val\ndef put(self, key: int, val: str) -> None:\n\"\"\" \u6dfb\u52a0\u64cd\u4f5c \"\"\"\npair = Entry(key, val)\nindex: int = self.hash_func(key)\nself.buckets[index] = pair\ndef remove(self, key: int) -> None:\n\"\"\" \u5220\u9664\u64cd\u4f5c \"\"\"\nindex: int = self.hash_func(key)\n# \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None\ndef entry_set(self) -> List[Entry]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 \"\"\"\nresult: List[Entry] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair)\nreturn result\ndef key_set(self) -> List[int]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u952e \"\"\"\nresult: List[int] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.key)\nreturn result\ndef value_set(self) -> List[str]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u503c \"\"\"\nresult: List[str] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.val)\nreturn result\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u54c8\u5e0c\u8868 \"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\n
     array_hash_map.go
    /* \u952e\u503c\u5bf9 int->String */\ntype entry struct {\nkey int\nval string\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\nbuckets []*entry\n}\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets := make([]*entry, 100)\nreturn &arrayHashMap{buckets: buckets}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\nindex := key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\nindex := a.hashFunc(key)\npair := a.buckets[index]\nif pair == nil {\nreturn \"Not Found\"\n}\nreturn pair.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\npair := &entry{key: key, val: val}\nindex := a.hashFunc(key)\na.buckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\nindex := a.hashFunc(key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\na.buckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) entrySet() []*entry {\nvar pairs []*entry\nfor _, pair := range a.buckets {\nif pair != nil {\npairs = append(pairs, pair)\n}\n}\nreturn pairs\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\nvar keys []int\nfor _, pair := range a.buckets {\nif pair != nil {\nkeys = append(keys, pair.key)\n}\n}\nreturn keys\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\nvar values []string\nfor _, pair := range a.buckets {\nif pair != nil {\nvalues = append(values, pair.val)\n}\n}\nreturn values\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\nfor _, pair := range a.buckets {\nif pair != nil {\nfmt.Println(pair.key, \"->\", pair.val)\n}\n}\n}\n
     array_hash_map.js
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Entry {\nconstructor(key, val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n#buckets;\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.#buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nlet index = this.#hashFunc(key);\nlet entry = this.#buckets[index];\nif (entry === null) return null;\nreturn entry.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nset(key, val) {\nlet index = this.#hashFunc(key);\nthis.#buckets[index] = new Entry(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\ndelete(key) {\nlet index = this.#hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.#buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nentries() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nkeys() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvalues() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nlet entrySet = this.entries();\nfor (const entry of entrySet) {\nif (!entry) continue;\nconsole.info(`${entry.key} -> ${entry.val}`);\n}\n}\n}\n
     array_hash_map.ts
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Entry {\npublic key: number;\npublic val: string;\nconstructor(key: number, val: string) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate readonly buckets: (Entry | null)[];\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.buckets = (new Array(100)).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic get(key: number): string | null {\nlet index = this.hashFunc(key);\nlet entry = this.buckets[index];\nif (entry === null) return null;\nreturn entry.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic set(key: number, val: string) {\nlet index = this.hashFunc(key);\nthis.buckets[index] = new Entry(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic delete(key: number) {\nlet index = this.hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic entries(): (Entry | null)[] {\nlet arr: (Entry | null)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic keys(): (number | undefined)[] {\nlet arr: (number | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic values(): (string | undefined)[] {\nlet arr: (string | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic print() {\nlet entrySet = this.entries();\nfor (const entry of entrySet) {\nif (!entry) continue;\nconsole.info(`${entry.key} -> ${entry.val}`);\n}\n}\n}\n
     array_hash_map.c
    [class]{entry}-[func]{}\n[class]{arrayHashMap}-[func]{}\n
     array_hash_map.cs
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry\n{\npublic int key;\npublic String val;\npublic Entry(int key, String val)\n{\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap\n{\nprivate List<Entry?> buckets;\npublic ArrayHashMap()\n{\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new();\nfor (int i = 0; i < 100; i++)\n{\nbuckets.Add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key)\n{\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String? get(int key)\n{\nint index = hashFunc(key);\nEntry? pair = buckets[index];\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val)\n{\nEntry pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key)\n{\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Entry> entrySet()\n{\nList<Entry> entrySet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nentrySet.Add(pair);\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<int> keySet()\n{\nList<int> keySet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nkeySet.Add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet()\n{\nList<String> valueSet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nvalueSet.Add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print()\n{\nforeach (Entry kv in entrySet())\n{\nConsole.WriteLine(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
     array_hash_map.swift
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry {\nvar key: Int\nvar val: String\ninit(key: Int, val: String) {\nself.key = key\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate var buckets: [Entry?] = []\ninit() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nfor _ in 0 ..< 100 {\nbuckets.append(nil)\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate func hashFunc(key: Int) -> Int {\nlet index = key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet pair = buckets[index]\nreturn pair?.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\nlet pair = Entry(key: key, val: val)\nlet index = hashFunc(key: key)\nbuckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nfunc entrySet() -> [Entry] {\nvar entrySet: [Entry] = []\nfor pair in buckets {\nif let pair = pair {\nentrySet.append(pair)\n}\n}\nreturn entrySet\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc keySet() -> [Int] {\nvar keySet: [Int] = []\nfor pair in buckets {\nif let pair = pair {\nkeySet.append(pair.key)\n}\n}\nreturn keySet\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc valueSet() -> [String] {\nvar valueSet: [String] = []\nfor pair in buckets {\nif let pair = pair {\nvalueSet.append(pair.val)\n}\n}\nreturn valueSet\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor entry in entrySet() {\nSwift.print(\"\\(entry.key) -> \\(entry.val)\")\n}\n}\n}\n
     array_hash_map.zig
    // \u952e\u503c\u5bf9 int->String\nconst Entry = struct {\nkey: usize = undefined,\nval: []const u8 = undefined,\npub fn init(key: usize, val: []const u8) Entry {\nreturn Entry {\n.key = key,\n.val = val,\n};\n}\n};\n// \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\nreturn struct {\nbuckets: ?std.ArrayList(?T) = null,\nmem_allocator: std.mem.Allocator = undefined,\nconst Self = @This();\n// \u6784\u9020\u65b9\u6cd5\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nself.mem_allocator = allocator;\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets = std.ArrayList(?T).init(self.mem_allocator);\nvar i: i32 = 0;\nwhile (i < 100) : (i += 1) {\ntry self.buckets.?.append(null);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\npub fn deinit(self: *Self) void {\nif (self.buckets != null) self.buckets.?.deinit();\n}\n// \u54c8\u5e0c\u51fd\u6570\nfn hashFunc(key: usize) usize {\nvar index = key % 100;\nreturn index;\n}\n// \u67e5\u8be2\u64cd\u4f5c\npub fn get(self: *Self, key: usize) []const u8 {\nvar index = hashFunc(key);\nvar pair = self.buckets.?.items[index];\nreturn pair.?.val;\n}\n// \u6dfb\u52a0\u64cd\u4f5c\npub fn put(self: *Self, key: usize, val: []const u8) !void {\nvar pair = Entry.init(key, val);\nvar index = hashFunc(key);\nself.buckets.?.items[index] = pair;\n}\n// \u5220\u9664\u64cd\u4f5c\npub fn remove(self: *Self, key: usize) !void {\nvar index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets.?.items[index] = null;\n}       // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\npub fn entrySet(self: *Self) !*std.ArrayList(T) {\nvar entry_set = std.ArrayList(T).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry entry_set.append(item.?);\n}\nreturn &entry_set;\n}  // \u83b7\u53d6\u6240\u6709\u952e\npub fn keySet(self: *Self) !*std.ArrayList(usize) {\nvar key_set = std.ArrayList(usize).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry key_set.append(item.?.key);\n}\nreturn &key_set;\n}  // \u83b7\u53d6\u6240\u6709\u503c\npub fn valueSet(self: *Self) !*std.ArrayList([]const u8) {\nvar value_set = std.ArrayList([]const u8).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry value_set.append(item.?.val);\n}\nreturn &value_set;\n}\n// \u6253\u5370\u54c8\u5e0c\u8868\npub fn print(self: *Self) !void {\nvar entry_set = try self.entrySet();\ndefer entry_set.deinit();\nfor (entry_set.items) |item| {\nstd.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n}\n}\n};\n}\n
    "},{"location":"chapter_hashing/hash_map/#614","title":"6.1.4. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u4f1a\u53d1\u73b0\uff0c\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x \\% 100\\) \u4f1a\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u5931\u6548\u3002\u5177\u4f53\u5730\uff0c\u5f53\u8f93\u5165\u7684 key \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u7ed3\u679c\u4e5f\u76f8\u540c\uff0c\u6307\u5411\u540c\u4e00\u4e2a value \u3002\u4f8b\u5982\uff0c\u5206\u522b\u67e5\u8be2\u4e24\u4e2a\u5b66\u53f7 \\(12836\\) \u548c \\(20336\\) \uff0c\u5219\u6709
     \\[ f(12836) = f(20336) = 36 \\] 
    \u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u660e\u663e\u662f\u4e0d\u5bf9\u7684\uff0c\u6211\u4eec\u5c06\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002\u5982\u4f55\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u7684\u95ee\u9898\u5c06\u88ab\u7559\u5728\u4e0b\u7ae0\u8ba8\u8bba\u3002
     
     
     Fig. \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b 
     
    \u7efc\u4e0a\u6240\u8ff0\uff0c\u4e00\u4e2a\u4f18\u79c0\u7684\u300c\u54c8\u5e0c\u51fd\u6570\u300d\u5e94\u8be5\u5177\u5907\u4ee5\u4e0b\u7279\u6027\uff1a
     
       
    • \u5c3d\u91cf\u5c11\u5730\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\uff1b
      •  
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff0c\u8ba1\u7b97\u5c3d\u53ef\u80fd\u9ad8\u6548\uff1b
      •  
    • \u7a7a\u95f4\u4f7f\u7528\u7387\u9ad8\uff0c\u5373\u201c\u952e\u503c\u5bf9\u5360\u7528\u7a7a\u95f4 / \u54c8\u5e0c\u8868\u603b\u5360\u7528\u7a7a\u95f4\u201d\u5c3d\u53ef\u80fd\u5927\uff1b
      •  
    "},{"location":"chapter_hashing/summary/","title":"6.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u4e00\u4e2a\u952e key \uff0c\u67e5\u8be2\u5230\u503c value \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
      •  
    • \u54c8\u5e0c\u8868\u7684\u5e38\u7528\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u4e0e\u5220\u9664\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u503c\u5bf9\u7b49\u3002
      •  
    • \u54c8\u5e0c\u51fd\u6570\u5c06 key \u6620\u5c04\u5230\u6876\uff08\u6570\u7ec4\uff09\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u503c value \u3002
      •  
    • \u4e24\u4e2a\u4e0d\u540c\u7684 key \u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u53ef\u80fd\u5f97\u5230\u76f8\u540c\u7684\u6876\u7d22\u5f15\uff0c\u8fdb\u800c\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\uff0c\u5bfc\u81f4\u67e5\u8be2\u9519\u8bef\u3002
      •  
    • \u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u7684\u9014\u5f84\u6709\u4e24\u79cd\uff1a\u54c8\u5e0c\u8868\u6269\u5bb9\u3001\u4f18\u5316\u54c8\u5e0c\u8868\u7684\u8868\u793a\u65b9\u5f0f\u3002
      •  
    • \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u69fd\u6570\u91cf\uff0c\u4f53\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7684\u539f\u7406\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u5f00\u9500\u4e5f\u5f88\u5927\u3002
      •  
    • \u94fe\u5f0f\u5730\u5740\u8003\u8651\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u6210\u4e00\u4e2a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u90fd\u5b58\u50a8\u5728\u4e00\u4e2a\u94fe\u8868\u4e2d\uff0c\u4ece\u800c\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u94fe\u8868\u8fc7\u957f\u4f1a\u5bfc\u81f4\u67e5\u8be2\u6548\u7387\u53d8\u4f4e\uff0c\u53ef\u4ee5\u901a\u8fc7\u628a\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\u6765\u89e3\u51b3\u3002
      •  
    • \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u5bf9\u7ebf\u6027\u63a2\u6d4b\u4e0d\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4ee3\u4ef7\u662f\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
      •  
    • \u5728\u5de5\u4e1a\u754c\u4e2d\uff0cJava \u7684 HashMap \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3001Python \u7684 Dict \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002
      •  
    "},{"location":"chapter_heap/build_heap/","title":"8.2. \u00a0 \u5efa\u5806\u64cd\u4f5c *","text":"
    \u5982\u679c\u6211\u4eec\u60f3\u8981\u6839\u636e\u8f93\u5165\u5217\u8868\u6765\u751f\u6210\u4e00\u4e2a\u5806\uff0c\u8fd9\u6837\u7684\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5efa\u5806\u300d\u3002
    "},{"location":"chapter_heap/build_heap/#821","title":"8.2.1. \u00a0 \u4e24\u79cd\u5efa\u5806\u65b9\u6cd5","text":""},{"location":"chapter_heap/build_heap/#_1","title":"\u501f\u52a9\u5165\u5806\u65b9\u6cd5\u5b9e\u73b0","text":"
    \u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u300c\u5143\u7d20\u5165\u5806\u300d\u65b9\u6cd5\uff0c\u5148\u5efa\u7acb\u4e00\u4e2a\u7a7a\u5806\uff0c\u518d\u5c06\u5217\u8868\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u5373\u53ef\u3002
     
    \u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u5165\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5728\u4f9d\u6b21\u5165\u5806\u65f6\uff0c\u5806\u7684\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8be5\u65b9\u6cd5\u7684\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
    "},{"location":"chapter_heap/build_heap/#_2","title":"\u57fa\u4e8e\u5806\u5316\u64cd\u4f5c\u5b9e\u73b0","text":"
    \u6709\u8da3\u7684\u662f\uff0c\u5b58\u5728\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u3002\u6211\u4eec\u5148\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\uff0c\u7136\u540e\u8fed\u4ee3\u5730\u5bf9\u5404\u4e2a\u7ed3\u70b9\u6267\u884c\u300c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u300d\u3002\u5f53\u7136\uff0c\u65e0\u9700\u5bf9\u53f6\u7ed3\u70b9\u6267\u884c\u5806\u5316\uff0c\u56e0\u4e3a\u5176\u6ca1\u6709\u5b50\u7ed3\u70b9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new ArrayList<>(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
     my_heap.cpp
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
     my_heap.py
    def __init__(self, nums: List[int]):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nself.max_heap = nums\n# \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor i in range(self.parent(self.size() - 1), -1, -1):\nself.sift_down(i)\n
     my_heap.go
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nh := &maxHeap{data: nums}\nfor i := len(h.data) - 1; i >= 0; i-- {\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nh.siftDown(i)\n}\nreturn h\n}\n
     my_heap.js
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.#maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (let i = this.#parent(this.size() - 1); i >= 0; i--) {\nthis.#siftDown(i);\n}\n}\n
     my_heap.ts
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (let i = this.parent(this.size() - 1); i >= 0; i--) {\nthis.siftDown(i);\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{newMaxHeap}\n
     my_heap.cs
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums)\n{\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new List<int>(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nvar size = parent(this.size() - 1);\nfor (int i = size; i >= 0; i--)\n{\nsiftDown(i);\n}\n}\n
     my_heap.swift
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\nsiftDown(i: i)\n}\n}\n
     my_heap.zig
    // \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\nif (self.max_heap != null) return;\nself.max_heap = std.ArrayList(T).init(allocator);\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\ntry self.max_heap.?.appendSlice(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nvar i: usize = parent(self.size() - 1) + 1;\nwhile (i > 0) : (i -= 1) {\ntry self.siftDown(i - 1);\n}\n}\n
    "},{"location":"chapter_heap/build_heap/#822","title":"8.2.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n)\\) \u5462\uff1f\u6211\u4eec\u6765\u5c55\u5f00\u63a8\u7b97\u4e00\u4e0b\u3002
     
       
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u53f6\u7ed3\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u5728\u6392\u9664\u53f6\u7ed3\u70b9\u540e\uff0c\u9700\u8981\u5806\u5316\u7ed3\u70b9\u6570\u91cf\u4e3a \\((n - 1)/2\\) \uff0c\u5373\u4e3a \\(O(n)\\) \uff1b
      •  
    • \u4ece\u9876\u81f3\u5e95\u5806\u5316\u4e2d\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u6700\u591a\u5806\u5316\u81f3\u53f6\u7ed3\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(O(\\log n)\\) \uff1b
      •  
     
    \u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u4e0d\u591f\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230 \u4e8c\u53c9\u6811\u5e95\u5c42\u7ed3\u70b9\u8fdc\u591a\u4e8e\u9876\u5c42\u7ed3\u70b9 \u7684\u6027\u8d28\u3002
     
    \u4e0b\u9762\u6211\u4eec\u6765\u5c55\u5f00\u8ba1\u7b97\u3002\u4e3a\u4e86\u51cf\u5c0f\u8ba1\u7b97\u96be\u5ea6\uff0c\u6211\u4eec\u5047\u8bbe\u6811\u662f\u4e00\u4e2a\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002\u8bbe\u4e8c\u53c9\u6811\uff08\u5373\u5806\uff09\u7ed3\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \u3002\u4e0a\u6587\u63d0\u5230\uff0c\u7ed3\u70b9\u5806\u5316\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u7ed3\u70b9\u5230\u53f6\u7ed3\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8fd9\u6b63\u662f\u201c\u7ed3\u70b9\u9ad8\u5ea6\u201d\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u7ed3\u70b9\u6570\u91cf 
     
    \u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5404\u5c42\u7684\u201c\u7ed3\u70b9\u6570\u91cf \\(\\times\\) \u7ed3\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5373\u53ef\u5f97\u5230 \u6240\u6709\u7ed3\u70b9\u7684\u5806\u5316\u7684\u8fed\u4ee3\u6b21\u6570\u603b\u548c\u3002
     \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{(h-1)}\\times1 \\] 
    \u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u6613\u5f97
     \\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\cdots + 2^{h}\\times1 \\newline \\end{aligned} \\] 
    \u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u4ee4\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97
     \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\cdots + 2^{h-1} + 2^h \\] 
    \u89c2\u5bdf\u4e0a\u5f0f\uff0c\\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a
     \\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h \\newline & = O(2^h) \\end{aligned} \\] 
    \u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\)\u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
    "},{"location":"chapter_heap/heap/","title":"8.1. \u00a0 \u5806","text":"
    \u300c\u5806 Heap\u300d\u662f\u4e00\u68f5\u9650\u5b9a\u6761\u4ef6\u4e0b\u7684\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u3002\u6839\u636e\u6210\u7acb\u6761\u4ef6\uff0c\u5806\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff1a
     
       
    • \u300c\u5927\u9876\u5806 Max Heap\u300d\uff0c\u4efb\u610f\u7ed3\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u7ed3\u70b9\u7684\u503c\uff1b
      •  
    • \u300c\u5c0f\u9876\u5806 Min Heap\u300d\uff0c\u4efb\u610f\u7ed3\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u7ed3\u70b9\u7684\u503c\uff1b
      •  
     
     
     Fig. \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 
    "},{"location":"chapter_heap/heap/#811","title":"8.1.1. \u00a0 \u5806\u672f\u8bed\u4e0e\u6027\u8d28","text":"
       
    • \u7531\u4e8e\u5806\u662f\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6700\u5e95\u5c42\u7ed3\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u5b83\u5c42\u7ed3\u70b9\u7686\u88ab\u586b\u6ee1\u3002
      •  
    • \u4e8c\u53c9\u6811\u4e2d\u7684\u6839\u7ed3\u70b9\u5bf9\u5e94\u300c\u5806\u9876\u300d\uff0c\u5e95\u5c42\u6700\u9760\u53f3\u7ed3\u70b9\u5bf9\u5e94\u300c\u5806\u5e95\u300d\u3002
      •  
    • \u5bf9\u4e8e\u5927\u9876\u5806 / \u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\uff08\u5373\u6839\u7ed3\u70b9\uff09\u7684\u503c\u6700\u5927 / \u6700\u5c0f\u3002
      •  
    "},{"location":"chapter_heap/heap/#812","title":"8.1.2. \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"
    \u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 Priority Queue\u300d\uff0c\u5176\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\u3002
     
    \u800c\u6070\u597d\uff0c\u5806\u7684\u5b9a\u4e49\u4e0e\u4f18\u5148\u961f\u5217\u7684\u64cd\u4f5c\u903b\u8f91\u5b8c\u5168\u543b\u5408\uff0c\u5927\u9876\u5806\u5c31\u662f\u4e00\u4e2a\u5143\u7d20\u4ece\u5927\u5230\u5c0f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u4f18\u5148\u961f\u5217\u300d\u548c\u300c\u5806\u300d\u7406\u89e3\u4e3a\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u6587\u4e0e\u4ee3\u7801\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u4f7f\u7528\u300c\u5806\u300d\u6765\u547d\u540d\u3002
     
    \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u786e\u5b9a\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002
     
    Tip
     
    \u7c7b\u4f3c\u4e8e\u6392\u5e8f\u4e2d\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u201c\u5927\u9876\u5806\u201d\u548c\u201c\u5c0f\u9876\u5806\u201d\u53ef\u4ec5\u901a\u8fc7\u4fee\u6539 Comparator \u6765\u4e92\u76f8\u8f6c\u6362\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig heap.java
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = heap.poll();  // 5\npeek = heap.poll();  // 4\npeek = heap.poll();  // 3\npeek = heap.poll();  // 2\npeek = heap.poll();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
     heap.cpp
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
     heap.py
    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\"\"\" \u5143\u7d20\u5165\u5806 \"\"\"\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\"\"\" \u83b7\u53d6\u5806\u9876\u5143\u7d20 \"\"\"\npeek: int = flag * max_heap[0] # 5\n\"\"\" \u5806\u9876\u5143\u7d20\u51fa\u5806 \"\"\"\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\"\"\" \u83b7\u53d6\u5806\u5927\u5c0f \"\"\"\nsize: int = len(max_heap)\n\"\"\" \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = not max_heap\n\"\"\" \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 \"\"\"\nmin_heap: List[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
     heap.go
    // Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n// Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n// \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n*h = append(*h, x.(int))\n}\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n// \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\nlast := (*h)[len(*h)-1]\n*h = (*h)[:len(*h)-1]\nreturn last\n}\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\nreturn len(*h)\n}\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n// \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\nreturn (*h)[i].(int) > (*h)[j].(int)\n}\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n(*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\nreturn (*h)[0]\n}\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nmaxHeap := &intHeap{}\nheap.Init(maxHeap)\n/* \u5143\u7d20\u5165\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\nheap.Push(maxHeap, 1)\nheap.Push(maxHeap, 3)\nheap.Push(maxHeap, 2)\nheap.Push(maxHeap, 4)\nheap.Push(maxHeap, 5)\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\ntop := maxHeap.Top()\nfmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\nheap.Pop(maxHeap) // 5\nheap.Pop(maxHeap) // 4\nheap.Pop(maxHeap) // 3\nheap.Pop(maxHeap) // 2\nheap.Pop(maxHeap) // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nsize := len(*maxHeap)\nfmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(*maxHeap) == 0\nfmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
     heap.js
    // JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.ts
    // TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.c
    \n
     heap.cs
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new PriorityQueue<int, int>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((x, y) => y - x));\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), });\n
     heap.swift
    // Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.zig
    \n
    "},{"location":"chapter_heap/heap/#813","title":"8.1.3. \u00a0 \u5806\u7684\u5b9e\u73b0","text":"
    \u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u300c\u5927\u9876\u5806\u300d\uff0c\u82e5\u60f3\u8f6c\u6362\u4e3a\u300c\u5c0f\u9876\u5806\u300d\uff0c\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u5373\u53ef\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_heap/heap/#_1","title":"\u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"
    \u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u6211\u4eec\u5b66\u8fc7\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u975e\u5e38\u9002\u5408\u4f7f\u7528\u300c\u6570\u7ec4\u300d\u6765\u8868\u793a\uff0c\u800c\u5806\u6070\u597d\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u800c\u6211\u4eec\u91c7\u7528\u300c\u6570\u7ec4\u300d\u6765\u5b58\u50a8\u300c\u5806\u300d\u3002
     
    \u4e8c\u53c9\u6811\u6307\u9488\u3002\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u7ed3\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u7ed3\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\uff0c\u800c\u7ed3\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002
     
    \u5177\u4f53\u5730\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u90a3\u4e48\u5176\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \u3001\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3001\u7236\u7ed3\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\) \uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u4ee3\u8868\u7a7a\u7ed3\u70b9\u6216\u7ed3\u70b9\u4e0d\u5b58\u5728\u3002
     
     
     Fig. \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 
     
    \u6211\u4eec\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u4ee5\u4fbf\u540e\u7eed\u4f7f\u7528\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.cpp
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n} /* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n
     my_heap.py
    def left(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n
     my_heap.go
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n// \u5411\u4e0b\u6574\u9664\nreturn (i - 1) / 2\n}\n
     my_heap.js
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\n#left(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\n#right(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\n#parent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.ts
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{left}\n[class]{maxHeap}-[func]{right}\n[class]{maxHeap}-[func]{parent}\n
     my_heap.cs
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i)\n{\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i)\n{\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i)\n{\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.swift
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.zig
    // \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\nreturn 2 * i + 1;\n}\n// \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\nreturn 2 * i + 2;\n}\n// \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n// return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\nreturn @divFloor(i - 1, 2);\n}\n
    "},{"location":"chapter_heap/heap/#_2","title":"\u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"
    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u7ed3\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap.get(0);\n}\n
     my_heap.cpp
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
     my_heap.py
    def peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u5806\u9876\u5143\u7d20 \"\"\"\nreturn self.max_heap[0]\n
     my_heap.go
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\nreturn h.data[0]\n}\n
     my_heap.js
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\nreturn this.#maxHeap[0];\n}\n
     my_heap.ts
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\nreturn this.maxHeap[0];\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{peek}\n
     my_heap.cs
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek()\n{\nreturn maxHeap[0];\n}\n
     my_heap.swift
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nmaxHeap[0]\n}\n
     my_heap.zig
    // \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\nreturn self.max_heap.?.items[0];\n}  
    "},{"location":"chapter_heap/heap/#_3","title":"\u5143\u7d20\u5165\u5806","text":"
    \u7ed9\u5b9a\u5143\u7d20 val \uff0c\u6211\u4eec\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u5b83\u5143\u7d20\uff0c\u6b64\u65f6\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u7ecf\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u7ed3\u70b9\u5230\u6839\u7ed3\u70b9\u8fd9\u6761\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u7ed3\u70b9\uff0c\u8be5\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 Heapify\u300d\u3002
     
    \u8003\u8651\u4ece\u5165\u5806\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5177\u4f53\u5730\uff0c\u6bd4\u8f83\u63d2\u5165\u7ed3\u70b9\u4e0e\u5176\u7236\u7ed3\u70b9\u7684\u503c\uff0c\u82e5\u63d2\u5165\u7ed3\u70b9\u66f4\u5927\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\uff1b\u5e76\u5faa\u73af\u4ee5\u4e0a\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u7ed3\u70b9\uff1b\u76f4\u81f3\u8d8a\u8fc7\u6839\u7ed3\u70b9\u65f6\u7ed3\u675f\uff0c\u6216\u5f53\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u7ed3\u70b9\u65f6\u63d0\u524d\u7ed3\u675f\u3002
     <1><2><3><4><5><6> 
     
     
     
     
     
     
    \u8bbe\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u6613\u5f97\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u56e0\u800c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.cpp
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.push_back(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p =  parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(maxHeap[i], maxHeap[p]);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.py
    def push(self, val: int):\n\"\"\" \u5143\u7d20\u5165\u5806 \"\"\"\n# \u6dfb\u52a0\u7ed3\u70b9\nself.max_heap.append(val)\n# \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1)\ndef sift_up(self, i: int):\n\"\"\" \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 \"\"\"\nwhile True:\n# \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\np = self.parent(i)\n# \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 or self.max_heap[i] <= self.max_heap[p]:\nbreak\n# \u4ea4\u6362\u4e24\u7ed3\u70b9\nself.swap(i, p)\n# \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n
     my_heap.go
    /* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n// \u6dfb\u52a0\u7ed3\u70b9\nh.data = append(h.data, val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nh.siftUp(len(h.data) - 1)\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\nfor true {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\np := h.parent(i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || h.data[i].(int) <= h.data[p].(int) {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nh.swap(i, p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
     my_heap.js
    /* \u5143\u7d20\u5165\u5806 */\npush(val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nthis.#maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.#siftUp(this.size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nconst p = this.#parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.#swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.ts
    /* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n// \u6dfb\u52a0\u7ed3\u70b9\nthis.maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.siftUp(this.size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nconst p = this.parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{push}\n[class]{maxHeap}-[func]{siftUp}\n
     my_heap.cs
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val)\n{\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.Add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i)\n{\nwhile (true)\n{\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p = parent(i);\n// \u82e5\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.swift
    /* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.append(val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(i: size() - 1)\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\nvar i = i\nwhile true {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nlet p = parent(i: i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || maxHeap[i] <= maxHeap[p] {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i: i, j: p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
     my_heap.zig
    // \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n// \u6dfb\u52a0\u7ed3\u70b9\ntry self.max_heap.?.append(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\ntry self.siftUp(self.size() - 1);\n}  // \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nvar p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\ntry self.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    "},{"location":"chapter_heap/heap/#_4","title":"\u5806\u9876\u5143\u7d20\u51fa\u5806","text":"
    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u6839\u7ed3\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\uff0c\u5982\u679c\u6211\u4eec\u76f4\u63a5\u5c06\u9996\u5143\u7d20\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff0c\u5219\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u90fd\u4f1a\u968f\u4e4b\u53d1\u751f\u79fb\u4f4d\uff08\u7d22\u5f15\u53d1\u751f\u53d8\u5316\uff09\uff0c\u8fd9\u6837\u540e\u7eed\u4f7f\u7528\u5806\u5316\u4fee\u590d\u5c31\u5f88\u9ebb\u70e6\u4e86\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u53d8\u52a8\uff0c\u91c7\u53d6\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\uff1a
     
       
    1. \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u5373\u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff09\uff1b
      2.  
    2. \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u56e0\u4e3a\u5df2\u7ecf\u4ea4\u6362\uff0c\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\uff1b
      3.  
    3. \u4ece\u6839\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\uff1b
      4.  
     
    \u987e\u540d\u601d\u4e49\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u4ece\u5e95\u81f3\u9876\u5806\u5316\u76f8\u53cd\uff0c\u6211\u4eec\u6bd4\u8f83\u6839\u7ed3\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u7ed3\u70b9\u7684\u503c\uff0c\u5c06\u6700\u5927\u7684\u5b50\u7ed3\u70b9\u4e0e\u6839\u7ed3\u70b9\u6267\u884c\u4ea4\u6362\uff0c\u5e76\u5faa\u73af\u4ee5\u4e0a\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u65f6\u7ed3\u675f\uff0c\u6216\u5f53\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u7ed3\u70b9\u65f6\u63d0\u524d\u7ed3\u675f\u3002
     <1><2><3><4><5><6><7><8><9><10> 
     
     
     
     
     
     
     
     
     
     
    \u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7c7b\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new EmptyStackException();\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nint val = maxHeap.remove(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap.get(l) > maxHeap.get(ma))\nma = l;\nif (r < size() && maxHeap.get(r) > maxHeap.get(ma))\nma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.cpp
    /* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n// \u5224\u7a7a\u5904\u7406\nif (empty()) {\nthrow out_of_range(\"\u5806\u4e3a\u7a7a\");\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap[0], maxHeap[size() - 1]);\n// \u5220\u9664\u7ed3\u70b9\nmaxHeap.pop_back();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (l < size() && maxHeap[l] > maxHeap[ma]) ma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\nswap(maxHeap[i], maxHeap[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.py
    def pop(self) -> int:\n\"\"\" \u5143\u7d20\u51fa\u5806 \"\"\"\n# \u5224\u7a7a\u5904\u7406\nassert not self.is_empty()\n# \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1)\n# \u5220\u9664\u7ed3\u70b9\nval = self.max_heap.pop()\n# \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0)\n# \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\ndef sift_down(self, i: int):\n\"\"\" \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 \"\"\"\nwhile True:\n# \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nl, r, ma = self.left(i), self.right(i), i\nif l < self.size() and self.max_heap[l] > self.max_heap[ma]:\nma = l\nif r < self.size() and self.max_heap[r] > self.max_heap[ma]:\nma = r\n# \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u7ed3\u70b9\nself.swap(i, ma)\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n
     my_heap.go
    /* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n// \u5224\u7a7a\u5904\u7406\nif h.isEmpty() {\nfmt.Println(\"error\")\nreturn nil\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nh.swap(0, h.size()-1)\n// \u5220\u9664\u7ed3\u70b9\nval := h.data[len(h.data)-1]\nh.data = h.data[:len(h.data)-1]\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nh.siftDown(0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\nfor true {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a max\nl, r, max := h.left(i), h.right(i), i\nif l < h.size() && h.data[l].(int) > h.data[max].(int) {\nmax = l\n}\nif r < h.size() && h.data[r].(int) > h.data[max].(int) {\nmax = r\n}\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif max == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nh.swap(i, max)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max\n}\n}\n
     my_heap.js
    /* \u5143\u7d20\u51fa\u5806 */\npop() {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new Error(\"\u5806\u4e3a\u7a7a\");\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.#swap(0, this.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nconst val = this.#maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.#siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.#left(i),\nr = this.#right(i);\nlet ma = i;\nif (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\nif (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.#swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.ts
    /* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new RangeError('Heap is empty.');\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.swap(0, this.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nconst val = this.maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.left(i),\nr = this.right(i);\nlet ma = i;\nif (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\nif (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{pop}\n[class]{maxHeap}-[func]{siftDown}\n
     my_heap.cs
    /* \u5143\u7d20\u51fa\u5806 */\nint pop()\n{\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfRangeException();\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nint val = maxHeap.Last();\nmaxHeap.RemoveAt(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i)\n{\nwhile (true)\n{\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u201c\u7ed3\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.swift
    /* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n// \u5224\u7a7a\u5904\u7406\nif isEmpty() {\nfatalError(\"\u5806\u4e3a\u7a7a\")\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(i: 0, j: size() - 1)\n// \u5220\u9664\u7ed3\u70b9\nlet val = maxHeap.remove(at: size() - 1)\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(i: 0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = left(i: i)\nlet r = right(i: i)\nvar ma = i\nif l < size(), maxHeap[l] > maxHeap[ma] {\nma = l\n}\nif r < size(), maxHeap[r] > maxHeap[ma] {\nma = r\n}\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i: i, j: ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n
     my_heap.zig
    // \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n// \u5224\u65ad\u5904\u7406\nif (self.isEmpty()) unreachable;\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\ntry self.swap(0, self.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nvar val = self.max_heap.?.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\ntry self.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n} // \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nvar l = left(i);\nvar r = right(i);\nvar ma = i;\nif (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\nif (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\ntry self.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    "},{"location":"chapter_heap/heap/#814","title":"8.1.4. \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"
       
    • \u4f18\u5148\u961f\u5217\u3002\u5806\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u7686\u975e\u5e38\u9ad8\u6548\u3002
      •  
    • \u5806\u6392\u5e8f\u3002\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u4f7f\u7528\u5176\u5efa\u5806\uff0c\u5e76\u4f9d\u6b21\u5168\u90e8\u5f39\u51fa\uff0c\u5219\u53ef\u4ee5\u5f97\u5230\u6709\u5e8f\u7684\u5e8f\u5217\u3002\u5f53\u7136\uff0c\u5806\u6392\u5e8f\u4e00\u822c\u65e0\u9700\u5f39\u51fa\u5143\u7d20\uff0c\u4ec5\u9700\u6bcf\u8f6e\u5c06\u5806\u9876\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u5c3e\u90e8\u5e76\u51cf\u5c0f\u5806\u7684\u957f\u5ea6\u5373\u53ef\u3002
      •  
    • \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002\u8fd9\u65e2\u662f\u4e00\u9053\u7ecf\u5178\u7b97\u6cd5\u9898\u76ee\uff0c\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u53d6\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u524d 10 \u9500\u91cf\u7684\u5546\u54c1\u7b49\u3002
      •  
    "},{"location":"chapter_heap/summary/","title":"8.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5806\u662f\u4e00\u68f5\u9650\u5b9a\u6761\u4ef6\u4e0b\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u6700\u5927\uff08\u5c0f\uff09\u3002
      •  
    • \u4f18\u5148\u961f\u5217\u5b9a\u4e49\u4e3a\u4e00\u79cd\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\u3002\u5806\u662f\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u6700\u5e38\u7528\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u548c\u5bf9\u5e94\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) \u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u3001\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
      •  
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u4e00\u822c\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
      •  
    • \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u4fee\u590d\u5806\u7684\u7279\u6027\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u4f7f\u7528\u5230\u3002
      •  
    • \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
      •  
    "},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1. \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"
    \u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\uff0c\u6211\u4eec\u4e00\u822c\u4f1a\u8054\u60f3\u5230\u6570\u5b66\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5927\u591a\u6570\u7b97\u6cd5\u5e76\u4e0d\u5305\u542b\u590d\u6742\u7684\u6570\u5b66\uff0c\u800c\u66f4\u50cf\u662f\u5728\u8003\u5bdf\u57fa\u672c\u903b\u8f91\uff0c\u800c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002
     
    \u5728\u6b63\u5f0f\u4ecb\u7ecd\u7b97\u6cd5\u4e4b\u524d\uff0c\u6211\u60f3\u544a\u8bc9\u4f60\u4e00\u4ef6\u6709\u8da3\u7684\u4e8b\uff1a\u5176\u5b9e\uff0c\u4f60\u5728\u8fc7\u53bb\u5df2\u7ecf\u5b66\u4f1a\u4e86\u5f88\u591a\u7b97\u6cd5\uff0c\u5e76\u4e14\u5df2\u7ecf\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u5c06\u4ecb\u7ecd\u4e24\u4e2a\u5177\u4f53\u4f8b\u5b50\u6765\u4f50\u8bc1\u3002
     
    \u4f8b\u4e00\uff1a\u62fc\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u6709\u8bb8\u591a\u90e8\u4ef6\u4e4b\u5916\uff0c\u8fd8\u4f1a\u9644\u9001\u8be6\u7ec6\u7684\u62fc\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e0a\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5373\u53ef\u62fc\u51fa\u590d\u6742\u7684\u79ef\u6728\u6a21\u578b\u3002
     
    \u5982\u679c\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u5927\u5927\u5c0f\u5c0f\u7684\u300c\u79ef\u6728\u300d\u5c31\u662f\u6570\u636e\u7ed3\u6784\uff0c\u800c\u300c\u62fc\u88c5\u8bf4\u660e\u4e66\u300d\u4e0a\u7684\u4e00\u7cfb\u5217\u6b65\u9aa4\u5c31\u662f\u7b97\u6cd5\u3002
     
    \u4f8b\u4e8c\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u4e2d\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u6709\u4e00\u4e2a\u5bf9\u5e94\u7684\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u7684\u82f1\u6587\u5b57\u6bcd\u8868\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u9700\u8981\u5728\u5b57\u5178\u4e2d\u67e5\u8be2\u4efb\u610f\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u4e00\u822c\u6211\u4eec\u4f1a\u8fd9\u6837\u505a\uff1a
     
       
    1. \u6253\u5f00\u5b57\u5178\u5927\u81f4\u4e00\u534a\u9875\u6570\u7684\u4f4d\u7f6e\uff0c\u67e5\u770b\u6b64\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff08\u5047\u8bbe\u4e3a \\(m\\) \uff09\uff1b
      2.  
    2. \u7531\u4e8e\u5728\u82f1\u6587\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u5728 \\(m\\) \u7684\u540e\u9762\uff0c\u56e0\u6b64\u5e94\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u4ec5\u5269\u540e\u534a\u90e8\u5206\uff1b
      3.  
    3. \u5faa\u73af\u6267\u884c\u6b65\u9aa4 1-2 \uff0c\u76f4\u5230\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u65f6\u7ec8\u6b62\u3002
      4.  
     <1><2><3><4><5> 
     
     
     
     
     
    \u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u7684\u6807\u914d\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u5927\u540d\u9f0e\u9f0e\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u3002\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u770b\u4f5c\u662f\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u300c\u6570\u7ec4\u300d\uff1b\u800c\u4ece\u7b97\u6cd5\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u6307\u4ee4\u770b\u4f5c\u662f\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7b97\u6cd5\u3002
     
    \u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\u3001\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u4e5f\u53ef\u4ee5\u7f16\u5199\u4ee3\u7801\u6765\u8c03\u7528 CPU, GPU \u6267\u884c\u7b97\u6cd5\uff0c\u4ece\u800c\u5c06\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u642c\u8fd0\u5230\u8ba1\u7b97\u673a\u4e2d\uff0c\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3\u5404\u5f0f\u5404\u6837\u7684\u590d\u6742\u95ee\u9898\u3002
     
    Tip
     
    \u8bfb\u5230\u8fd9\u91cc\uff0c\u5982\u679c\u4f60\u611f\u5230\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u6b64\u7c7b\u6982\u5ff5\u4e00\u77e5\u534a\u89e3\uff0c\u90a3\u4e48\u5c31\u592a\u597d\u4e86\uff01\u56e0\u4e3a\u8fd9\u6b63\u662f\u672c\u4e66\u5b58\u5728\u7684\u4ef7\u503c\uff0c\u63a5\u4e0b\u6765\uff0c\u672c\u4e66\u5c06\u4f1a\u4e00\u6b65\u6b65\u5730\u5f15\u5bfc\u4f60\u8fdb\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002
    "},{"location":"chapter_introduction/summary/","title":"1.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u7b97\u6cd5\u5728\u751f\u6d3b\u4e2d\u968f\u5904\u53ef\u89c1\uff0c\u5e76\u4e0d\u9ad8\u6df1\u83ab\u6d4b\u3002\u6211\u4eec\u5df2\u7ecf\u4e0d\u77e5\u4e0d\u89c9\u5730\u5b66\u4e60\u5230\u8bb8\u591a\u201c\u7b97\u6cd5\u201d\uff0c\u7528\u4e8e\u89e3\u51b3\u751f\u6d3b\u4e2d\u5927\u5927\u5c0f\u5c0f\u7684\u95ee\u9898\u3002
      •  
    • \u201c\u67e5\u5b57\u5178\u201d\u7684\u539f\u7406\u548c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4e00\u81f4\u3002\u4e8c\u5206\u4f53\u73b0\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
      •  
    • \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e24\u8005\u7d27\u5bc6\u8054\u7cfb\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u5e95\u5ea7\uff0c\u7b97\u6cd5\u662f\u53d1\u6325\u6570\u636e\u7ed3\u6784\u7684\u821e\u53f0\u3002
      •  
    • \u4e50\u9ad8\u79ef\u6728\u5bf9\u5e94\u6570\u636e\uff0c\u79ef\u6728\u5f62\u72b6\u548c\u8fde\u63a5\u5f62\u5f0f\u5bf9\u5e94\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6d41\u7a0b\u6b65\u9aa4\u5bf9\u5e94\u7b97\u6cd5\u3002
      •  
    "},{"location":"chapter_introduction/what_is_dsa/","title":"1.2. \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1. \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"
    \u300c\u7b97\u6cd5 Algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\u3002\u7b97\u6cd5\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a
     
       
    • \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u9700\u8981\u62e5\u6709\u660e\u786e\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
      •  
    • \u89e3\u5177\u6709\u786e\u5b9a\u6027\uff0c\u5373\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u65f6\uff0c\u8f93\u51fa\u4e00\u5b9a\u76f8\u540c\u3002
      •  
    • \u5177\u6709\u53ef\u884c\u6027\uff0c\u53ef\u5728\u6709\u9650\u6b65\u9aa4\u3001\u6709\u9650\u65f6\u95f4\u3001\u6709\u9650\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
      •  
    • \u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\uff0c\u5373\u53ef\u7528\u591a\u79cd\u8bed\u8a00\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"
    \u300c\u6570\u636e\u7ed3\u6784 Data Structure\u300d\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u4e3a\u4e86\u63d0\u9ad8\u6570\u636e\u5b58\u50a8\u548c\u64cd\u4f5c\u6027\u80fd\uff0c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u539f\u5219\u6709\uff1a
     
       
    • \u7a7a\u95f4\u5360\u7528\u5c3d\u53ef\u80fd\u5c0f\uff0c\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
      •  
    • \u6570\u636e\u64cd\u4f5c\u5c3d\u91cf\u5feb\uff0c\u5305\u62ec\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
      •  
    • \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002
      •  
     
    \u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\uff0c\u8fd9\u610f\u5473\u7740\u5982\u679c\u83b7\u5f97\u67d0\u65b9\u9762\u7684\u4f18\u52bf\uff0c\u5219\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u505a\u51fa\u59a5\u534f\u3002\u4f8b\u5982\uff0c\u94fe\u8868\u76f8\u5bf9\u4e8e\u6570\u7ec4\uff0c\u6570\u636e\u6dfb\u52a0\u5220\u9664\u64cd\u4f5c\u66f4\u52a0\u65b9\u4fbf\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u901f\u5ea6\uff1b\u56fe\u76f8\u5bf9\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u591a\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
    "},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3. \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"
    \u300c\u6570\u636e\u7ed3\u6784\u300d\u4e0e\u300c\u7b97\u6cd5\u300d\u662f\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u5d4c\u5408\u7684\uff0c\u4f53\u73b0\u5728\uff1a
     
       
    • \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u5e95\u5ea7\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u5bf9\u5e94\u65b9\u6cd5\u3002
      •  
    • \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u53ef\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
      •  
    • \u7b97\u6cd5\u6709\u5bf9\u5e94\u6700\u4f18\u7684\u6570\u636e\u7ed3\u6784\u3002\u7ed9\u5b9a\u7b97\u6cd5\uff0c\u4e00\u822c\u53ef\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u800c\u6700\u7ec8\u6267\u884c\u6548\u7387\u5f80\u5f80\u76f8\u5dee\u5f88\u5927\u3002
      •  
     
     
     Fig. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb 
     
    \u5982\u679c\u5c06\u300cLEGO \u4e50\u9ad8\u300d\u7c7b\u6bd4\u5230\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\uff0c\u90a3\u4e48\u53ef\u4ee5\u5f97\u5230\u4e0b\u8868\u6240\u793a\u7684\u5bf9\u5e94\u5173\u7cfb\u3002
     \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 LEGO \u4e50\u9ad8 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b 
    \u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0
     
    \u5728\u5b9e\u9645\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\u76f4\u63a5\u7b80\u79f0\u4e3a\u300c\u7b97\u6cd5\u300d\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u719f\u79f0\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u5bdf\u4e86\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u90e8\u5206\u77e5\u8bc6\u3002
    "},{"location":"chapter_preface/about_the_book/","title":"0.1. \u00a0 \u5173\u4e8e\u672c\u4e66","text":"
    \u672c\u9879\u76ee\u81f4\u529b\u4e8e\u6784\u5efa\u4e00\u672c\u5f00\u6e90\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\u3002
     
       
    • \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\uff1b
      •  
    • \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00\uff1b
      •  
    • \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\uff1b
      •  
    "},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1. \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u521d\u5b66\u8005\u300d\uff0c\u5b8c\u5168\u6ca1\u6709\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u5c11\u91cf\u5237\u9898\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6726\u80e7\u7684\u7406\u89e3\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u8fd9\u672c\u4e66\u5c31\u662f\u4e3a\u4f60\u800c\u5199\uff01
     
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u8001\u624b\u300d\uff0c\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u63a5\u89e6\u8fc7\u5927\u591a\u6570\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u4ee5\u5e2e\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002
     
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u5927\u4f6c\u300d\uff0c\u5e0c\u671b\u53ef\u4ee5\u5f97\u5230\u4f60\u7684\u5b9d\u8d35\u610f\u89c1\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002
     
    \u524d\u7f6e\u6761\u4ef6
     
    \u60a8\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002
    "},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2. \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"
    \u672c\u4e66\u4e3b\u8981\u5185\u5bb9\u6709\uff1a
     
       
    • \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3001\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u4f30\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5305\u62ec\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\uff1a\u5e38\u7528\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u5728\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3001\u6570\u636e\u7ed3\u6784\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u6563\u5217\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f18\u52a3\u52bf\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
      •  
    • \u7b97\u6cd5\uff1a\u67e5\u627e\u7b97\u6cd5\u3001\u6392\u5e8f\u7b97\u6cd5\u3001\u641c\u7d22\u4e0e\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u5206\u6cbb\u7b97\u6cd5\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f7f\u7528\u573a\u666f\u3001\u4f18\u52a3\u52bf\u3001\u65f6\u7a7a\u6548\u7387\u3001\u5b9e\u73b0\u65b9\u6cd5\u3001\u793a\u4f8b\u9898\u76ee\u7b49\u3002
      •  
     
     
     Fig. Hello \u7b97\u6cd5\u5185\u5bb9\u7ed3\u6784 
    "},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3. \u00a0 \u81f4\u8c22","text":"
    \u672c\u4e66\u7684\u6210\u4e66\u8fc7\u7a0b\u4e2d\uff0c\u6211\u83b7\u5f97\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a
     
       
    • \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u65f6\u60a8\u544a\u8bc9\u6211\u201c\u89c9\u5f97\u5e94\u8be5\u505a\u5c31\u53bb\u505a\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\u3002
      •  
    • \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u62c5\u4efb\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89c6\u89d2\u63d0\u51fa\u4e86\u8bb8\u591a\u5efa\u8bae\uff0c\u4f7f\u8fd9\u672c\u4e66\u66f4\u52a0\u9002\u5408\u521d\u5b66\u8005\u6765\u9605\u8bfb\u3002
      •  
    • \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e2a\u597d\u542c\u53c8\u6709\u6897\u540d\u5b57\uff0c\u76f4\u63a5\u5524\u8d77\u6211\u6700\u521d\u6572\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u56de\u5fc6\u3002
      •  
    • \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u5c01\u9762\u548c LOGO \uff0c\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u524d\u540e\u591a\u6b21\u5e2e\u5fd9\u4fee\u6539\uff0c\u8c22\u8c22\u4f60\u7684\u8010\u5fc3\u3002
      •  
    • \u611f\u8c22 @squidfunk \u7ed9\u51fa\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4f18\u79c0\u5f00\u6e90\u9879\u76ee Material-for-MkDocs \u3002
      •  
     
    \u672c\u4e66\u9f13\u52b1\u201c\u624b\u8111\u5e76\u7528\u201d\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u70b9\u4e0a\u53d7\u5230\u4e86\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u5f88\u5927\u5f71\u54cd\uff0c\u4e5f\u5728\u6b64\u5411\u5404\u4f4d\u540c\u5b66\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u8457\u4f5c\uff0c\u5305\u62ec\u4e2d\u6587\u7248\u3001\u82f1\u6587\u7248\u3001\u674e\u6c90\u8001\u5e08 bilibili \u4e3b\u9875\u3002
     
    \u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u4e0e\u6587\u7ae0\uff0c\u8fd9\u4e9b\u8457\u4f5c\u4e3a\u672c\u4e66\u4f5c\u51fa\u4e86\u5f88\u597d\u7684\u699c\u6837\uff0c\u4fdd\u8bc1\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u6b63\u786e\u6027\u4e0e\u8d28\u91cf\uff0c\u611f\u8c22\u5404\u4f4d\u8001\u5e08\u4e0e\u524d\u8f88\u7684\u7cbe\u5f69\u521b\u4f5c\uff01
     
    \u611f\u8c22\u7236\u6bcd\uff0c\u4f60\u4eec\u4e00\u8d2f\u7684\u652f\u6301\u4e0e\u9f13\u52b1\u7ed9\u4e86\u6211\u81ea\u7531\u5ea6\u6765\u505a\u8fd9\u4e9b\u6709\u8da3\u7684\u4e8b\u3002
    "},{"location":"chapter_preface/suggestions/","title":"0.2. \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"
    \u5efa\u8bae\u901a\u8bfb\u672c\u8282\u5185\u5bb9\uff0c\u4ee5\u83b7\u53d6\u6700\u4f73\u9605\u8bfb\u4f53\u9a8c\u3002
    "},{"location":"chapter_preface/suggestions/#021","title":"0.2.1. \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"
    \u603b\u4f53\u4e0a\u770b\uff0c\u6211\u8ba4\u4e3a\u53ef\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002
     
       
    1. \u7b97\u6cd5\u5165\u95e8\u3002\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u3001\u7528\u6cd5\uff0c\u5b66\u4e60\u5404\u79cd\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u3001\u6548\u7387\u7b49\u3002
      2.  
    2. \u5237\u7b97\u6cd5\u9898\u3002\u53ef\u4ee5\u5148\u4ece\u70ed\u95e8\u9898\u5355\u5f00\u5237\uff0c\u63a8\u8350\u5251\u6307 Offer\u3001LeetCode Hot 100\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u91cf\uff0c\u719f\u6089\u5927\u591a\u6570\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521a\u5f00\u59cb\u5237\u9898\u65f6\uff0c\u201c\u9057\u5fd8\u201d\u662f\u6700\u5927\u7684\u56f0\u6270\u70b9\uff0c\u4f46\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\uff0c\u8bf7\u4e0d\u8981\u62c5\u5fc3\u3002\u5b66\u4e60\u4e2d\u6709\u4e00\u79cd\u6982\u5ff5\u53eb\u201c\u5468\u671f\u6027\u56de\u987e\u201d\uff0c\u540c\u4e00\u9053\u9898\u9694\u6bb5\u65f6\u95f4\u505a\u4e00\u6b21\uff0c\u5728\u91cd\u590d 3 \u8f6e\u4ee5\u4e0a\u540e\uff0c\u5f80\u5f80\u5c31\u80fd\u7262\u8bb0\u4e8e\u5fc3\u4e86\u3002
      3.  
    3. \u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u3001\u7b97\u6cd5\u6559\u6750\uff0c\u4e0d\u65ad\u5730\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5f00\u59cb\u91c7\u7528\u8fdb\u9636\u5237\u9898\u65b9\u6848\uff0c\u4f8b\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u4e2d\u627e\u5230\u3002
      4.  
     
    \u4f5c\u4e3a\u4e00\u672c\u5165\u95e8\u6559\u7a0b\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u5bf9\u5e94\u201c\u7b2c\u4e00\u9636\u6bb5\u201d\uff0c\u81f4\u529b\u4e8e\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5f00\u5c55\u7b2c\u4e8c\u3001\u4e09\u9636\u6bb5\u7684\u5b66\u4e60\u3002
     
     
     Fig. \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf 
    "},{"location":"chapter_preface/suggestions/#022","title":"0.2.2. \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"
    \u6807\u9898\u540e\u6807\u6ce8 * \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u8f83\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002
     
    \u6587\u7ae0\u4e2d\u7684\u91cd\u8981\u540d\u8bcd\u4f1a\u7528 \u300c\u62ec\u53f7\u300d \u6807\u6ce8\uff0c\u4f8b\u5982 \u300c\u6570\u7ec4 Array\u300d \u3002\u5efa\u8bae\u8bb0\u4f4f\u8fd9\u4e9b\u540d\u8bcd\uff0c\u5305\u62ec\u82f1\u6587\u7ffb\u8bd1\uff0c\u4ee5\u4fbf\u540e\u7eed\u9605\u8bfb\u6587\u732e\u65f6\u4f7f\u7528\u3002
     
    \u91cd\u70b9\u5185\u5bb9\u3001\u603b\u8d77\u53e5\u3001\u603b\u7ed3\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u6b64\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
     
    \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 \u201c\u53cc\u5f15\u53f7\u201d \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002
     
    \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    \"\"\" \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 \"\"\"\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    // \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n// \u591a\u884c\n// \u6ce8\u91ca\n
    "},{"location":"chapter_preface/suggestions/#023","title":"0.2.3. \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"
    \u89c6\u9891\u548c\u56fe\u7247\u76f8\u6bd4\u4e8e\u6587\u5b57\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\u66f4\u9ad8\uff0c\u66f4\u5bb9\u6613\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u77e5\u8bc6\u91cd\u96be\u70b9\u4f1a\u4e3b\u8981\u4ee5\u52a8\u753b\u3001\u56fe\u89e3\u7684\u5f62\u5f0f\u5448\u73b0\uff0c\u800c\u6587\u5b57\u7684\u4f5c\u7528\u5219\u662f\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002
     
    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u82e5\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u5efa\u8bae\u4f60\u4ee5\u56fe\u4e3a\u4e3b\u7ebf\uff0c\u5c06\u6587\u5b57\u5185\u5bb9\uff08\u4e00\u822c\u5728\u56fe\u7684\u4e0a\u65b9\uff09\u5bf9\u9f50\u5230\u56fe\u4e2d\u5185\u5bb9\uff0c\u7efc\u5408\u6765\u7406\u89e3\u3002
     
     
     Fig. \u52a8\u753b\u56fe\u89e3\u793a\u4f8b 
    "},{"location":"chapter_preface/suggestions/#024","title":"0.2.4. \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"
    \u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728GitHub \u4ed3\u5e93\uff0c\u6e90\u4ee3\u7801\u5305\u542b\u8be6\u7ec6\u6ce8\u91ca\uff0c\u914d\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4ee5\u76f4\u63a5\u8fd0\u884c\u3002
     
       
    • \u82e5\u5b66\u4e60\u65f6\u95f4\u7d27\u5f20\uff0c\u5efa\u8bae\u81f3\u5c11\u5c06\u6240\u6709\u4ee3\u7801\u901a\u8bfb\u5e76\u8fd0\u884c\u4e00\u904d\u3002
      •  
    • \u82e5\u65f6\u95f4\u5141\u8bb8\uff0c\u5f3a\u70c8\u5efa\u8bae\u5bf9\u7167\u7740\u4ee3\u7801\u81ea\u5df1\u6572\u4e00\u904d\u3002\u76f8\u6bd4\u4e8e\u8bfb\u4ee3\u7801\uff0c\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u65b0\u7684\u6536\u83b7\u3002
      •  
     
     
     Fig. \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b 
     
    \u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u53c2\u7167\u9644\u5f55\u6559\u7a0b\uff0c\u5982\u679c\u5df2\u6709\u53ef\u76f4\u63a5\u8df3\u8fc7\u3002
     
    \u7b2c\u4e8c\u6b65\uff1a\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u547d\u4ee4\u884c\u6765\u514b\u9686\u4ee3\u7801\u4ed3\u3002
     
    git clone https://github.com/krahets/hello-algo.git\n
     
    \u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201cDownload ZIP\u201d\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002
     
     
     Fig. \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 
     
    \u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u82e5\u4ee3\u7801\u5757\u7684\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\uff0c\u5219\u53ef\u5728\u4ed3\u5e93 codes \u6587\u4ef6\u5939\u4e2d\u627e\u5230\u5bf9\u5e94\u7684 \u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4ee5\u5e2e\u52a9\u4f60\u7701\u53bb\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u5c06\u7cbe\u529b\u96c6\u4e2d\u5728\u5b66\u4e60\u5185\u5bb9\u4e0a\u3002
     
     
     Fig. \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 
    "},{"location":"chapter_preface/suggestions/#025","title":"0.2.5. \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"
    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u201c\u60ef\u7740\u201d\u90a3\u4e9b\u5f04\u4e0d\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u7559\u4e0b\u4f60\u7684\u95ee\u9898\uff0c\u5c0f\u4f19\u4f34\u4eec\u548c\u6211\u90fd\u4f1a\u7ed9\u4e88\u89e3\u7b54\uff0c\u60a8\u4e00\u822c 2 \u65e5\u5185\u4f1a\u5f97\u5230\u56de\u590d\u3002
     
    \u540c\u65f6\uff0c\u4e5f\u5e0c\u671b\u4f60\u53ef\u4ee5\u591a\u82b1\u65f6\u95f4\u901b\u901b\u8bc4\u8bba\u533a\u3002\u4e00\u65b9\u9762\uff0c\u53ef\u4ee5\u770b\u770b\u5927\u5bb6\u9047\u5230\u4e86\u4ec0\u4e48\u95ee\u9898\uff0c\u53cd\u8fc7\u6765\u67e5\u6f0f\u8865\u7f3a\uff0c\u8fd9\u5f80\u5f80\u53ef\u4ee5\u5f15\u8d77\u66f4\u52a0\u6df1\u5ea6\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e5f\u5e0c\u671b\u4f60\u53ef\u4ee5\u6177\u6168\u5730\u89e3\u7b54\u5c0f\u4f19\u4f34\u4eec\u7684\u95ee\u9898\u3001\u5206\u4eab\u81ea\u5df1\u7684\u89c1\u89e3\uff0c\u5927\u5bb6\u4e92\u76f8\u5b66\u4e60\u4e0e\u8fdb\u6b65\uff01
     
     
     Fig. \u8bc4\u8bba\u533a\u793a\u4f8b 
    "},{"location":"chapter_preface/summary/","title":"0.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u672c\u4e66\u4e3b\u8981\u9762\u5411\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5bf9\u4e8e\u5df2\u7ecf\u6709\u4e00\u5b9a\u79ef\u7d2f\u7684\u540c\u5b66\uff0c\u8fd9\u672c\u4e66\u53ef\u4ee5\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u6e90\u4ee3\u7801\u53ef\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6765\u4f7f\u7528\u3002
      •  
    • \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5206\u4e3a\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u8986\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
      •  
    • \u5bf9\u4e8e\u7b97\u6cd5\u5c0f\u767d\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u662f\u975e\u5e38\u6709\u5fc5\u8981\u7684\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
      •  
    • \u4e66\u5185\u7684\u52a8\u753b\u548c\u56fe\u89e3\u5f80\u5f80\u4ecb\u7ecd\u7684\u662f\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\uff0c\u5728\u9605\u8bfb\u65f6\u5e94\u8be5\u591a\u52a0\u5173\u6ce8\u3002
      •  
    • \u5b9e\u8df5\u662f\u5b66\u4e60\u7f16\u7a0b\u7684\u6700\u4f73\u65b9\u5f0f\uff0c\u5f3a\u70c8\u63a8\u8350\u8fd0\u884c\u6e90\u4ee3\u7801\uff0c\u52a8\u624b\u6572\u4ee3\u7801\u3002
      •  
    • \u672c\u4e66\u63d0\u4f9b\u4e86\u8ba8\u8bba\u533a\uff0c\u9047\u5230\u7591\u60d1\u53ef\u4ee5\u968f\u65f6\u63d0\u95ee\u3002
      •  
    "},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"
    [1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).
     
    [2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).
     
    [3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08 C \u8bed\u8a00\u7248\uff09.
     
    [4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08 C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.
     
    [5] \u9a6c\u514b\u00b7\u827e\u4f26\u00b7\u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.
     
    [6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.
     
    [7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.
     
    [8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition). 
     
    [9] Aston Zhang, et al. Dive into Deep Learning.
    "},{"location":"chapter_searching/binary_search/","title":"10.2. \u00a0 \u4e8c\u5206\u67e5\u627e","text":"
    \u300c\u4e8c\u5206\u67e5\u627e Binary Search\u300d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u67e5\u627e\u76ee\u6807\u5143\u7d20\u3002
     
    \u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u6709\u4e24\u4e2a\u524d\u7f6e\u6761\u4ef6\uff1a
     
       
    • \u8981\u6c42\u8f93\u5165\u6570\u636e\u662f\u6709\u5e8f\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u5224\u65ad\u5927\u5c0f\u5173\u7cfb\u6765\u6392\u9664\u4e00\u534a\u7684\u641c\u7d22\u533a\u95f4\uff1b
      •  
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u4f7f\u7528\u6548\u7387\u5f88\u4f4e\uff0c\u56e0\u4e3a\u5176\u5728\u5faa\u73af\u4e2d\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\u3002
      •  
    "},{"location":"chapter_searching/binary_search/#1021","title":"10.2.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6392\u5e8f\u6570\u7ec4 nums \uff0c\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\u3002\u6570\u7ec4\u7684\u7d22\u5f15\u53d6\u503c\u8303\u56f4\u4e3a
     \\[ 0, 1, 2, \\cdots, n-1 \\] 
    \u4f7f\u7528\u300c\u533a\u95f4\u300d\u6765\u8868\u793a\u8fd9\u4e2a\u53d6\u503c\u8303\u56f4\u7684\u65b9\u6cd5\u4e3b\u8981\u6709\u4e24\u79cd\uff1a
     
       
    1. \u53cc\u95ed\u533a\u95f4 \\([0, n-1]\\) \uff0c\u5373\u4e24\u4e2a\u8fb9\u754c\u90fd\u5305\u542b\u81ea\u8eab\uff1b\u6b64\u65b9\u6cd5\u4e0b\uff0c\u533a\u95f4 \\([0, 0]\\) \u4ecd\u5305\u542b\u4e00\u4e2a\u5143\u7d20\uff1b
      2.  
    2. \u5de6\u95ed\u53f3\u5f00 \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\u3001\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\uff1b\u6b64\u65b9\u6cd5\u4e0b\uff0c\u533a\u95f4 \\([0, 0)\\) \u4e3a\u7a7a\uff1b
      3.  
    "},{"location":"chapter_searching/binary_search/#_1","title":"\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5b9e\u73b0","text":"
    \u9996\u5148\uff0c\u6211\u4eec\u5148\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u8868\u793a\uff0c\u5728\u6570\u7ec4 nums \u4e2d\u67e5\u627e\u76ee\u6807\u5143\u7d20 target \u7684\u5bf9\u5e94\u7d22\u5f15\u3002
     <1><2><3><4><5><6><7> 
     
     
     
     
     
     
     
    \u4e8c\u5206\u67e5\u627e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e0b\u7684\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int>& nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.size() - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.py
    def binary_search(nums: List[int], target: int) -> int:\n\"\"\" \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 \"\"\"\n# \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j = 0, len(nums) - 1\nwhile i <= j:\nm = (i + j) // 2        # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:    # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\nelif nums[m] > target:  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\nelse:\nreturn m            # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1                   # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j := 0, len(nums)-1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nfor i <= j {\nm := (i + j) / 2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nconst m = parseInt((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u5728 JS \u4e2d\u9700\u4f7f\u7528 parseInt \u51fd\u6570\u53d6\u6574\nif (nums[m] < target)          // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target)     // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse\nreturn m;                  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nconst m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {        // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
     binary_search.c
    [class]{}-[func]{binarySearch}\n
     binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target)\n{\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.Length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j)\n{\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i = 0\nvar j = nums.count - 1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j {\nlet m = (i + j) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i: usize = 0;\nvar j: usize = nums.items.len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = (i + j) / 2;                    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    "},{"location":"chapter_searching/binary_search/#_2","title":"\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5b9e\u73b0","text":"
    \u5f53\u7136\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5199\u51fa\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(vector<int>& nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.size();\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.py
    def binary_search1(nums: List[int], target: int) -> int:\n\"\"\" \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 \"\"\"\n# \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j = 0, len(nums)\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j:\nm = (i + j) // 2        # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:    # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\nelif nums[m] > target:  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\nelse:                   # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\nreturn -1                   # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearch1(nums []int, target int) int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j := 0, len(nums)\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nfor i < j {\nm := (i + j) / 2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearch1(nums, target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nconst m = parseInt((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u5728 JS \u4e2d\u9700\u4f7f\u7528 parseInt \u51fd\u6570\u53d6\u6574\nif (nums[m] < target)          // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target)     // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                           // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearch1(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nconst m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {        // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
     binary_search.c
    [class]{}-[func]{binarySearch1}\n
     binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(int[] nums, int target)\n{\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.Length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j)\n{\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearch1(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i = 0\nvar j = nums.count\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j {\nlet m = (i + j) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\nfn binarySearch1(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i: usize = 0;\nvar j: usize = nums.items.len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = (i + j) / 2;                    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    "},{"location":"chapter_searching/binary_search/#_3","title":"\u4e24\u79cd\u8868\u793a\u5bf9\u6bd4","text":"
    \u5bf9\u6bd4\u4e0b\u6765\uff0c\u4e24\u79cd\u8868\u793a\u7684\u4ee3\u7801\u5199\u6cd5\u6709\u4ee5\u4e0b\u4e0d\u540c\u70b9\uff1a
     \u8868\u793a\u65b9\u6cd5 \u521d\u59cb\u5316\u6307\u9488 \u7f29\u5c0f\u533a\u95f4 \u5faa\u73af\u7ec8\u6b62\u6761\u4ef6 \u53cc\u95ed\u533a\u95f4 \\([0, n-1]\\) \\(i = 0\\) , \\(j = n-1\\) \\(i = m + 1\\) , \\(j = m - 1\\) \\(i > j\\) \u5de6\u95ed\u53f3\u5f00 \\([0, n)\\) \\(i = 0\\) , \\(j = n\\) \\(i = m + 1\\) , \\(j = m\\) \\(i = j\\) 
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\uff0c\u7531\u4e8e\u5bf9\u5de6\u53f3\u4e24\u8fb9\u754c\u7684\u5b9a\u4e49\u662f\u76f8\u540c\u7684\uff0c\u56e0\u6b64\u7f29\u5c0f\u533a\u95f4\u7684 \\(i\\) , \\(j\\) \u5904\u7406\u65b9\u6cd5\u4e5f\u662f\u5bf9\u79f0\u7684\uff0c\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\u3002\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002
    "},{"location":"chapter_searching/binary_search/#_4","title":"\u5927\u6570\u8d8a\u754c\u5904\u7406","text":"
    \u5f53\u6570\u7ec4\u957f\u5ea6\u5f88\u5927\u65f6\uff0c\u52a0\u6cd5 \\(i + j\\) \u7684\u7ed3\u679c\u6709\u53ef\u80fd\u4f1a\u8d85\u51fa int \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u5728\u6b64\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u6362\u4e00\u79cd\u8ba1\u7b97\u4e2d\u70b9\u7684\u5199\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    # Python \u4e2d\u7684\u6570\u5b57\u7406\u8bba\u4e0a\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\n# \u56e0\u6b64\u65e0\u9700\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nm := (i + j) / 2\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nm := i + (j - i) / 2\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nlet m = parseInt((i + j) / 2);\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = parseInt(i + (j - i) / 2);\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa Number \u7684\u53d6\u503c\u8303\u56f4\nlet m = Math.floor((i + j) / 2);\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = Math.floor(i + (j - i) / 2);\n
     
    \n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nlet m = (i + j) / 2\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = i + (j - 1) / 2\n
     
    \n
    "},{"location":"chapter_searching/binary_search/#1022","title":"10.2.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\uff1b\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u7684\u533a\u95f4\uff0c\u56e0\u6b64\u5faa\u73af\u8f6e\u6570\u4e3a \\(\\log_2 n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 i , j \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002
    "},{"location":"chapter_searching/binary_search/#1023","title":"10.2.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u4e8c\u5206\u67e5\u627e\u6548\u7387\u5f88\u9ad8\uff0c\u4f53\u73b0\u5728\uff1a
     
       
    • \u4e8c\u5206\u67e5\u627e\u65f6\u95f4\u590d\u6742\u5ea6\u4f4e\u3002\u5bf9\u6570\u9636\u5728\u6570\u636e\u91cf\u5f88\u5927\u65f6\u5177\u6709\u5de8\u5927\u4f18\u52bf\uff0c\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700\u8981 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u4e0d\u9700\u8981\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u5bf9\u4e8e\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u6765\u5b9e\u73b0\u67e5\u627e\u7684\u7b97\u6cd5\u6765\u8bf4\uff0c\u5176\u66f4\u52a0\u8282\u7ea6\u7a7a\u95f4\u4f7f\u7528\u3002
      •  
     
    \u4f46\u5e76\u4e0d\u610f\u5473\u7740\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u5e94\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u65e0\u5e8f\u7684\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u6267\u884c\u6570\u636e\u6392\u5e8f\uff0c\u90a3\u4e48\u662f\u5f97\u4e0d\u507f\u5931\u7684\uff0c\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e00\u822c\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u5dee\u3002\u518d\u4f8b\u5982\uff0c\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4e86\u4fdd\u6301\u6570\u7ec4\u7684\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u7531\u4e8e\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u8bbf\u95ee\u7d22\u5f15\u662f \u201c\u975e\u8fde\u7eed\u201d \u7684\uff0c\u56e0\u6b64\u94fe\u8868\u6216\u8005\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u90fd\u65e0\u6cd5\u4f7f\u7528\u3002
      •  
    • \u5728\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u7684\u6027\u80fd\u66f4\u597d\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700\u8981 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5728\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
      •  
    "},{"location":"chapter_searching/hashing_search/","title":"10.3. \u00a0 \u54c8\u5e0c\u67e5\u627e","text":"
    Question
     
    \u5728\u6570\u636e\u91cf\u5f88\u5927\u65f6\uff0c\u300c\u7ebf\u6027\u67e5\u627e\u300d\u592a\u6162\uff1b\u800c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u8981\u6c42\u6570\u636e\u5fc5\u987b\u662f\u6709\u5e8f\u7684\uff0c\u5e76\u4e14\u53ea\u80fd\u5728\u6570\u7ec4\u4e2d\u5e94\u7528\u3002\u90a3\u4e48\u662f\u5426\u6709\u65b9\u6cd5\u53ef\u4ee5\u540c\u65f6\u907f\u514d\u4e0a\u8ff0\u7f3a\u70b9\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\uff0c\u6b64\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u54c8\u5e0c\u67e5\u627e\u300d\u3002
     
    \u300c\u54c8\u5e0c\u67e5\u627e Hash Searching\u300d\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u9700\u8981\u7684\u300c\u952e\u503c\u5bf9 Key Value Pair\u300d\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u5b9e\u73b0\u201c\u952e \\(\\rightarrow\\) \u503c\u201d\u6620\u5c04\u67e5\u627e\uff0c\u4f53\u73b0\u7740\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002
    "},{"location":"chapter_searching/hashing_search/#1031","title":"10.3.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u5982\u679c\u6211\u4eec\u60f3\u8981\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 target \uff0c\u83b7\u53d6\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u90a3\u4e48\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u5b9e\u73b0\u67e5\u627e\u3002
     
     
     Fig. \u54c8\u5e0c\u67e5\u627e\u6570\u7ec4\u7d22\u5f15 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hashing_search.java
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(Map<Integer, Integer> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.getOrDefault(target, -1);\n}\n
     hashing_search.cpp
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(unordered_map<int, int> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nif (map.find(target) == map.end())\nreturn -1;\nreturn map[target];\n}\n
     hashing_search.py
    def hashing_search_array(mapp: Dict[int, int], target: int) -> int:\n\"\"\" \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 \"\"\"\n# \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n# \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn mapp.get(target, -1)\n
     hashing_search.go
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc hashingSearchArray(m map[int]int, target int) int {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nif index, ok := m[target]; ok {\nreturn index\n} else {\nreturn -1\n}\n}\n
     hashing_search.js
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction hashingSearchArray(map, target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.has(target) ? map.get(target) : -1;\n}\n
     hashing_search.ts
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction hashingSearchArray(map: Map<number, number>, target: number): number {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.has(target) ? (map.get(target) as number) : -1;\n}\n
     hashing_search.c
    [class]{}-[func]{hashingSearchArray}\n
     hashing_search.cs
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(Dictionary<int, int> map, int target)\n{\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.GetValueOrDefault(target, -1);\n}\n
     hashing_search.swift
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc hashingSearchArray(map: [Int: Int], target: Int) -> Int {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map[target, default: -1]\n}\n
     hashing_search.zig
    // \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09\nfn hashingSearchArray(comptime T: type, map: std.AutoHashMap(T, T), target: T) T {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1   \nif (map.getKey(target) == null) return -1;\nreturn map.get(target).?;\n}\n
     
    \u518d\u6bd4\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u7ed9\u5b9a\u4e00\u4e2a\u76ee\u6807\u7ed3\u70b9\u503c target \uff0c\u83b7\u53d6\u5bf9\u5e94\u7684\u94fe\u8868\u7ed3\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u4e5f\u53ef\u4ee5\u4f7f\u7528\u54c8\u5e0c\u67e5\u627e\u5b9e\u73b0\u3002
     
     
     Fig. \u54c8\u5e0c\u67e5\u627e\u94fe\u8868\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hashing_search.java
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode hashingSearchLinkedList(Map<Integer, ListNode> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.getOrDefault(target, null);\n}\n
     hashing_search.cpp
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode* hashingSearchLinkedList(unordered_map<int, ListNode*> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de nullptr\nif (map.find(target) == map.end())\nreturn nullptr;\nreturn map[target];\n}\n
     hashing_search.py
    def hashing_search_linkedlist(mapp: Dict[int, ListNode], target: int) -> Optional[ListNode]:\n\"\"\" \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 \"\"\"\n# \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n# \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de None\nreturn mapp.get(target, None)\n
     hashing_search.go
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc hashingSearchLinkedList(m map[int]*ListNode, target int) *ListNode {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de nil\nif node, ok := m[target]; ok {\nreturn node\n} else {\nreturn nil\n}\n}\n
     hashing_search.js
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunction hashingSearchLinkedList(map, target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.has(target) ? map.get(target) : null;\n}\n
     hashing_search.ts
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunction hashingSearchLinkedList(map: Map<number, ListNode>, target: number): ListNode | null {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.has(target) ? (map.get(target) as ListNode) : null;\n}\n
     hashing_search.c
    [class]{}-[func]{hashingSearchLinkedList}\n
     hashing_search.cs
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode? hashingSearchLinkedList(Dictionary<int, ListNode> map, int target)\n{\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.GetValueOrDefault(target);\n}\n
     hashing_search.swift
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc hashingSearchLinkedList(map: [Int: ListNode], target: Int) -> ListNode? {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map[target]\n}\n
     hashing_search.zig
    // \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09\nfn hashingSearchLinkedList(comptime T: type, map: std.AutoHashMap(T, *inc.ListNode(T)), target: T) ?*inc.ListNode(T) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null \nif (map.getKey(target) == null) return null;\nreturn map.get(target);\n}\n
    "},{"location":"chapter_searching/hashing_search/#1032","title":"10.3.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u54c8\u5e0c\u8868\u7684\u67e5\u627e\u64cd\u4f5c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\u3002
    "},{"location":"chapter_searching/hashing_search/#1033","title":"10.3.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \uff0c\u8fd9\u610f\u5473\u7740\u65e0\u8bba\u662f\u9ad8\u9891\u589e\u5220\u8fd8\u662f\u9ad8\u9891\u67e5\u627e\u573a\u666f\uff0c\u54c8\u5e0c\u67e5\u627e\u7684\u6027\u80fd\u8868\u73b0\u90fd\u975e\u5e38\u597d\u3002\u5f53\u7136\uff0c\u4e00\u5207\u7684\u524d\u63d0\u662f\u4fdd\u8bc1\u54c8\u5e0c\u8868\u672a\u9000\u5316\u3002
     
    \u5373\u4f7f\u5982\u6b64\uff0c\u54c8\u5e0c\u67e5\u627e\u4ecd\u5b58\u5728\u4e00\u4e9b\u95ee\u9898\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u9700\u8981\u6839\u636e\u60c5\u51b5\u7075\u6d3b\u9009\u62e9\u65b9\u6cd5\u3002
     
       
    • \u8f85\u52a9\u54c8\u5e0c\u8868 \u9700\u8981\u4f7f\u7528 \\(O(n)\\) \u7684\u989d\u5916\u7a7a\u95f4\uff0c\u610f\u5473\u7740\u9700\u8981\u9884\u7559\u66f4\u591a\u7684\u8ba1\u7b97\u673a\u5185\u5b58\uff1b
      •  
    • \u5efa\u7acb\u548c\u7ef4\u62a4\u54c8\u5e0c\u8868\u9700\u8981\u65f6\u95f4\uff0c\u56e0\u6b64\u54c8\u5e0c\u67e5\u627e \u4e0d\u9002\u5408\u9ad8\u9891\u589e\u5220\u3001\u4f4e\u9891\u67e5\u627e\u7684\u4f7f\u7528\u573a\u666f\uff1b
      •  
    • \u5f53\u54c8\u5e0c\u51b2\u7a81\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \uff1b
      •  
    • \u5f53\u6570\u636e\u91cf\u5f88\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u6bd4\u54c8\u5e0c\u67e5\u627e\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u8ba1\u7b97\u54c8\u5e0c\u6620\u5c04\u51fd\u6570\u53ef\u80fd\u6bd4\u904d\u5386\u4e00\u4e2a\u5c0f\u578b\u6570\u7ec4\u66f4\u6162\uff1b
      •  
    "},{"location":"chapter_searching/linear_search/","title":"10.1. \u00a0 \u7ebf\u6027\u67e5\u627e","text":"
    \u300c\u7ebf\u6027\u67e5\u627e Linear Search\u300d\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u67e5\u627e\u65b9\u6cd5\uff0c\u5176\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u4f9d\u6b21\u8bbf\u95ee\u6bcf\u4e2a\u5143\u7d20\uff0c\u76f4\u5230\u53e6\u4e00\u7aef\u540e\u505c\u6b62\u3002
    "},{"location":"chapter_searching/linear_search/#1011","title":"10.1.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u7ebf\u6027\u67e5\u627e\u5b9e\u8d28\u4e0a\u5c31\u662f\u904d\u5386\u6570\u636e\u7ed3\u6784 + \u5224\u65ad\u6761\u4ef6\u3002\u6bd4\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4 nums \u4e2d\u67e5\u627e\u76ee\u6807\u5143\u7d20 target \u7684\u5bf9\u5e94\u7d22\u5f15\uff0c\u90a3\u4e48\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u7ebf\u6027\u67e5\u627e\u3002
     
     
     Fig. \u5728\u6570\u7ec4\u4e2d\u7ebf\u6027\u67e5\u627e\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linear_search.java
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(int[] nums, int target) {\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.cpp
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(vector<int>& nums, int target) {\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.size(); i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.py
    def linear_search_array(nums: List[int], target: int) -> int:\n\"\"\" \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 \"\"\"\n# \u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\nif nums[i] == target:  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn i\nreturn -1                  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     linear_search.go
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc linearSearchArray(nums []int, target int) int {\n// \u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif nums[i] == target {\nreturn i\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     linear_search.js
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction linearSearchArray(nums, target) {\n// \u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] === target) {\nreturn i;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1; }\n
     linear_search.ts
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09*/\nfunction linearSearchArray(nums: number[], target: number): number {\n// \u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] === target) {\nreturn i;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.c
    [class]{}-[func]{linearSearchArray}\n
     linear_search.cs
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(int[] nums, int target)\n{\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.swift
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc linearSearchArray(nums: [Int], target: Int) -> Int {\n// \u904d\u5386\u6570\u7ec4\nfor i in nums.indices {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif nums[i] == target {\nreturn i\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     linear_search.zig
    // \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09\nfn linearSearchArray(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u904d\u5386\u6570\u7ec4\nfor (nums.items) |num, i| {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c \u8fd4\u56de\u5176\u7d22\u5f15\nif (num == target) {\nreturn @intCast(T, i);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     
    \u518d\u6bd4\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5728\u7ed9\u5b9a\u4e00\u4e2a\u76ee\u6807\u7ed3\u70b9\u503c target \uff0c\u8fd4\u56de\u6b64\u7ed3\u70b9\u5bf9\u8c61\uff0c\u4e5f\u53ef\u4ee5\u5728\u94fe\u8868\u4e2d\u8fdb\u884c\u7ebf\u6027\u67e5\u627e\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linear_search.java
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode linearSearchLinkedList(ListNode head, int target) {\n// \u904d\u5386\u94fe\u8868\nwhile (head != null) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val == target)\nreturn head;\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.cpp
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode* linearSearchLinkedList(ListNode* head, int target) {\n// \u904d\u5386\u94fe\u8868\nwhile (head != nullptr) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head->val == target)\nreturn head;\nhead = head->next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de nullptr\nreturn nullptr;\n}\n
     linear_search.py
    def linear_search_linkedlist(head: ListNode, target: int) -> Optional[ListNode]:\n\"\"\" \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 \"\"\"\n# \u904d\u5386\u94fe\u8868\nwhile head:\nif head.val == target: # \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nreturn head\nhead = head.next\nreturn None                # \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de None\n
     linear_search.go
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc linearSearchLinkedList(node *ListNode, target int) *ListNode {\n// \u904d\u5386\u94fe\u8868\nfor node != nil {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif node.Val == target {\nreturn node\n}\nnode = node.Next\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de nil\nreturn nil\n}\n
     linear_search.js
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09*/\nfunction linearSearchLinkedList(head, target) {\n// \u904d\u5386\u94fe\u8868\nwhile(head) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif(head.val === target) {\nreturn head;\n}\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.ts
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09*/\nfunction linearSearchLinkedList(head: ListNode | null, target: number): ListNode | null {\n// \u904d\u5386\u94fe\u8868\nwhile (head) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val === target) {\nreturn head;\n}\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.c
    [class]{}-[func]{linearSearchLinkedList}\n
     linear_search.cs
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode? linearSearchLinkedList(ListNode head, int target)\n{\n// \u904d\u5386\u94fe\u8868\nwhile (head != null)\n{\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val == target)\nreturn head;\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.swift
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc linearSearchLinkedList(head: ListNode?, target: Int) -> ListNode? {\nvar head = head\n// \u904d\u5386\u94fe\u8868\nwhile head != nil {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif head?.val == target {\nreturn head\n}\nhead = head?.next\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn nil\n}\n
     linear_search.zig
    // \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09\nfn linearSearchLinkedList(comptime T: type, node: ?*inc.ListNode(T), target: T) ?*inc.ListNode(T) {\nvar head = node;\n// \u904d\u5386\u94fe\u8868\nwhile (head != null) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.?.val == target) return head;\nhead = head.?.next;\n}\nreturn null;\n}\n
    "},{"location":"chapter_searching/linear_search/#1012","title":"10.1.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u65e0\u9700\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\u3002
    "},{"location":"chapter_searching/linear_search/#1013","title":"10.1.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u7ebf\u6027\u67e5\u627e\u7684\u901a\u7528\u6027\u6781\u4f73\u3002\u7531\u4e8e\u7ebf\u6027\u67e5\u627e\u662f\u4f9d\u6b21\u8bbf\u95ee\u5143\u7d20\u7684\uff0c\u5373\u6ca1\u6709\u8df3\u8dc3\u8bbf\u95ee\u5143\u7d20\uff0c\u56e0\u6b64\u6570\u7ec4\u6216\u94fe\u8868\u7686\u9002\u7528\u3002
     
    \u7ebf\u6027\u67e5\u627e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u592a\u9ad8\u3002\u5728\u6570\u636e\u91cf \\(n\\) \u5f88\u5927\u65f6\uff0c\u67e5\u627e\u6548\u7387\u5f88\u4f4e\u3002
    "},{"location":"chapter_searching/summary/","title":"10.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u67e5\u627e\u65b9\u6cd5\uff0c\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784 + \u5224\u65ad\u6761\u4ef6\u5b9e\u73b0\u67e5\u627e\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u4e0d\u65ad\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u5b9e\u73b0\u67e5\u627e\uff0c\u5176\u8981\u6c42\u8f93\u5165\u6570\u636e\u662f\u6709\u5e8f\u7684\uff0c\u5e76\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u54c8\u5e0c\u67e5\u627e\u501f\u52a9\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u5e38\u6570\u9636\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u67e5\u627e\u64cd\u4f5c\uff0c\u4f53\u73b0\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u7684\u7b97\u6cd5\u601d\u60f3\u3002
      •  
    • \u4e0b\u8868\u603b\u7ed3\u5bf9\u6bd4\u4e86\u67e5\u627e\u7b97\u6cd5\u7684\u5404\u79cd\u7279\u6027\u548c\u65f6\u95f4\u590d\u6742\u5ea6\u3002
      •  
     \u7ebf\u6027\u67e5\u627e \u4e8c\u5206\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u9002\u7528\u6570\u636e\u7ed3\u6784 \u6570\u7ec4\u3001\u94fe\u8868 \u6570\u7ec4 \u6570\u7ec4\u3001\u94fe\u8868 \u8f93\u5165\u6570\u636e\u8981\u6c42 \u65e0 \u6709\u5e8f \u65e0 \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u67e5\u627e / \u63d2\u5165 / \u5220\u9664 \\(O(n)\\) / \\(O(1)\\) / \\(O(n)\\) \\(O(\\log n)\\) / \\(O(n)\\) / \\(O(n)\\) \\(O(1)\\) / \\(O(1)\\) / \\(O(1)\\) \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u67e5\u627e / \u63d2\u5165 / \u5220\u9664 \\(O(n)\\) / \\(O(1)\\) / \\(O(n)\\) \\(O(\\log n)\\) / \\(O(n)\\) / \\(O(n)\\) \\(O(n)\\) / \\(O(n)\\) / \\(O(n)\\) \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\)"},{"location":"chapter_sorting/bubble_sort/","title":"11.2. \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"
    \u300c\u5192\u6ce1\u6392\u5e8f Bubble Sort\u300d\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u975e\u5e38\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u4e2a\u5b66\u4e60\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u987e\u540d\u601d\u4e49\uff0c\u300c\u5192\u6ce1\u300d\u662f\u8be5\u7b97\u6cd5\u7684\u6838\u5fc3\u64cd\u4f5c\u3002
     
    \u4e3a\u4ec0\u4e48\u53eb\u201c\u5192\u6ce1\u201d
     
    \u5728\u6c34\u4e2d\uff0c\u8d8a\u5927\u7684\u6ce1\u6ce1\u6d6e\u529b\u8d8a\u5927\uff0c\u6240\u4ee5\u6700\u5927\u7684\u6ce1\u6ce1\u4f1a\u6700\u5148\u6d6e\u5230\u6c34\u9762\u3002
     
    \u300c\u5192\u6ce1\u300d\u64cd\u4f5c\u5219\u662f\u5728\u6a21\u62df\u4e0a\u8ff0\u8fc7\u7a0b\uff0c\u5177\u4f53\u505a\u6cd5\u4e3a\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u5bf9\u6bd4\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u82e5 \u5de6\u5143\u7d20 > \u53f3\u5143\u7d20 \u5219\u5c06\u5b83\u4fe9\u4ea4\u6362\uff0c\u6700\u7ec8\u53ef\u5c06\u6700\u5927\u5143\u7d20\u79fb\u52a8\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\u3002
     
    \u5b8c\u6210\u6b64\u6b21\u5192\u6ce1\u64cd\u4f5c\u540e\uff0c\u6570\u7ec4\u6700\u5927\u5143\u7d20\u5df2\u5728\u6b63\u786e\u4f4d\u7f6e\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u6392\u5e8f\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u3002
     <1><2><3><4><5><6><7> 
     
     
     
     
     
     
    "},{"location":"chapter_sorting/bubble_sort/#1121","title":"11.2.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5b8c\u6210\u7b2c\u4e00\u8f6e\u300c\u5192\u6ce1\u300d\u540e\uff0c\u6570\u7ec4\u6700\u5927\u5143\u7d20\u5df2\u5728\u6b63\u786e\u4f4d\u7f6e\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u6392\u5e8f\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u3002
      2.  
    2. \u540c\u7406\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u300c\u5192\u6ce1\u300d\uff0c\u53ef\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c\u56e0\u800c\u5f85\u6392\u5e8f\u5143\u7d20\u53ea\u5269 \\(n - 2\\) \u4e2a\u3002
      3.  
    3. \u4ee5\u6b64\u7c7b\u63a8\u2026\u2026 \u5faa\u73af \\(n - 1\\) \u8f6e\u300c\u5192\u6ce1\u300d\uff0c\u5373\u53ef\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002
      4.  
     
     Fig. \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\n}\n}\n}\n}\n
     bubble_sort.py
    def bubble_sort(nums: List[int]) -> None:\n\"\"\" \u5192\u6ce1\u6392\u5e8f \"\"\"\nn: int = len(nums)\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(n - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\n
     bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\n}\n}\n}\n}\n
     bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.c
    [class]{}-[func]{bubbleSort}\n
     bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\n}\n}\n}\n}\n
     bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1122","title":"11.2.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u5404\u8f6e\u300c\u5192\u6ce1\u300d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\) , \\(n - 2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488\u53d8\u91cf\u4ec5\u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0d\u4ea4\u6362\u76f8\u7b49\u5143\u7d20\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5f15\u5165 flag \u4f18\u5316\u540e\uff08\u89c1\u4e0b\u6587\uff09\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002
    "},{"location":"chapter_sorting/bubble_sort/#1123","title":"11.2.3. \u00a0 \u6548\u7387\u4f18\u5316","text":"
    \u6211\u4eec\u53d1\u73b0\uff0c\u82e5\u5728\u67d0\u8f6e\u300c\u5192\u6ce1\u300d\u4e2d\u672a\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u5219\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u8003\u8651\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d flag \u6765\u76d1\u542c\u8be5\u60c5\u51b5\uff0c\u82e5\u51fa\u73b0\u5219\u76f4\u63a5\u8fd4\u56de\u3002
     
    \u4f18\u5316\u540e\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u800c\u5728\u8f93\u5165\u6570\u7ec4 \u5df2\u6392\u5e8f \u65f6\uff0c\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\nboolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.py
    def bubble_sort_with_flag(nums: List[int]) -> None:\n\"\"\" \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 \"\"\"\nn: int = len(nums)\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(n - 1, 0, -1):\nflag: bool = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\nflag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\nif not flag:\nbreak            # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
     bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\nflag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
     bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.c
    [class]{}-[func]{bubbleSortWithFlag}\n
     bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\nvar flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
     bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;\n}\n}\nif (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    "},{"location":"chapter_sorting/bucket_sort/","title":"Bucket sort","text":""},{"location":"chapter_sorting/bucket_sort/#_1","title":"\u62d3\u5c55\u5230\u6876\u6392\u5e8f","text":"
    \u5982\u679c\u6211\u4eec\u628a\u4e0a\u8ff0 bucket \u4e2d\u7684\u6bcf\u4e2a\u7d22\u5f15\u60f3\u8c61\u6210\u4e00\u4e2a\u6876\uff0c\u90a3\u4e48\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u7406\u89e3\u4e3a\u628a \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\uff0c\u518d\u6839\u636e\u6876\u4e0e\u6876\u4e4b\u95f4\u5929\u7136\u7684\u6709\u5e8f\u6027\u6765\u5b9e\u73b0\u6392\u5e8f\u3002
     
    \u4ee5\u4e0a\u89e3\u8bfb\u4fbf\u662f\u300c\u6876\u6392\u5e8f Bucket Sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u3002\u5177\u4f53\u5730\uff0c\u6876\u6392\u5e8f\u8003\u8651\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u6839\u636e\u5927\u5c0f\u8303\u56f4\u5747\u5300\u5730\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\uff0c\u7531\u4e8e\u6876\u4e4b\u95f4\u662f\u6709\u5e8f\u7684\uff0c\u56e0\u6b64\u4ec5\u9700\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u6267\u884c\u6392\u5e8f\uff0c\u6700\u7ec8\u6309\u7167\u6876\u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u5c06\u5143\u7d20\u4f9d\u6b21\u6392\u5217\uff0c\u5373\u53ef\u5f97\u5230\u6392\u5e8f\u7ed3\u679c\u3002
     
    \u5047\u8bbe\u4f7f\u7528\u300c\u5feb\u901f\u6392\u5e8f\u300d\u6765\u6392\u5e8f\u5404\u4e2a\u6876\u5185\u7684\u5143\u7d20\uff0c\u6bcf\u4e2a\u6876\u5185\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \uff0c\u5219\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u63a5\u8fd1 \\(n\\) \u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002
     
    \uff08\u56fe\uff09
     
    \u7406\u8bba\u4e0a\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \uff0c\u4f46\u524d\u63d0\u662f\u9700\u8981\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u800c\u8fd9\u662f\u4e0d\u592a\u5bb9\u6613\u505a\u5230\u7684\u3002\u5047\u8bbe\u6211\u4eec\u8981\u628a\u6dd8\u5b9d\u4e2d\u7684 \\(100\\) \u4e07\u4ef6\u5546\u54c1\u6839\u636e\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 \\(100\\) \u4e2a\u6876\u4e2d\uff0c\u7531\u4e8e\u5546\u54c1\u4ef7\u683c\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\uff0c\u6bd4\u5982 \\(1\\) ~ \\(100\\) \u5143\u7684\u5546\u54c1\u975e\u5e38\u591a\u3001\\(1\\) \u4e07\u5143\u4ee5\u4e0a\u7684\u5546\u54c1\u975e\u5e38\u5c11\u7b49\uff0c\u56e0\u6b64\u96be\u4ee5\u7b80\u5355\u5730\u8bbe\u5b9a\u5404\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u89e3\u51b3\u65b9\u6848\u6709\uff1a
     
       
    • \u5148\u521d\u6b65\u8bbe\u7f6e\u4e00\u4e2a\u5206\u754c\u7ebf\uff0c\u5c06\u5143\u7d20\u5206\u914d\u5b8c\u540e\uff0c\u628a\u5143\u7d20\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a\u591a\u4e2a\u6876\uff0c\u76f4\u81f3\u6bcf\u4e2a\u6876\u5185\u5143\u7d20\u6570\u91cf\u5408\u7406\u4e3a\u6b62\uff1b\u8be5\u505a\u6cd5\u4e00\u822c\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\uff1b
      •  
    • \u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u5df2\u77e5\u5206\u5e03\u6765\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\uff1b\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u4e0d\u4e00\u5b9a\u9700\u8981 case-by-case \u5730\u7edf\u8ba1\uff0c\u6709\u65f6\u53ef\u4ee5\u91c7\u7528\u4e00\u4e9b\u5e38\u89c1\u5206\u5e03\u6765\u8fd1\u4f3c\uff0c\u4f8b\u5982\u81ea\u7136\u754c\u7684\u6b63\u6001\u5206\u5e03\uff1b
      •  
     
    \uff08\u56fe\uff09
    "},{"location":"chapter_sorting/counting_sort/","title":"11.6. \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"
    \u524d\u9762\u4ecb\u7ecd\u7684\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e \u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5373\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u4e4b\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5b66\u4e60\u4e00\u79cd \u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5 \uff0c\u540d\u4e3a\u300c\u8ba1\u6570\u6392\u5e8f Counting Sort\u300d\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u3002
    "},{"location":"chapter_sorting/counting_sort/#1161","title":"11.6.1. \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"
    \u5148\u770b\u4e00\u4e2a\u7b80\u5355\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5143\u7d20\u7686\u4e3a \u975e\u8d1f\u6574\u6570\u3002\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u4e3a\uff1a
     
       
    1. \u7edf\u8ba1\u6570\u7ec4\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u5e76\u5efa\u7acb\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 counter \uff1b
      2.  
    2. \u501f\u52a9 counter \u7edf\u8ba1 nums \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d counter[num] \u5bf9\u5e94\u6570\u5b57 num \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 nums \uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a num\uff09\uff0c\u6bcf\u8f6e\u5c06 counter[num] \u81ea\u589e \\(1\\) \u5373\u53ef\u3002
      3.  
    3. \u7531\u4e8e counter \u7684\u5404\u4e2a\u7d22\u5f15\u662f\u5929\u7136\u6709\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u88ab\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 counter \uff0c\u6839\u636e\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5c06\u5404\u6570\u5b57\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 nums \u5373\u53ef\u3002
      4.  
     <1><2><3> 
     
     
     
    \u4ee5\u4e0b\u662f\u5b9e\u73b0\u4ee3\u7801\uff0c\u8ba1\u6570\u6392\u5e8f\u540d\u526f\u5176\u5b9e\uff0c\u786e\u5b9e\u662f\u901a\u8fc7\u201c\u7edf\u8ba1\u6570\u91cf\u201d\u6765\u5b9e\u73b0\u6392\u5e8f\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
     counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int>& nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
     counting_sort.py
    def counting_sort_naive(nums: List[int]) -> None:\n\"\"\" \u8ba1\u6570\u6392\u5e8f \"\"\"\n# \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = 0\nfor num in nums:\nm = max(m, num)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\ni = 0\nfor num in range(m + 1):\nfor _ in range(counter[num]):\nnums[i] = num\ni += 1\n
     counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nfor i, num := 0, 0; num < m+1; num++ {\nfor j := 0; j < counter[num]; j++ {\nnums[i] = num\ni++\n}\n}\n}\n
     counting_sort.js
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.ts
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.c
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.cs
    [class]{counting_sort}-[func]{countingSortNaive}\n
     counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nvar i = 0\nfor num in stride(from: 0, to: m + 1, by: 1) {\nfor _ in stride(from: 0, to: counter[num], by: 1) {\nnums[i] = num\ni += 1\n}\n}\n}\n
     counting_sort.zig
    [class]{}-[func]{countingSortNaive}\n
    "},{"location":"chapter_sorting/counting_sort/#1162","title":"11.6.2. \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u53d1\u73b0\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 3. \u5c31\u5931\u6548\u4e86\u3002\u4f8b\u5982\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u8981\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002
     
    \u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 counter \u7684\u300c\u524d\u7f00\u548c\u300d\uff0c\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 i \u5904\u7684\u524d\u7f00\u548c prefix[i] \u7b49\u4e8e\u6570\u7ec4\u524d i \u4e2a\u5143\u7d20\u4e4b\u548c\uff0c\u5373
     \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] 
    \u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u610f\u4e49\uff0cprefix[num] - 1 \u4ee3\u8868\u5143\u7d20 num \u5728\u7ed3\u679c\u6570\u7ec4 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u5f88\u5173\u952e\uff0c\u56e0\u4e3a\u5176\u7ed9\u51fa\u4e86\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 nums \u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\uff1a
     
       
    1. \u5c06 num \u586b\u5165\u6570\u7ec4 res \u7684\u7d22\u5f15 prefix[num] - 1 \u5904\uff1b
      2.  
    2. \u4ee4\u524d\u7f00\u548c prefix[num] \u81ea\u51cf \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\uff1b
      3.  
     
    \u5b8c\u6210\u904d\u5386\u540e\uff0c\u6570\u7ec4 res \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 res \u8986\u76d6\u539f\u6570\u7ec4 nums \u5373\u53ef\uff1b
     <1><2><3><4><5><6><7><8> 
     
     
     
     
     
     
     
     
    \u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
     counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int>& nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.size();\nvector<int> res(n);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums = res;\n}\n
     counting_sort.py
    def counting_sort(nums: List[int]) -> None:\n\"\"\" \u8ba1\u6570\u6392\u5e8f \"\"\"\n# \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = max(nums)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n# \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in range(m):\ncounter[i + 1] += counter[i]\n# 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n# \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn = len(nums)\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nnum = nums[i]\nres[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n# \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\n
     counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i := 0; i < m; i++ {\ncounter[i+1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn := len(nums)\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nnum := nums[i]\n// \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\nres[counter[num]-1] = num\n// \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\ncounter[num]--\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\ncopy(nums, res)\n}\n
     counting_sort.js
    [class]{}-[func]{countingSort}\n
     counting_sort.ts
    [class]{}-[func]{countingSort}\n
     counting_sort.c
    [class]{}-[func]{countingSort}\n
     counting_sort.cs
    [class]{counting_sort}-[func]{countingSort}\n
     counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in stride(from: 0, to: m, by: 1) {\ncounter[i + 1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nvar res = Array(repeating: 0, count: nums.count)\nfor i in stride(from: nums.count - 1, through: 0, by: -1) {\nlet num = nums[i]\nres[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in stride(from: 0, to: nums.count, by: 1) {\nnums[i] = res[i]\n}\n}\n
     counting_sort.zig
    [class]{}-[func]{countingSort}\n
    "},{"location":"chapter_sorting/counting_sort/#1163","title":"11.6.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 nums \u548c\u904d\u5386 counter \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u6b64\u65f6\u4f7f\u7528\u7ebf\u6027 \\(O(n)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6570\u7ec4 res \u548c counter \u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) , \\(m\\) \u3002
     
    \u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u8f85\u52a9\u6570\u7ec4 counter \u548c\u7ed3\u679c\u6570\u7ec4 res \u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u5012\u5e8f\u904d\u5386 nums \u4fdd\u6301\u4e86\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002
     
    \u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u4e0e\u5143\u7d20\u5206\u5e03\u65e0\u5173\u3002
     
    \u4e3a\u4ec0\u4e48\u662f\u7a33\u5b9a\u6392\u5e8f\uff1f
     
    \u7531\u4e8e\u5411 res \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 nums \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u201c\u7a33\u5b9a\u6392\u5e8f\u201d\uff1b\u5176\u5b9e\u6b63\u5e8f\u904d\u5386 nums \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u201c\u975e\u7a33\u5b9a\u201d\u3002
    "},{"location":"chapter_sorting/counting_sort/#1164","title":"11.6.4. \u00a0 \u5c40\u9650\u6027","text":"
    \u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u592a\u5999\u4e86\uff0c\u5494\u5494\u4e00\u901a\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4e0b\u6765\u4e86\u3002\u4f46\u5b9e\u9645\u4e0a\u4e0e\u5176\u5b83\u7b97\u6cd5\u4e00\u6837\uff0c\u8ba1\u6570\u6392\u5e8f\u4e5f\u65e0\u6cd5\u6446\u8131\u201c\u6b64\u6d88\u5f7c\u957f\u201d\u7684\u5bbf\u547d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u7684\u4ee3\u4ef7\u662f\u901a\u7528\u578b\u53d8\u5dee\u3002
     
    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u8981\u7528\u5728\u5176\u4ed6\u7c7b\u578b\u6570\u636e\u4e0a\uff0c\u5219\u8981\u6c42\u8be5\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u88ab\u8f6c\u5316\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u5373\u53ef\u3002
     
    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u6570\u636e\u8303\u56f4\u4e0d\u5927\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u5360\u7528\u7a7a\u95f4\u592a\u591a\uff1b\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u6709\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002
    "},{"location":"chapter_sorting/insertion_sort/","title":"11.3. \u00a0 \u63d2\u5165\u6392\u5e8f","text":"
    \u300c\u63d2\u5165\u6392\u5e8f Insertion Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e \u6570\u7ec4\u63d2\u5165\u64cd\u4f5c \u7684\u6392\u5e8f\u7b97\u6cd5\u3002
     
    \u300c\u63d2\u5165\u64cd\u4f5c\u300d\u539f\u7406\uff1a\u9009\u5b9a\u67d0\u4e2a\u5f85\u6392\u5e8f\u5143\u7d20\u4e3a\u57fa\u51c6\u6570 base\uff0c\u5c06 base \u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u5143\u7d20\u4f9d\u6b21\u5bf9\u6bd4\u5927\u5c0f\uff0c\u5e76\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u3002
     
    \u56de\u5fc6\u6570\u7ec4\u63d2\u5165\u64cd\u4f5c\uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 base \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u5c06 base \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002
     
     
     Fig. \u5355\u6b21\u63d2\u5165\u64cd\u4f5c 
    "},{"location":"chapter_sorting/insertion_sort/#1131","title":"11.3.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u7b2c 1 \u8f6e\u5148\u9009\u53d6\u6570\u7ec4\u7684 \u7b2c 2 \u4e2a\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      2.  
    2. \u7b2c 2 \u8f6e\u9009\u53d6 \u7b2c 3 \u4e2a\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6570\u7ec4\u524d 3 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      3.  
    3. \u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u6700\u540e\u4e00\u8f6e\u9009\u53d6 \u6570\u7ec4\u5c3e\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6240\u6709\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      4.  
     
     Fig. \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig insertion_sort.java
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.cpp
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.size(); i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.py
    def insertion_sort(nums: List[int]) -> None:\n\"\"\" \u63d2\u5165\u6392\u5e8f \"\"\"\n# \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]   \nfor i in range(1, len(nums)):\nbase: int = nums[i]\nj: int = i - 1\n# \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0 and nums[j] > base:\nnums[j + 1] = nums[j]  # 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\nnums[j + 1] = base         # 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
     insertion_sort.go
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := 1; i < len(nums); i++ {\nbase := nums[i]\nj := i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nfor j >= 0 && nums[j] > base {\nnums[j+1] = nums[j] // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--\n}\nnums[j+1] = base // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.js
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (let i = 1; i < nums.length; i++) {\nlet base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.ts
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (let i = 1; i < nums.length; i++) {\nconst base = nums[i];\nlet j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.c
    [class]{}-[func]{insertionSort}\n
     insertion_sort.cs
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.Length; i++)\n{\nint bas = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > bas)\n{\nnums[j + 1] = nums[j]; // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = bas;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.swift
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor i in stride(from: 1, to: nums.count, by: 1) {\nlet base = nums[i]\nvar j = i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0, nums[j] > base {\nnums[j + 1] = nums[j] // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\n}\nnums[j + 1] = base // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.zig
    // \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nvar i: usize = 1;\nwhile (i < nums.len) : (i += 1) {\nvar base = nums[i];\nvar j: usize = i;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 1 and nums[j - 1] > base) : (j -= 1) {\nnums[j] = nums[j - 1];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n}\nnums[j] = base;             // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    "},{"location":"chapter_sorting/insertion_sort/#1132","title":"11.3.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5404\u8f6e\u63d2\u5165\u64cd\u4f5c\u5faa\u73af \\(n - 1\\) , \\(n-2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488\u53d8\u91cf\u4ec5\u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0d\u4ea4\u6362\u76f8\u7b49\u5143\u7d20\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u6700\u4f73\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)  \u3002
    "},{"location":"chapter_sorting/insertion_sort/#1133-vs","title":"11.3.3. \u00a0 \u63d2\u5165\u6392\u5e8f vs \u5192\u6ce1\u6392\u5e8f","text":"
    Question
     
    \u867d\u7136\u300c\u63d2\u5165\u6392\u5e8f\u300d\u548c\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u901f\u5ea6\u5374\u6709\u5f88\u5927\u5dee\u522b\uff0c\u8fd9\u662f\u4e3a\u4ec0\u4e48\u5462\uff1f
     
    \u56de\u987e\u590d\u6742\u5ea6\u5206\u6790\uff0c\u4e24\u4e2a\u65b9\u6cd5\u7684\u5faa\u73af\u6b21\u6570\u90fd\u662f \\(\\frac{(n - 1) n}{2}\\) \u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u300c\u5192\u6ce1\u64cd\u4f5c\u300d\u662f\u5728\u505a \u5143\u7d20\u4ea4\u6362\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\u5b9e\u73b0\uff0c\u5171 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u800c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u662f\u5728\u505a \u8d4b\u503c\uff0c\u53ea\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u53ef\u4ee5\u7c97\u7565\u4f30\u8ba1\u51fa\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u7ea6\u4e3a\u63d2\u5165\u6392\u5e8f\u7684 3 \u500d\u3002
     
    \u63d2\u5165\u6392\u5e8f\u8fd0\u884c\u901f\u5ea6\u5feb\uff0c\u5e76\u4e14\u5177\u6709\u539f\u5730\u3001\u7a33\u5b9a\u3001\u81ea\u9002\u5e94\u7684\u4f18\u70b9\uff0c\u56e0\u6b64\u5f88\u53d7\u6b22\u8fce\u3002\u5b9e\u9645\u4e0a\uff0c\u5305\u62ec Java \u5728\u5185\u7684\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u7684\u5b9e\u73b0\u90fd\u7528\u5230\u4e86\u63d2\u5165\u6392\u5e8f\u3002\u5e93\u51fd\u6570\u7684\u5927\u81f4\u601d\u8def\uff1a
     
       
    • \u5bf9\u4e8e \u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u300c\u5feb\u901f\u6392\u5e8f\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \uff1b
      •  
    • \u5bf9\u4e8e \u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u300c\u63d2\u5165\u6392\u5e8f\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1b
      •  
     
    \u5728\u6570\u7ec4\u8f83\u77ed\u65f6\uff0c\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\uff08\u5373\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\uff09\u5360\u4e3b\u5bfc\u4f5c\u7528\uff0c\u6b64\u65f6\u63d2\u5165\u6392\u5e8f\u8fd0\u884c\u5730\u66f4\u5feb\u3002\u8fd9\u4e2a\u73b0\u8c61\u4e0e\u300c\u7ebf\u6027\u67e5\u627e\u300d\u548c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7684\u60c5\u51b5\u7c7b\u4f3c\u3002
    "},{"location":"chapter_sorting/intro_to_sort/","title":"11.1. \u00a0 \u6392\u5e8f\u7b80\u4ecb","text":"
    \u300c\u6392\u5e8f\u7b97\u6cd5 Sorting Algorithm\u300d\u4f7f\u5f97\u5217\u8868\u4e2d\u7684\u6240\u6709\u5143\u7d20\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002
     
       
    • \u5f85\u6392\u5e8f\u7684\u5217\u8868\u7684 \u5143\u7d20\u7c7b\u578b \u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u3001\u6216\u5b57\u7b26\u4e32\uff1b
      •  
    • \u6392\u5e8f\u7b97\u6cd5\u53ef\u4ee5\u6839\u636e\u9700\u8981\u8bbe\u5b9a \u5224\u65ad\u89c4\u5219\uff0c\u4f8b\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u3001\u81ea\u5b9a\u4e49\u89c4\u5219\uff1b
      •  
     
     
     Fig. \u6392\u5e8f\u4e2d\u4e0d\u540c\u7684\u5143\u7d20\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219 
    "},{"location":"chapter_sorting/intro_to_sort/#1111","title":"11.1.1. \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"
    \u6392\u5e8f\u7b97\u6cd5\u4e3b\u8981\u53ef\u6839\u636e \u7a33\u5b9a\u6027 \u3001\u5c31\u5730\u6027 \u3001\u81ea\u9002\u5e94\u6027 \u3001\u6bd4\u8f83\u7c7b \u6765\u5206\u7c7b\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_1","title":"\u7a33\u5b9a\u6027","text":"
       
    • \u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u4e0d\u6539\u53d8 \u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
    • \u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u4f4d\u7f6e \u53ef\u80fd\u88ab\u6539\u53d8\u3002
      •  
     
    \u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1, 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u90a3\u4e48\u5728\u4ee5\u4e0b\u793a\u4f8b\u4e2d\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u4f1a\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e22\u5931\u3002\u56e0\u6b64\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u662f\u5f88\u597d\u7684\u7279\u6027\uff0c\u5728\u591a\u7ea7\u6392\u5e8f\u4e2d\u662f\u5fc5\u987b\u7684\u3002
     
    # \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n('A', 19)\n('B', 18)\n('C', 21)\n('D', 19)\n('E', 23)\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n('B', 18)\n('D', 19)\n('A', 19)\n('C', 21)\n('E', 23)\n
    "},{"location":"chapter_sorting/intro_to_sort/#_2","title":"\u5c31\u5730\u6027","text":"
       
    • \u300c\u539f\u5730\u6392\u5e8f\u300d\u65e0\u9700\u8f85\u52a9\u6570\u636e\uff0c\u4e0d\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff1b
      •  
    • \u300c\u975e\u539f\u5730\u6392\u5e8f\u300d\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u636e\uff0c\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff1b
      •  
     
    \u300c\u539f\u5730\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff0c\u53ef\u4ee5\u8282\u7ea6\u5185\u5b58\uff1b\u5e76\u4e14\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7531\u4e8e\u6570\u636e\u64cd\u4f5c\u51cf\u5c11\uff0c\u539f\u5730\u6392\u5e8f\u7684\u8fd0\u884c\u6548\u7387\u4e5f\u66f4\u9ad8\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_3","title":"\u81ea\u9002\u5e94\u6027","text":"
       
    • \u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d7\u8f93\u5165\u6570\u636e\u5f71\u54cd\uff0c\u5373\u6700\u4f73 / \u6700\u5dee / \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u76f8\u7b49\u3002
      •  
    • \u300c\u975e\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6052\u5b9a\uff0c\u4e0e\u8f93\u5165\u6570\u636e\u65e0\u5173\u3002
      •  
     
    \u6211\u4eec\u5e0c\u671b \u6700\u5dee = \u5e73\u5747\uff0c\u5373\u4e0d\u5e0c\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u5728\u67d0\u4e9b\u8f93\u5165\u6570\u636e\u4e0b\u53d1\u751f\u52a3\u5316\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_4","title":"\u6bd4\u8f83\u7c7b","text":"
       
    • \u300c\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u57fa\u4e8e\u5143\u7d20\u4e4b\u95f4\u7684\u6bd4\u8f83\u7b97\u5b50\uff08\u5c0f\u4e8e\u3001\u76f8\u7b49\u3001\u5927\u4e8e\uff09\u6765\u51b3\u5b9a\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
    • \u300c\u975e\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u4e0d\u57fa\u4e8e\u5143\u7d20\u4e4b\u95f4\u7684\u6bd4\u8f83\u7b97\u5b50\u6765\u51b3\u5b9a\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
     
    \u300c\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6700\u4f18\u4e3a \\(O(n \\log n)\\) \uff1b\u800c\u300c\u975e\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u4f46\u901a\u7528\u6027\u8f83\u5dee\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#1112","title":"11.1.2. \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"
       
    • \u8fd0\u884c\u5feb\uff0c\u5373\u65f6\u95f4\u590d\u6742\u5ea6\u4f4e\uff1b
      •  
    • \u7a33\u5b9a\u6392\u5e8f\uff0c\u5373\u6392\u5e8f\u540e\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\u5316\uff1b
      •  
    • \u539f\u5730\u6392\u5e8f\uff0c\u5373\u8fd0\u884c\u4e2d\u4e0d\u4f7f\u7528\u989d\u5916\u7684\u8f85\u52a9\u7a7a\u95f4\uff1b
      •  
    • \u6b63\u5411\u81ea\u9002\u5e94\u6027\uff0c\u5373\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u4e0d\u4f1a\u5728\u67d0\u4e9b\u8f93\u5165\u6570\u636e\u4e0b\u53d1\u751f\u52a3\u5316\uff1b
      •  
     
    \u7136\u800c\uff0c\u6ca1\u6709\u6392\u5e8f\u7b97\u6cd5\u540c\u65f6\u5177\u5907\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u3002\u6392\u5e8f\u7b97\u6cd5\u7684\u9009\u578b\u4f7f\u7528\u53d6\u51b3\u4e8e\u5177\u4f53\u7684\u5217\u8868\u7c7b\u578b\u3001\u5217\u8868\u957f\u5ea6\u3001\u5143\u7d20\u5206\u5e03\u7b49\u56e0\u7d20\u3002
    "},{"location":"chapter_sorting/merge_sort/","title":"11.5. \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"
    \u300c\u5f52\u5e76\u6392\u5e8f Merge Sort\u300d\u662f\u7b97\u6cd5\u4e2d\u201c\u5206\u6cbb\u601d\u60f3\u201d\u7684\u5178\u578b\u4f53\u73b0\uff0c\u5176\u6709\u300c\u5212\u5206\u300d\u548c\u300c\u5408\u5e76\u300d\u4e24\u4e2a\u9636\u6bb5\uff1a
     
       
    1. \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad \u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u4f4d\u7f6e\u5212\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u5316\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\uff1b
      2.  
    2. \u5408\u5e76\u9636\u6bb5\uff1a\u5212\u5206\u5230\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u5f00\u59cb\u5411\u4e0a\u5408\u5e76\uff0c\u4e0d\u65ad\u5c06 \u5de6\u3001\u53f3\u4e24\u4e2a\u77ed\u6392\u5e8f\u6570\u7ec4 \u5408\u5e76\u4e3a \u4e00\u4e2a\u957f\u6392\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u5408\u5e76\u81f3\u539f\u6570\u7ec4\u65f6\u5b8c\u6210\u6392\u5e8f\uff1b
      3.  
     
     
     Fig. \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 
    "},{"location":"chapter_sorting/merge_sort/#1151","title":"11.5.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
    \u300c\u9012\u5f52\u5212\u5206\u300d \u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730 \u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u957f\u5ea6\u4e3a 1 \uff1b
     
       
    1. \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 mid \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [left, mid] \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [mid + 1, right] \uff09\uff1b
      2.  
    2. \u9012\u5f52\u6267\u884c 1. \u6b65\u9aa4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5212\u5206\uff1b
      3.  
     
    \u300c\u56de\u6eaf\u5408\u5e76\u300d \u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a \u6709\u5e8f\u6570\u7ec4 \uff1b
     
    \u9700\u8981\u6ce8\u610f\uff0c\u7531\u4e8e\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u6240\u4ee5 \u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002\u56e0\u6b64\uff0c\u5408\u5e76\u4efb\u52a1\u672c\u8d28\u662f\u8981 \u5c06\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002
     <1><2><3><4><5><6><7><8><9><10> 
     
     
     
     
     
     
     
     
     
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u9012\u5f52\u987a\u5e8f\u5c31\u662f\u4e8c\u53c9\u6811\u7684\u300c\u540e\u5e8f\u904d\u5386\u300d\u3002
     
       
    • \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\u3001\u518d\u9012\u5f52\u53f3\u5b50\u6811\u3001\u6700\u540e\u5904\u7406\u6839\u7ed3\u70b9\u3002
      •  
    • \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\u3001\u518d\u9012\u5f52\u53f3\u5b50\u6811\u3001\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig merge_sort.java
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = Arrays.copyOfRange(nums, left, right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.cpp
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(vector<int>& nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvector<int> tmp(nums.begin() + left, nums.begin() + right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int>& nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.py
    def merge(nums: List[int], left: int, mid: int, right: int) -> None:\n\"\"\" \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 \"\"\"\n# \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n# \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\n# \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy\u6a21\u5757\ntmp: List[int] = nums[left:right + 1].copy()\n# \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleft_start: int = 0\nleft_end: int = mid - left\n# \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nright_start: int = mid + 1 - left\nright_end: int = right - left\n# i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni: int = left_start\nj: int = right_start\n# \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in range(left, right + 1):\n# \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > left_end:\nnums[k] = tmp[j]\nj += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelif j > right_end or tmp[i] <= tmp[j]:\nnums[k] = tmp[i]\ni += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse:\nnums[k] = tmp[j]\nj += 1\ndef merge_sort(nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5f52\u5e76\u6392\u5e8f \"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif left >= right:\nreturn                        # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n# \u5212\u5206\u9636\u6bb5\nmid: int = (left + right) // 2    # \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid)       # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n# \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n
     merge_sort.go
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums []int, left, mid, right int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy \u6a21\u5757\ntmp := make([]int, right-left+1)\nfor i := left; i <= right; i++ {\ntmp[i-left] = nums[i]\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleftStart, leftEnd := left-left, mid-left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nrightStart, rightEnd := mid+1-left, right-left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni, j := leftStart, rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k := left; k <= right; k++ {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj++\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni++\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j]\nj++\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right {\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nmid := (left + right) / 2\nmergeSort(nums, left, mid)\nmergeSort(nums, mid+1, right)\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n}\n
     merge_sort.js
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums, left, mid, right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nlet leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nlet rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2);    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.ts
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.c
    \n
     merge_sort.cs
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right)\n{\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = nums[left..(right + 1)];\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++)\n{\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right)\n{\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.swift
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = Array(nums[left ..< (right + 1)])\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left\nlet leftEnd = mid - left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left\nlet rightEnd = right - left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart\nvar j = rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in left ... right {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse {\nnums[k] = tmp[j]\nj += 1\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums: &nums, left: left, mid: mid, right: right)\n}\n
     merge_sort.zig
    // \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\ndefer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar tmp = try mem_allocator.alloc(i32, right + 1 - left);\nstd.mem.copy(i32, tmp, nums[left..right+1]);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nvar leftStart = left - left;\nvar leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nvar rightStart = mid + 1 - left;\nvar rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart;\nvar j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nvar k = left;\nwhile (k <= right) : (k += 1) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j];\nj += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if  (j > rightEnd or tmp[i] <= tmp[j]) {\nnums[k] = tmp[i];\ni += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j];\nj += 1;\n}\n}\n}\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nvar mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\ntry mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\ntry mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\ntry merge(nums, left, mid, right);\n}\n
     
    \u4e0b\u9762\u91cd\u70b9\u89e3\u91ca\u4e00\u4e0b\u5408\u5e76\u65b9\u6cd5 merge() \u7684\u6d41\u7a0b\uff1a
     
       
    1. \u521d\u59cb\u5316\u4e00\u4e2a\u8f85\u52a9\u6570\u7ec4 tmp \u6682\u5b58\u5f85\u5408\u5e76\u533a\u95f4 [left, right] \u5185\u7684\u5143\u7d20\uff0c\u540e\u7eed\u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u7684\u5143\u7d20\u6765\u5b9e\u73b0\u5408\u5e76\uff1b
      2.  
    2. \u521d\u59cb\u5316\u6307\u9488 i , j , k \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u3001\u539f\u6570\u7ec4\u7684\u9996\u5143\u7d20\uff1b
      3.  
    3. \u5faa\u73af\u5224\u65ad tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\uff0c\u5c06\u8f83\u5c0f\u7684\u5148\u8986\u76d6\u81f3 nums[k] \uff0c\u6307\u9488 i , j \u6839\u636e\u5224\u65ad\u7ed3\u679c\u4ea4\u66ff\u524d\u8fdb\uff08\u6307\u9488 k \u4e5f\u524d\u8fdb\uff09\uff0c\u76f4\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u90fd\u904d\u5386\u5b8c\uff0c\u5373\u53ef\u5b8c\u6210\u5408\u5e76\u3002
      4.  
     
    \u5408\u5e76\u65b9\u6cd5 merge() \u4ee3\u7801\u4e2d\u7684\u4e3b\u8981\u96be\u70b9\uff1a
     
       
    • nums \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a [left, right] \uff0c\u800c\u56e0\u4e3a tmp \u53ea\u590d\u5236\u4e86 nums \u8be5\u533a\u95f4\u5143\u7d20\uff0c\u6240\u4ee5 tmp \u5bf9\u5e94\u533a\u95f4\u4e3a [0, right - left] \uff0c\u9700\u8981\u7279\u522b\u6ce8\u610f\u4ee3\u7801\u4e2d\u5404\u4e2a\u53d8\u91cf\u7684\u542b\u4e49\u3002
      •  
    • \u5224\u65ad tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\u7684\u64cd\u4f5c\u4e2d\uff0c\u8fd8 \u9700\u8003\u8651\u5f53\u5b50\u6570\u7ec4\u904d\u5386\u5b8c\u6210\u540e\u7684\u7d22\u5f15\u8d8a\u754c\u95ee\u9898\uff0c\u5373 i > leftEnd \u548c j > rightEnd \u7684\u60c5\u51b5\uff0c\u7d22\u5f15\u8d8a\u754c\u7684\u4f18\u5148\u7ea7\u662f\u6700\u9ad8\u7684\uff0c\u4f8b\u5982\u5982\u679c\u5de6\u5b50\u6570\u7ec4\u5df2\u7ecf\u88ab\u5408\u5e76\u5b8c\u4e86\uff0c\u90a3\u4e48\u4e0d\u7528\u7ee7\u7eed\u5224\u65ad\uff0c\u76f4\u63a5\u5408\u5e76\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u5373\u53ef\u3002
      •  
    "},{"location":"chapter_sorting/merge_sort/#1152","title":"11.5.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
       
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \uff1a\u5212\u5206\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002
      •  
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u9700\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\u5408\u5e76\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff1b\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
      •  
    • \u975e\u539f\u5730\u6392\u5e8f\uff1a\u8f85\u52a9\u6570\u7ec4\u9700\u8981\u4f7f\u7528 \\(O(n)\\) \u989d\u5916\u7a7a\u95f4\u3002
      •  
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u65f6\u53ef\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\u3002
      •  
    • \u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5bf9\u4e8e\u4efb\u610f\u8f93\u5165\u6570\u636e\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u76f8\u540c\u3002
      •  
    "},{"location":"chapter_sorting/merge_sort/#1153","title":"11.5.3. \u00a0 \u94fe\u8868\u6392\u5e8f *","text":"
    \u5f52\u5e76\u6392\u5e8f\u6709\u4e00\u4e2a\u5f88\u7279\u522b\u7684\u4f18\u52bf\uff0c\u7528\u4e8e\u6392\u5e8f\u94fe\u8868\u65f6\u6709\u5f88\u597d\u7684\u6027\u80fd\u8868\u73b0\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u7531\u4e8e\u94fe\u8868\u53ef\u4ec5\u901a\u8fc7\u6539\u53d8\u6307\u9488\u6765\u5b9e\u73b0\u7ed3\u70b9\u589e\u5220\uff0c\u56e0\u6b64\u201c\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\u201d\u65e0\u9700\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff0c\u5373\u56de\u6eaf\u5408\u5e76\u9636\u6bb5\u4e0d\u7528\u50cf\u6392\u5e8f\u6570\u7ec4\u4e00\u6837\u5efa\u7acb\u8f85\u52a9\u6570\u7ec4 tmp \uff1b
      •  
    • \u901a\u8fc7\u4f7f\u7528\u300c\u8fed\u4ee3\u300d\u4ee3\u66ff\u300c\u9012\u5f52\u5212\u5206\u300d\uff0c\u53ef\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\uff1b
      •  
     
    \u8be6\u60c5\u53c2\u8003\uff1a148. \u6392\u5e8f\u94fe\u8868
    "},{"location":"chapter_sorting/quick_sort/","title":"11.4. \u00a0 \u5feb\u901f\u6392\u5e8f","text":"
    \u300c\u5feb\u901f\u6392\u5e8f Quick Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u201c\u5206\u6cbb\u601d\u60f3\u201d\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u901f\u5ea6\u5f88\u5feb\u3001\u5e94\u7528\u5f88\u5e7f\u3002
     
    \u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u4e3a\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5176\u76ee\u6807\u4e3a\uff1a\u9009\u53d6\u6570\u7ec4\u67d0\u4e2a\u5143\u7d20\u4e3a \u57fa\u51c6\u6570\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u52a8\u81f3\u5176\u5de6\u8fb9\uff0c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u52a8\u81f3\u5176\u53f3\u8fb9\u3002\u300c\u54e8\u5175\u5212\u5206\u300d\u7684\u5b9e\u73b0\u6d41\u7a0b\u4e3a\uff1a
     
       
    1. \u4ee5\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 i , j \u6307\u5411\u6570\u7ec4\u4e24\u7aef\uff1b
      2.  
    2. \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4e2d\u4f7f\u7528 i / j \u5206\u522b\u5bfb\u627e\u9996\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927 / \u5c0f\u7684\u5143\u7d20\uff0c\u5e76\u4ea4\u6362\u6b64\u4e24\u5143\u7d20\uff1b
      3.  
    3. \u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 2. \uff0c\u76f4\u81f3 i , j \u76f8\u9047\u65f6\u8df3\u51fa\uff0c\u6700\u7ec8\u628a\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\uff1b
      4.  
     
    \u300c\u54e8\u5175\u5212\u5206\u300d\u6267\u884c\u5b8c\u6bd5\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e24\u4e2a\u90e8\u5206\uff0c\u5373 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3 \u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 < \u57fa\u51c6\u6570 < \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u53ea\u9700\u8981\u6392\u5e8f\u4e24\u4e2a\u5b50\u6570\u7ec4\u5373\u53ef\u3002
     <1><2><3><4><5><6><7><8><9> 
     
     
     
     
     
     
     
     
     
    \u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u601d\u60f3
     
    \u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06 \u4e00\u4e2a\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898 \u7b80\u5316\u4e3a \u4e24\u4e2a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.cpp
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int>& nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int>& nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.py
    def partition(self, nums: List[int], left: int, right: int) -> int:\n\"\"\" \u54e8\u5175\u5212\u5206 \"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
     quick_sort.go
    /* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.js
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.ts
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.c
    [class]{quickSort}-[func]{partition}\n
     quick_sort.cs
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j)\n{\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right)\n{\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j)\n{\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.swift
    /* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\nlet tmp = nums[i]\nnums[i] = nums[j]\nnums[j] = tmp\n}\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left\nvar j = right\nwhile i < j {\nwhile i < j, nums[j] >= nums[left] {\nj -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i < j, nums[i] <= nums[left] {\ni += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.zig
    // \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\nvar tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     
    \u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f
     
    \u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002
     
    \u54e8\u5175\u5212\u5206 partition() \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 nums[left] \u548c nums[i] \uff0c\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <= \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d nums[left] >= nums[i] \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 i == j \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd nums[j] == nums[i] > nums[left] \uff1b\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002
     
    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 [0, 0, 0, 0, 1] \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a [1, 0, 0, 0, 0] \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u5bf9\u7684\u3002
     
    \u518d\u6df1\u60f3\u4e00\u6b65\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 nums[right] \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002
    "},{"location":"chapter_sorting/quick_sort/#1141","title":"11.4.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u9996\u5148\uff0c\u5bf9\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5f97\u5230\u5f85\u6392\u5e8f\u7684 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4\uff1b
      2.  
    2. \u63a5\u4e0b\u6765\uff0c\u5bf9 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4 \u5206\u522b \u9012\u5f52\u6267\u884c\u300c\u54e8\u5175\u5212\u5206\u300d\u2026\u2026
      3.  
    3. \u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6 \u7ec8\u6b62\u9012\u5f52\uff0c\u5373\u53ef\u5b8c\u6210\u5bf9\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\uff1b
      4.  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5feb\u901f\u6392\u5e8f\u548c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7684\u539f\u7406\u7c7b\u4f3c\uff0c\u90fd\u662f\u4ee5\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u7f29\u5c0f\u5904\u7406\u533a\u95f4\u3002
     
     
     Fig. \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int>& nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.py
    def quick_sort(self, nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5feb\u901f\u6392\u5e8f \"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right:\nreturn\n# \u54e8\u5175\u5212\u5206\npivot: int = self.partition(nums, left, right)\n# \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nself.quick_sort(nums, left, pivot - 1)\nself.quick_sort(nums, pivot + 1, right)\n
     quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\npivot := q.partition(nums, left, right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nq.quickSort(nums, left, pivot-1)\nq.quickSort(nums, pivot+1, right)\n}\n
     quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
     quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
     quick_sort.c
    [class]{quickSort}-[func]{quickSort}\n
     quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right)\n{\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums: &nums, left: left, right: pivot - 1)\nquickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
     quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nvar pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1142","title":"11.4.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \uff1a\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002
     
    \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \u5c42\uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u53ea\u5728\u9012\u5f52\u4e2d\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
     
    \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u53ef\u80fd\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002
    "},{"location":"chapter_sorting/quick_sort/#1143","title":"11.4.3. \u00a0 \u5feb\u6392\u4e3a\u4ec0\u4e48\u5feb\uff1f","text":"
    \u4ece\u547d\u540d\u80fd\u591f\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u4e00\u5b9a\u201c\u6709\u4e24\u628a\u5237\u5b50\u201d\u3002\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u867d\u7136\u4e0e\u300c\u5f52\u5e76\u6392\u5e8f\u300d\u548c\u300c\u5806\u6392\u5e8f\u300d\u4e00\u81f4\uff0c\u4f46\u5b9e\u9645 \u6548\u7387\u66f4\u9ad8\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4e0d\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u4f46\u7edd\u5927\u90e8\u5206\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u4ee5\u8fbe\u5230 \\(O(n \\log n)\\) \u7684\u590d\u6742\u5ea6\u3002
      •  
    • \u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5165\u7f13\u5b58\u4e2d\uff0c\u8bbf\u95ee\u5143\u7d20\u6548\u7387\u5f88\u9ad8\u3002\u800c\u8bf8\u5982\u300c\u5806\u6392\u5e8f\u300d\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u5177\u6709\u6b64\u7279\u6027\u3002
      •  
    • \u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u4f4e\uff1a\u5728\u63d0\u53ca\u7684\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684 \u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362 \u4e09\u79cd\u64cd\u4f5c\u7684\u603b\u4f53\u6570\u91cf\u6700\u5c11\uff08\u7c7b\u4f3c\u4e8e\u300c\u63d2\u5165\u6392\u5e8f\u300d\u5feb\u4e8e\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u7684\u539f\u56e0\uff09\u3002
      •  
    "},{"location":"chapter_sorting/quick_sort/#1144","title":"11.4.4. \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"
    \u666e\u901a\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53d8\u5dee\u3002\u4e3e\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u53d6\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u4ece\u800c \u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u8fd9\u6837\u8fdb\u4e00\u6b65\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e86\u3002
     
    \u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u4e00\u4e0b\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u9996\u5148\uff0c\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5 \u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u4f46\u5982\u679c\u8fd0\u6c14\u5f88\u5dee\uff0c\u6bcf\u6b21\u90fd\u9009\u62e9\u5230\u6bd4\u8f83\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6548\u7387\u4f9d\u7136\u4e0d\u597d\u3002
     
    \u8fdb\u4e00\u6b65\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6 3 \u4e2a\u5019\u9009\u5143\u7d20\uff08\u4e00\u822c\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u8fd9\u6837\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u5927\u4e5f\u4e0d\u5c0f\u201d\u7684\u6982\u7387\u5c31\u5927\u5927\u63d0\u5347\u4e86\u3002\u5f53\u7136\uff0c\u5982\u679c\u6570\u7ec4\u5f88\u957f\u7684\u8bdd\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u6765\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u53d6\u8be5\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u6781\u4f4e\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.cpp
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int>& nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int>& nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.py
    def median_three(self, nums: List[int], left: int, mid: int, right: int) -> int:\n\"\"\" \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 \"\"\"\n# \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n# \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\nreturn left\nelif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\nreturn mid\nreturn right\ndef partition(self, nums: List[int], left: int, right: int) -> int:\n\"\"\" \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 \"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed: int = self.median_three(nums, left, (left + right) // 2, right)\n# \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
     quick_sort.go
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n}\nreturn right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed := q.medianThree(nums, left, (left+right)/2, right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n//\u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n//\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.js
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right])) return mid;\nelse return right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(nums, left, Math.floor((left + right) / 2), right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.ts
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums: number[], left: number, mid: number, right: number): number {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\nreturn left;\n} else if (Number(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(nums, left, Math.floor((left + right) / 2), right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.c
    [class]{quickSortMedian}-[func]{medianThree}\n[class]{quickSortMedian}-[func]{partition}\n
     quick_sort.cs
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right)\n{\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right)\n{\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j)\n{\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.swift
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n} else {\nreturn right\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums: &nums, i: left, j: med)\nreturn partition(nums: &nums, left: left, right: right)\n}\n
     quick_sort.zig
    // \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\nreturn left;\n} else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nvar med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1145","title":"11.4.5. \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"
    \u666e\u901a\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u7a7a\u95f4\u6548\u7387\u53d8\u5dee\u3002\u4ecd\u7136\u4ee5\u5b8c\u5168\u5012\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u7531\u4e8e\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 0 \uff0c\u90a3\u4e48\u5c06\u5f62\u6210\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(n - 1\\) \u7684\u9012\u5f52\u6811\uff0c\u6b64\u65f6\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u5927\u5c0f\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
     
    \u4e3a\u4e86\u907f\u514d\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u5224\u65ad\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u5927\u5c0f\uff0c\u4ec5\u9012\u5f52\u6392\u5e8f\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u3002\u7531\u4e8e\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8fd9\u6837\u505a\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u5373\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u88ab\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int>& nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.py
    def quick_sort(self, nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 \"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right:\n# \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot: int = self.partition(nums, left, right)\n# \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot - left < right - pivot:\nself.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1     # \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\nelse:\nself.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1    # \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
     quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nfor left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot := q.partition(nums, left, right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot-left < right-pivot {\nq.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1                 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nq.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1                 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.c
    [class]{quickSortTailCall}-[func]{quickSort}\n
     quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right)\n{\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right)\n{\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot)\n{\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n}\nelse\n{\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\nvar left = left\nvar right = right\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left) < (right - pivot) {\nquickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\nvar left = left_;\nvar right = right_;\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nvar pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                   // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    "},{"location":"chapter_sorting/summary/","title":"11.7. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u589e\u52a0\u6807\u5fd7\u4f4d\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(N)\\) \u3002
      •  
    • \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u5f85\u6392\u5e8f\u533a\u95f4\u5185\u5143\u7d20\u63d2\u5165\u81f3\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u6392\u5e8f\u3002\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u867d\u4e3a \\(O(N^2)\\) \uff0c\u4f46\u56e0\u4e3a\u603b\u4f53\u64cd\u4f5c\u5c11\u800c\u5f88\u53d7\u6b22\u8fce\uff0c\u4e00\u822c\u7528\u4e8e\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u5de5\u4f5c\u3002
      •  
    • \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u4ece\u800c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(N^2)\\) \uff0c\u901a\u8fc7\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u5927\u5927\u964d\u4f4e\u52a3\u5316\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u51cf\u5c0f\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log N)\\) \u3002
      •  
    • \u5f52\u5e76\u6392\u5e8f\u5305\u542b\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u662f\u5206\u800c\u6cbb\u4e4b\u7684\u6807\u51c6\u4f53\u73b0\u3002\u5bf9\u4e8e\u5f52\u5e76\u6392\u5e8f\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff1b\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
      •  
    • \u4e0b\u56fe\u603b\u7ed3\u5bf9\u6bd4\u4e86\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u4e0e\u7279\u6027\u3002\u5176\u4e2d\uff0c\u6876\u6392\u5e8f\u4e2d \\(k\\) \u4e3a\u6876\u7684\u6570\u91cf\uff1b\u57fa\u6570\u6392\u5e8f\u4ec5\u9002\u7528\u4e8e\u6b63\u6574\u6570\u3001\u5b57\u7b26\u4e32\u3001\u7279\u5b9a\u683c\u5f0f\u7684\u6d6e\u70b9\u6570\uff0c\\(k\\) \u4e3a\u6700\u5927\u6570\u5b57\u7684\u4f4d\u6570\u3002
      •  
     
     Fig. \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 
     
       
    • \u603b\u4f53\u6765\u770b\uff0c\u6211\u4eec\u8ffd\u6c42\u8fd0\u884c\u5feb\u3001\u7a33\u5b9a\u3001\u539f\u5730\u3001\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7684\u6392\u5e8f\u3002\u663e\u7136\uff0c\u5982\u540c\u5176\u5b83\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e00\u6837\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e9b\u6761\u4ef6\u7684\u6392\u5e8f\u7b97\u6cd5\u5e76\u4e0d\u5b58\u5728\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u95ee\u9898\u7279\u70b9\u6765\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u3002
      •  
    "},{"location":"chapter_stack_and_queue/deque/","title":"5.3. \u00a0 \u53cc\u5411\u961f\u5217","text":"
    \u5bf9\u4e8e\u961f\u5217\uff0c\u6211\u4eec\u53ea\u80fd\u5728\u5934\u90e8\u5220\u9664\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u300c\u53cc\u5411\u961f\u5217 Deque\u300d\u66f4\u52a0\u7075\u6d3b\uff0c\u5728\u5176\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u80fd\u6267\u884c\u5143\u7d20\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002
     
     
     Fig. \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c 
    "},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1. \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"
    \u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u8bed\u8a00\u6765\u786e\u5b9a\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) 
    \u76f8\u540c\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u53cc\u5411\u961f\u5217\u7c7b\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig deque.java
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
     deque.cpp
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
     deque.py
    \"\"\" \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 \"\"\"\ndeque: Deque[int] = collections.deque()\n\"\"\" \u5143\u7d20\u5165\u961f \"\"\"\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n\"\"\" \u5143\u7d20\u51fa\u961f \"\"\"\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(deque)\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(deque) == 0\n
     deque_test.go
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
     deque.js
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
     deque.ts
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
     deque.c
    \n
     deque.cs
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new LinkedList<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
     deque.swift
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
     deque.zig
    \n
    "},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2. \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"
    \u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53cc\u5411\u961f\u5217\u540c\u6837\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u6216\u6570\u7ec4\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/deque/#_1","title":"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u56de\u5fc6\u4e0a\u8282\u5185\u5bb9\uff0c\u7531\u4e8e\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u94fe\u8868\u5934\u7ed3\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\uff0c\u4ee5\u53ca\u5728\u94fe\u8868\u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0\u65b0\u7ed3\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\u3002
     
    \u800c\u53cc\u5411\u961f\u5217\u7684\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u4e0e\u51fa\u961f\u64cd\u4f5c\uff0c\u6362\u8a00\u4e4b\uff0c\u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c\u662f\u201c\u9996\u5c3e\u5bf9\u79f0\u201d\u7684\uff0c\u4e5f\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u4f7f\u7528\u300c\u53cc\u5411\u94fe\u8868\u300d\u6765\u5b9e\u73b0\u3002
     
    \u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u7ed3\u70b9\u548c\u5c3e\u7ed3\u70b9\u5206\u522b\u770b\u4f5c\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u5e76\u4e14\u5b9e\u73b0\u5728\u4e24\u7aef\u90fd\u80fd\u6dfb\u52a0\u4e0e\u5220\u9664\u7ed3\u70b9\u3002
     LinkedListDequepushLast()pushFirst()popLast()popFirst() 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_deque.java
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nint val;       // \u7ed3\u70b9\u503c\nListNode next; // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode prev; // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(int val) {\nthis.val = val;\nprev = next = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode front, rear; // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear\nprivate int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, boolean isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate Integer pop(boolean isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty())\nreturn null;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear.val;  // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic Integer popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic Integer popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic Integer peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic Integer peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_deque.cpp
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nstruct DoublyListNode {\nint val;               // \u7ed3\u70b9\u503c\nDoublyListNode *next;  // \u540e\u7ee7\u7ed3\u70b9\u6307\u9488\nDoublyListNode *prev;  // \u524d\u9a71\u7ed3\u70b9\u6307\u9488\nDoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {}\n};\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate:\nDoublyListNode *front, *rear; // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear\nint queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nLinkedListDeque() : front(nullptr), rear(nullptr) {}\n/* \u6790\u6784\u65b9\u6cd5 */\n~LinkedListDeque() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nDoublyListNode *pre, *cur = front;\nwhile (cur != nullptr) {\npre = cur;\ncur = cur->next;\ndelete pre;\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nDoublyListNode *node = new DoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront->prev = node;\nnode->next = front;\nfront = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear->next = node;\nnode->prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de -1\nif (isEmpty())\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front->val; // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nDoublyListNode *fNext = front->next;\nif (fNext != nullptr) {\nfNext->prev = nullptr;\nfront->next = nullptr;\ndelete front;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear->val; // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nDoublyListNode *rPrev = rear->prev;\nif (rPrev != nullptr) {\nrPrev->next = nullptr;\nrear->prev = nullptr;\ndelete rear;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nreturn isEmpty() ? -1 : front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nreturn isEmpty() ? -1 : rear->val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\nDoublyListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_deque.py
    class ListNode:\n\"\"\" \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 \"\"\"\ndef __init__(self, val: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.val: int = val\nself.next: Optional[ListNode] = None  # \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nself.prev: Optional[ListNode] = None  # \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nclass LinkedListDeque:\n\"\"\" \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 \"\"\"\ndef __init__(self) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.front: Optional[ListNode] = None # \u5934\u7ed3\u70b9 front\nself.rear: Optional[ListNode] = None  # \u5c3e\u7ed3\u70b9 rear\nself.__size: int = 0        # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.size() == 0\ndef push(self, num: int, is_front: bool) -> None:\n\"\"\" \u5165\u961f\u64cd\u4f5c \"\"\"\nnode = ListNode(num)\n# \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif self.is_empty():\nself.front = self.rear = node\n# \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelif is_front:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.prev = node\nnode.next = self.front\nself.front = node  # \u66f4\u65b0\u5934\u7ed3\u70b9\n# \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.next = node\nnode.prev = self.rear\nself.rear = node  # \u66f4\u65b0\u5c3e\u7ed3\u70b9\nself.__size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\ndef push_first(self, num: int) -> None:\n\"\"\" \u961f\u9996\u5165\u961f \"\"\"\nself.push(num, True)\ndef push_last(self, num: int) -> None:\n\"\"\" \u961f\u5c3e\u5165\u961f \"\"\"\nself.push(num, False)\ndef pop(self, is_front: bool) -> int:\n\"\"\" \u51fa\u961f\u64cd\u4f5c \"\"\"\n# \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\nif self.is_empty():\nreturn None\n# \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front:\nval: int = self.front.val  # \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n# \u5220\u9664\u5934\u7ed3\u70b9\nfnext: Optional[ListNode] = self.front.next\nif fnext != None:\nfnext.prev = None\nself.front.next = None\nself.front = fnext  # \u66f4\u65b0\u5934\u7ed3\u70b9\n# \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse:\nval: int = self.rear.val  # \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n# \u5220\u9664\u5c3e\u7ed3\u70b9\nrprev: Optional[ListNode] = self.rear.prev\nif rprev != None:\nrprev.next = None\nself.rear.prev = None\nself.rear = rprev  # \u66f4\u65b0\u5c3e\u7ed3\u70b9\nself.__size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\ndef pop_first(self) -> int:\n\"\"\" \u961f\u9996\u51fa\u961f \"\"\"\nreturn self.pop(True)\ndef pop_last(self) -> int:\n\"\"\" \u961f\u5c3e\u51fa\u961f \"\"\"\nreturn self.pop(False)\ndef peek_first(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nreturn None if self.is_empty() else self.front.val\ndef peek_last(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \"\"\"\nreturn None if self.is_empty() else self.rear.val\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 \"\"\"\nnode: Optional[ListNode] = self.front\nres: List[int] = [0] * self.size()\nfor i in range(self.size()):\nres[i] = node.val\nnode = node.next\nreturn res\n
     linkedlist_deque.go
    /* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\nreturn &linkedListDeque{\ndata: list.New(),\n}\n}\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\ns.data.PushFront(value)\n}\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\ns.data.PushBack(value)\n}\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_deque.js
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nprev;   // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nnext;   // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nval;    // \u7ed3\u70b9\u503c\nconstructor(val) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n#front;  // \u5934\u7ed3\u70b9 front\n#rear;   // \u5c3e\u7ed3\u70b9 rear\n#queSize;    // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\nthis.#queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.#rear.next = node;\nnode.prev = this.#rear;\nthis.#rear = node; // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.#front.prev = node;\nnode.next = this.#front;\nthis.#front = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#rear.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet temp = this.#rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.#rear.prev = null;\n}\nthis.#rear = temp;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#front.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet temp = this.#front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.#front.next = null;\n}\nthis.#front = temp;   // \u66f4\u65b0\u5934\u7ed3\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nreturn this.#queSize === 0 ? null : this.#rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nreturn this.#queSize === 0 ? null : this.#front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint() {\nconst arr = [];\nlet temp = this.#front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log(\"[\" + arr.join(\", \") + \"]\");\n}\n}\n
     linkedlist_deque.ts
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nprev: ListNode;     // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nnext: ListNode;     // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nval: number;        // \u7ed3\u70b9\u503c\nconstructor(val: number) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate front: ListNode;    // \u5934\u7ed3\u70b9 front\nprivate rear: ListNode;     // \u5c3e\u7ed3\u70b9 rear\nprivate queSize: number;        // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.front = null;\nthis.rear = null;\nthis.queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.rear.next = node;\nnode.prev = this.rear;\nthis.rear = node; // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.front.prev = node;\nnode.next = this.front;\nthis.front = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet temp: ListNode = this.rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.rear.prev = null;\n}\nthis.rear = temp;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.front.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet temp: ListNode = this.front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.front.next = null;\n}\nthis.front = temp;   // \u66f4\u65b0\u5934\u7ed3\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nreturn this.queSize === 0 ? null : this.rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nreturn this.queSize === 0 ? null : this.front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint(): void {\nconst arr: number[] = [];\nlet temp: ListNode = this.front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log(\"[\" + arr.join(\", \") + \"]\");\n}\n}\n
     linkedlist_deque.c
    [class]{ListNode}-[func]{}\n[class]{LinkedListDeque}-[func]{}\n
     linkedlist_deque.cs
    [class]{ListNode}-[func]{}\n[class]{LinkedListDeque}-[func]{}\n
     linkedlist_deque.swift
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nvar prev: ListNode? // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\ninit(val: Int) {\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate var front: ListNode? // \u5934\u7ed3\u70b9 front\nprivate var rear: ListNode? // \u5c3e\u7ed3\u70b9 rear\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ninit() {\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate func push(num: Int, isFront: Bool) {\nlet node = ListNode(val: num)\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif isEmpty() {\nfront = node\nrear = node\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if isFront {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront?.prev = node\nnode.next = front\nfront = node // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear?.next = node\nnode.prev = rear\nrear = node // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\npush(num: num, isFront: true)\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\npush(num: num, isFront: false)\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate func pop(isFront: Bool) -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nlet val: Int\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif isFront {\nval = front!.val // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet fNext = front?.next\nif fNext != nil {\nfNext?.prev = nil\nfront?.next = nil\n}\nfront = fNext // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear!.val // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet rPrev = rear?.prev\nif rPrev != nil {\nrPrev?.next = nil\nrear?.prev = nil\n}\nrear = rPrev // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\npop(isFront: true)\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\npop(isFront: false)\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int? {\nisEmpty() ? nil : front?.val\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int? {\nisEmpty() ? nil : rear?.val\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_deque.zig
    // \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\nfn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = undefined,     // \u7ed3\u70b9\u503c\nnext: ?*Self = null,    // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nprev: ?*Self = null,    // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\n// Initialize a list node with specific value\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*ListNode(T) = null,                    // \u5934\u7ed3\u70b9 front\nrear: ?*ListNode(T) = null,                     // \u5c3e\u7ed3\u70b9 rear\nque_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u5165\u961f\u64cd\u4f5c\npub fn push(self: *Self, num: T, is_front: bool) !void {\nvar node = try self.mem_allocator.create(ListNode(T));\nnode.init(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (self.isEmpty()) {\nself.front = node;\nself.rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n} else if (is_front) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.?.prev = node;\nnode.next = self.front;\nself.front = node;  // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.?.next = node;\nnode.prev = self.rear;\nself.rear = node;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nself.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n} // \u961f\u9996\u5165\u961f\npub fn pushFirst(self: *Self, num: T) !void {\ntry self.push(num, true);\n} // \u961f\u5c3e\u5165\u961f\npub fn pushLast(self: *Self, num: T) !void {\ntry self.push(num, false);\n} // \u51fa\u961f\u64cd\u4f5c\npub fn pop(self: *Self, is_front: bool) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nvar val: T = undefined;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (is_front) {\nval = self.front.?.val;     // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nvar fNext = self.front.?.next;\nif (fNext != null) {\nfNext.?.prev = null;\nself.front.?.next = null;\n}\nself.front = fNext;         // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = self.rear.?.val;      // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nvar rPrev = self.rear.?.prev;\nif (rPrev != null) {\nrPrev.?.next = null;\nself.rear.?.prev = null;\n}\nself.rear = rPrev;          // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nself.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n} // \u961f\u9996\u51fa\u961f\npub fn popFirst(self: *Self) T {\nreturn self.pop(true);\n} // \u961f\u5c3e\u51fa\u961f\npub fn popLast(self: *Self) T {\nreturn self.pop(false);\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peekFirst(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\npub fn peekLast(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.rear.?.val;\n}\n// \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002\u5728\u5b9e\u73b0\u961f\u5217\u7684\u57fa\u7840\u4e0a\uff0c\u589e\u52a0\u5b9e\u73b0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u65b9\u6cd5\u5373\u53ef\u3002
     ArrayDequepushLast()pushFirst()popLast()popFirst() 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_deque.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new EmptyStackException();\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate:\nvector<int> nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;         // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;       // \u53cc\u5411\u961f\u5217\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nnums.resize(capacity);\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn nums.size();\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> res(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n};\n
     array_deque.py
    class ArrayDeque:\n\"\"\" \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 \"\"\"\ndef __init__(self, capacity: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__nums: List[int] = [0] * capacity\nself.__front: int = 0\nself.__size: int = 0\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf \"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__size == 0\ndef index(self, i: int) -> int:\n\"\"\" \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 \"\"\"\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + self.capacity()) % self.capacity()\ndef push_first(self, num: int) -> None:\n\"\"\" \u961f\u9996\u5165\u961f \"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself.__front = self.index(self.__front - 1)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself.__nums[self.__front] = num\nself.__size += 1\ndef push_last(self, num: int) -> None:\n\"\"\" \u961f\u5c3e\u5165\u961f \"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear = self.index(self.__front + self.__size)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop_first(self) -> int:\n\"\"\" \u961f\u9996\u51fa\u961f \"\"\"\nnum = self.peek_first()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself.__front = self.index(self.__front + 1)\nself.__size -= 1\nreturn num\ndef pop_last(self) -> int:\n\"\"\" \u961f\u5c3e\u51fa\u961f \"\"\"\nnum = self.peek_last()\nself.__size -= 1\nreturn num\ndef peek_first(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\"\nreturn self.__nums[self.__front]\ndef peek_last(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\"\n# \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast = self.index(self.__front + self.__size - 1)\nreturn self.__nums[last]\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 \"\"\"\n# \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres = []\nfor i in range(self.__size):\nres.append(self.__nums[self.index(self.__front + i)])\nreturn res\n
     array_deque.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\nreturn &arrayDeque{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + q.queCapacity) % q.queCapacity\n}\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nq.front = q.index(q.front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[q.front] = num\nq.queSize++\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear := q.index(q.front + q.queSize)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[rear] = num\nq.queSize++\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\nnum := q.peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nq.front = q.index(q.front + 1)\nq.queSize--\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\nnum := q.peekLast()\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\nif q.isEmpty() {\nreturn nil\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast := q.index(q.front + q.queSize - 1)\nreturn q.nums[last]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres := make([]int, q.queSize)\nfor i, j := 0, q.front; i < q.queSize; i++ {\nres[i] = q.nums[q.index(j)]\nj++\n}\nreturn res\n}\n
     array_deque.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n#nums;       // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front;      // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize;    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\nthis.#front = 0;\nthis.#queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.#front = this.index(this.#front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.#nums[this.#front] = num;\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear = this.index(this.#front + this.#queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst() {\nconst num = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.#front = this.index(this.#front + 1);\nthis.#queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast() {\nconst num = this.peekLast();\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\nreturn this.#nums[this.#front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.#front + this.#queSize - 1);\nreturn this.#nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res = [];\nfor (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\nres[i] = this.#nums[this.index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate nums: number[];     // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number;      // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number;    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = 0;\nthis.queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i: number): number {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.front = this.index(this.front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.nums[this.front] = num;\nthis.queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear: number = this.index(this.front + this.queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst(): number {\nconst num: number = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.front = this.index(this.front + 1);\nthis.queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast(): number {\nconst num: number = this.peekLast();\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\nreturn this.nums[this.front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.front + this.queSize - 1);\nreturn this.nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res: number[] = [];\nfor (let i = 0, j = this.front; i < this.queSize; i++, j++) {\nres[i] = this.nums[this.index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.c
    [class]{ArrayDeque}-[func]{}\n
     array_deque.cs
    [class]{ArrayDeque}-[func]{}\n
     array_deque.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(capacity: Int) {\nnums = Array(repeating: 0, count: capacity)\nfront = 0\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate func index(i: Int) -> Int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n(i + capacity()) % capacity()\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(i: front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num\nqueSize += 1\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = index(i: front + size())\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\nlet num = peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(i: front + 1)\nqueSize -= 1\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\nlet num = peekLast()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = index(i: front + size() - 1)\nreturn nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: size())\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[index(i: j)]\n}\nreturn res\n}\n}\n
     array_deque.zig
    [class]{ArrayDeque}-[func]{}\n
    "},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3. \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"
    \u53cc\u5411\u961f\u5217\u540c\u65f6\u8868\u73b0\u51fa\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002
     
    \u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u9700\u8981\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1b\u7cfb\u7edf\u628a\u6bcf\u4e00\u6b21\u66f4\u6539\u64cd\u4f5c push \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 pop \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u6709\u9650\uff0c\u8f6f\u4ef6\u4e00\u822c\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\uff0c\u90a3\u4e48\u5f53\u6808\u7684\u957f\u5ea6 \\(> 50\\) \u65f6\uff0c\u8f6f\u4ef6\u5c31\u9700\u8981\u5728\u6808\u5e95\uff08\u5373\u961f\u9996\uff09\u6267\u884c\u5220\u9664\uff0c\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u662f\u6808\u7684\u5148\u5165\u540e\u51fa\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/","title":"5.2. \u00a0 \u961f\u5217","text":"
    \u300c\u961f\u5217 Queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u300c\u5148\u5165\u5148\u51fa first in, first out\u300d\u6570\u636e\u64cd\u4f5c\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u7684\u662f\u6392\u961f\u73b0\u8c61\uff0c\u5373\u5916\u9762\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u5904\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u4e0d\u65ad\u5730\u79bb\u5f00\u3002
     
    \u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u300c\u961f\u9996\u300d\uff0c\u961f\u5217\u5c3e\u90e8\u79f0\u4e3a\u300c\u961f\u5c3e\u300d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u961f\u300d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u961f\u300d\u3002
     
     
     Fig. \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 
    "},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1. \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"
    \u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\u3002\u9700\u8981\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u662f\u4e0d\u540c\u7684\uff0c\u5728\u8fd9\u91cc\u6211\u4eec\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u961f\u5217\u7c7b\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig queue.java
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
     queue.cpp
    /* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
     queue.py
    \"\"\" \u521d\u59cb\u5316\u961f\u5217 \"\"\"\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u770b\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u5efa\u8bae\nque: Deque[int] = collections.deque()\n\"\"\" \u5143\u7d20\u5165\u961f \"\"\"\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nfront: int = que[0];\n\"\"\" \u5143\u7d20\u51fa\u961f \"\"\"\npop: int = que.popleft()\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(que)\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(que) == 0\n
     queue_test.go
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
     queue.js
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
     queue.ts
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
     queue.c
    \n
     queue.cs
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count() == 0;\n
     queue.swift
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
     queue.zig
    \n
    "},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2. \u00a0 \u961f\u5217\u5b9e\u73b0","text":"
    \u961f\u5217\u9700\u8981\u4e00\u79cd\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u7684\u6570\u636e\u7ed3\u6784\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u6216\u6570\u7ec4\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u6211\u4eec\u5c06\u94fe\u8868\u7684\u300c\u5934\u7ed3\u70b9\u300d\u548c\u300c\u5c3e\u7ed3\u70b9\u300d\u5206\u522b\u770b\u4f5c\u662f\u961f\u9996\u548c\u961f\u5c3e\uff0c\u5e76\u89c4\u5b9a\u961f\u5c3e\u53ea\u53ef\u6dfb\u52a0\u7ed3\u70b9\uff0c\u961f\u9996\u53ea\u53ef\u5220\u9664\u7ed3\u70b9\u3002
     LinkedListQueuepush()pop() 
     
     
     
    \u4ee5\u4e0b\u662f\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_queue.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode front, rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new EmptyStackException();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate:\nListNode *front, *rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nint queSize;\npublic:\nLinkedListQueue() {\nfront = nullptr;\nrear = nullptr;\nqueSize = 0;\n}\n~LinkedListQueue() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(front);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode* node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == nullptr) {\nfront = node;\nrear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse {\nrear->next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nListNode *tmp = front;\nfront = front->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp; queSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (size() == 0)\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn front->val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode* node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_queue.py
    class LinkedListQueue:\n\"\"\" \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__front: Optional[ListNode] = None  # \u5934\u7ed3\u70b9 front\nself.__rear: Optional[ListNode] = None   # \u5c3e\u7ed3\u70b9 rear\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn not self.__front\ndef push(self, num: int) -> None:\n\"\"\" \u5165\u961f \"\"\"\n# \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nnode = ListNode(num)\n# \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif self.__front is None:\nself.__front = node\nself.__rear = node\n# \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse:\nself.__rear.next = node\nself.__rear = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u961f \"\"\"\nnum = self.peek()\n# \u5220\u9664\u5934\u7ed3\u70b9\nself.__front = self.__front.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nif self.size() == 0:\nprint(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn False\nreturn self.__front.val\ndef to_list(self) -> List[int]:\n\"\"\" \u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nqueue = []\ntemp = self.__front\nwhile temp:\nqueue.append(temp.val)\ntemp = temp.next\nreturn queue\n
     linkedlist_queue.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\nreturn &linkedListQueue{\ndata: list.New(),\n}\n}\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\ns.data.PushBack(value)\n}\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_queue.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n#front;  // \u5934\u7ed3\u70b9 #front\n#rear;   // \u5c3e\u7ed3\u70b9 #rear\n#queSize = 0;\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (!this.#front) {\nthis.#front = node;\nthis.#rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nthis.#rear.next = node;\nthis.#rear = node;\n}\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nthis.#front = this.#front.next;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.size === 0)\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.#front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#front;\nconst res = new Array(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate front: ListNode | null; // \u5934\u7ed3\u70b9 front\nprivate rear: ListNode | null; // \u5c3e\u7ed3\u70b9 rear\nprivate queSize: number = 0;\nconstructor() {\nthis.front = null;\nthis.rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (!this.front) {\nthis.front = node;\nthis.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nthis.rear!.next = node;\nthis.rear = node;\n}\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\nif (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n// \u5220\u9664\u5934\u7ed3\u70b9\nthis.front = this.front.next;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.front;\nconst res = new Array<number>(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.c
    [class]{linkedListQueue}-[func]{}\n
     linkedlist_queue.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue\n{\nprivate ListNode? front, rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue()\n{\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num)\n{\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == null)\n{\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n}\nelse if (rear != null)\n{\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop()\n{\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front?.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek()\n{\nif (size() == 0 || front == null)\nthrow new Exception();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nif (front == null)\nreturn Array.Empty<int>();\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++)\n{\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate var front: ListNode? // \u5934\u7ed3\u70b9\nprivate var rear: ListNode? // \u5c3e\u7ed3\u70b9\nprivate var _size = 0\ninit() {}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nlet node = ListNode(x: num)\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif front == nil {\nfront = node\nrear = node\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse {\nrear?.next = node\nrear = node\n}\n_size += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn front!.val\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_queue.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*inc.ListNode(T) = null,                // \u5934\u7ed3\u70b9 front\nrear: ?*inc.ListNode(T) = null,                 // \u5c3e\u7ed3\u70b9 rear\nque_size: usize = 0,                             // \u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u5165\u961f\npub fn push(self: *Self, num: T) !void {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (self.front == null) {\nself.front = node;\nself.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nself.rear.?.next = node;\nself.rear = node;\n}\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nself.front = self.front.?.next;\nself.que_size -= 1;\nreturn num;\n} // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/queue/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u6570\u7ec4\u7684\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u4f4e\u4e0b\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u4e0b\u8ff0\u7684\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002
     
    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u53d8\u91cf front \u6765\u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u53d8\u91cf queSize \u6765\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u6211\u4eec\u5b9a\u4e49 rear = front + queSize \uff0c\u8be5\u516c\u5f0f\u8ba1\u7b97\u51fa\u6765\u7684 rear \u6307\u5411\u201c\u961f\u5c3e\u5143\u7d20\u7d22\u5f15 \\(+1\\) \u201d\u7684\u4f4d\u7f6e\u3002
     
    \u5728\u8be5\u8bbe\u8ba1\u4e0b\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a [front, rear - 1] \uff0c\u8fdb\u800c
     
       
    • \u5bf9\u4e8e\u5165\u961f\u64cd\u4f5c\uff0c\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 rear \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 queSize \u81ea\u589e \\(1\\) \u5373\u53ef\uff1b
      •  
    • \u5bf9\u4e8e\u51fa\u961f\u64cd\u4f5c\uff0c\u4ec5\u9700\u5c06 front \u81ea\u589e \\(1\\) \uff0c\u5e76\u5c06 queSize \u81ea\u51cf \\(1\\) \u5373\u53ef\uff1b
      •  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5165\u961f\u4e0e\u51fa\u961f\u64cd\u4f5c\u90fd\u4ec5\u9700\u5355\u6b21\u64cd\u4f5c\u5373\u53ef\u5b8c\u6210\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(1)\\) \u3002
     ArrayQueuepush()pop() 
     
     
     
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u5165\u961f\u4e0e\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0cfront \u548c rear \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5728\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u540e\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u8003\u8651\u5c06\u6570\u7ec4\u770b\u4f5c\u662f\u9996\u5c3e\u76f8\u63a5\u7684\uff0c\u8fd9\u6837\u7684\u6570\u7ec4\u88ab\u79f0\u4e3a\u300c\u73af\u5f62\u6570\u7ec4\u300d\u3002
     
    \u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u4ee4 front \u6216 rear \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u76f4\u63a5\u7ed5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u63a5\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u300c\u53d6\u4f59\u64cd\u4f5c\u300d\u6765\u5b9e\u73b0\uff0c\u8be6\u60c5\u8bf7\u89c1\u4ee5\u4e0b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_queue.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % capacity()];\n}\nreturn res;\n}\n}\n
     array_queue.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate:\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u961f\u5217\u957f\u5ea6\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\npublic:\nArrayQueue(int capacity) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = new int[capacity];\nqueCapacity = capacity;\nfront = queSize = 0;\n}\n~ArrayQueue() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (queSize == queCapacity) {\ncout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % queCapacity;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (empty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> arr(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\narr[i] = nums[j % queCapacity];\n}\nreturn arr;\n}\n};\n
     array_queue.py
    class ArrayQueue:\n\"\"\" \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 \"\"\"\ndef __init__(self, size: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__nums: List[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nself.__front: int = 0          # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nself.__size: int = 0           # \u961f\u5217\u957f\u5ea6\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf \"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__size == 0\ndef push(self, num: int) -> None:\n\"\"\" \u5165\u961f \"\"\"\nassert self.__size < self.capacity(), \"\u961f\u5217\u5df2\u6ee1\"\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear: int = (self.__front + self.__size) % self.capacity()\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u961f \"\"\"\nnum: int = self.peek()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.__front = (self.__front + 1) % self.capacity()\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u961f\u5217\u4e3a\u7a7a\"\nreturn self.__nums[self.__front]\ndef to_list(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nres: List[int] = [0] * self.size()\nj: int = self.__front\nfor i in range(self.size()):\nres[i] = self.__nums[(j % self.capacity())]\nj += 1\nreturn res\n
     array_queue.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\nreturn &arrayQueue{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n// \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\nif q.queSize == q.queCapacity {\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear := (q.front + q.queSize) % q.queCapacity\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nq.nums[rear] = num\nq.queSize++\n}\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\nnum := q.peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nq.front = (q.front + 1) % q.queCapacity\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\nrear := (q.front + q.queSize)\nif rear >= q.queCapacity {\nrear %= q.queCapacity\nreturn append(q.nums[q.front:], q.nums[:rear]...)\n}\nreturn q.nums[q.front:rear]\n}\n
     array_queue.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n#nums;         // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front = 0;    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize = 0;  // \u961f\u5217\u957f\u5ea6\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#queSize == 0;\n}\n/* \u5165\u961f */\npush(num) {\nif (this.size == this.capacity) {\nconsole.log(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.#front + this.size) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.#front = (this.#front + 1) % this.capacity;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.empty())\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.#nums[this.#front];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.#front; i < this.size; i++, j++) {\narr[i] = this.#nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
     array_queue.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate nums: number[];  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = this.queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.queSize == 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\nif (this.size == this.capacity) {\nconsole.log(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.front + this.queSize) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.front = (this.front + 1) % this.capacity;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.empty())\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.nums[this.front];\n}\n/* \u8fd4\u56de Array */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.front; i < this.size; i++, j++) {\narr[i] = this.nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
     array_queue.c
    [class]{arrayQueue}-[func]{}\n
     array_queue.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue\n{\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity)\n{\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity()\n{\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num)\n{\nif (queSize == capacity())\n{\nConsole.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop()\n{\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek()\n{\nif (isEmpty())\nthrow new Exception();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray()\n{\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++)\n{\nres[i] = nums[j % this.capacity()];\n}\nreturn res;\n}\n}\n
     array_queue.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize = 0 // \u961f\u5217\u957f\u5ea6\ninit(capacity: Int) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = Array(repeating: 0, count: capacity)\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nqueSize == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\nif size() == capacity() {\nprint(\"\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (front + queSize) % capacity()\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: queSize)\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[j % capacity()]\n}\nreturn res\n}\n}\n
     array_queue.zig
    // \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \ncap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\nfront: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nque_size: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.cap = cap;\nself.nums = try self.mem_allocator.alloc(T, self.cap);\nstd.mem.set(T, self.nums, @as(T, 0));\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.cap;\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.que_size == 0;\n}\n// \u5165\u961f\npub fn push(self: *Self, num: T) !void {\nif (self.size() == self.capacity()) {\nstd.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nvar rear = (self.front + self.que_size) % self.capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.nums[rear] = num;\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.capacity();\nself.que_size -= 1;\nreturn num;\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.nums[self.front];\n} // \u8fd4\u56de\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nvar j: usize = self.front;\nwhile (i < self.size()) : ({ i += 1; j += 1; }) {\nres[i] = self.nums[j % self.capacity()];\n}\nreturn res;\n}\n};\n}\n
     
    \u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u5b58\u5728\u5c40\u9650\u6027\uff0c\u5373\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u4e0d\u8fc7\u8fd9\u4e2a\u95ee\u9898\u5f88\u5bb9\u6613\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u5217\u8868\uff08\u5373\u52a8\u6001\u6570\u7ec4\uff09\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"
    \u4e0e\u6808\u7684\u7ed3\u8bba\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#524","title":"5.2.4. \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"
       
    • \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c31\u88ab\u52a0\u5165\u5230\u961f\u5217\u4e4b\u4e2d\uff0c\u968f\u540e\u7cfb\u7edf\u518d\u6839\u636e\u987a\u5e8f\u4f9d\u6b21\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u65f6\uff0c\u5728\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u7684\u8ba2\u5355\uff0c\u5982\u4f55\u5904\u7406\u300c\u9ad8\u5e76\u53d1\u300d\u5219\u662f\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u601d\u8003\u7684\u95ee\u9898\u3002
      •  
    • \u5404\u79cd\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u7684\u529f\u80fd\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\u7b49\u3002
      •  
    "},{"location":"chapter_stack_and_queue/stack/","title":"5.1. \u00a0 \u6808","text":"
    \u300c\u6808 Stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u300c\u5148\u5165\u540e\u51fa first in, last out\u300d\u6570\u636e\u64cd\u4f5c\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u653e\u5728\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u9700\u8981\u62ff\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u53d6\u51fa\u3002
     
    \u201c\u76d8\u5b50\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u6bd4\u55bb\uff0c\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u79cd\u5143\u7d20\uff08\u4f8b\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u6570\u636e\u7ed3\u6784\u3002
     
    \u6211\u4eec\u5c06\u8fd9\u4e00\u645e\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u300c\u6808\u9876\u300d\uff0c\u5c06\u5e95\u90e8\u79f0\u4e3a\u300c\u6808\u5e95\u300d\uff0c\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u6808\u300d\uff0c\u5c06\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u6808\u300d\u3002
     
     
     Fig. \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 
    "},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1. \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"
    \u6808\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\uff0c\u6b64\u5904\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 push , pop , peek \u4e3a\u4f8b\u3002
     \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u6808\u7c7b\u3002 \u67d0\u4e9b\u8bed\u8a00\u5e76\u672a\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u4f46\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u628a\u8be5\u8bed\u8a00\u7684\u300c\u6570\u7ec4\u300d\u6216\u300c\u94fe\u8868\u300d\u770b\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u901a\u8fc7\u201c\u8111\u8865\u201d\u6765\u5c4f\u853d\u65e0\u5173\u64cd\u4f5c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig stack.java
    /* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
     stack.cpp
    /* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
     stack.py
    \"\"\" \u521d\u59cb\u5316\u6808 \"\"\"\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: List[int] = []\n\"\"\" \u5143\u7d20\u5165\u6808 \"\"\"\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\npeek: int = stack[-1]\n\"\"\" \u5143\u7d20\u51fa\u6808 \"\"\"\npop: int = stack.pop()\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(stack)\n\"\"\" \u5224\u65ad\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(stack) == 0\n
     stack_test.go
    /* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
     stack.js
    /* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
     stack.ts
    /* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
     stack.c
    \n
     stack.cs
    /* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new ();\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count()==0;\n
     stack.swift
    /* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
     stack.zig
    \n
    "},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2. \u00a0 \u6808\u7684\u5b9e\u73b0","text":"
    \u4e3a\u4e86\u66f4\u52a0\u6e05\u6670\u5730\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u81ea\u5df1\u52a8\u624b\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002
     
    \u6808\u89c4\u5b9a\u5143\u7d20\u662f\u5148\u5165\u540e\u51fa\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u6216\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64 \u6808\u53ef\u88ab\u770b\u4f5c\u662f\u4e00\u79cd\u53d7\u7ea6\u675f\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u8a00\u4e4b\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u4e4b\u5bf9\u5916\u7684\u8868\u73b0\u903b\u8f91\u7b26\u5408\u6808\u7684\u89c4\u5b9a\u5373\u53ef\u3002
    "},{"location":"chapter_stack_and_queue/stack/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u4f7f\u7528\u300c\u94fe\u8868\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u5c06\u94fe\u8868\u7684\u5934\u7ed3\u70b9\u770b\u4f5c\u6808\u9876\uff0c\u5c06\u5c3e\u7ed3\u70b9\u770b\u4f5c\u6808\u5e95\u3002
     
    \u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5230\u94fe\u8868\u5934\u90e8\u5373\u53ef\uff0c\u8fd9\u79cd\u7ed3\u70b9\u6dfb\u52a0\u65b9\u5f0f\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u5219\u5c06\u5934\u7ed3\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002
     LinkedListStackpush()pop() 
     
     
     
    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_stack.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode stackPeek;  // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new EmptyStackException();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate:\nListNode* stackTop; // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize;        // \u6808\u7684\u957f\u5ea6\npublic:\nLinkedListStack() {\nstackTop = nullptr;\nstkSize = 0;\n}\n~LinkedListStack() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(stackTop);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nListNode* node = new ListNode(num);\nnode->next = stackTop;\nstackTop = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint num = top();\nListNode *tmp = stackTop;\nstackTop = stackTop->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nstkSize--;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (size() == 0)\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stackTop->val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode* node = stackTop;\nvector<int> res(size());\nfor (int i = res.size() - 1; i >= 0; i--) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_stack.py
    class LinkedListStack:\n\"\"\" \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__peek: Optional[ListNode] = None\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn not self.__peek\ndef push(self, val: int) -> None:\n\"\"\" \u5165\u6808 \"\"\"\nnode = ListNode(val)\nnode.next = self.__peek\nself.__peek = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u6808 \"\"\"\nnum: int = self.peek()\nself.__peek = self.__peek.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\n# \u5224\u7a7a\u5904\u7406\nif not self.__peek: return None\nreturn self.__peek.val\ndef to_list(self) -> List[int]:\n\"\"\" \u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\narr: List[int] = []\nnode = self.__peek\nwhile node:\narr.append(node.val)\nnode = node.next\narr.reverse()\nreturn arr\n
     linkedlist_stack.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\nreturn &linkedListStack{\ndata: list.New(),\n}\n}\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\ns.data.PushBack(value)\n}\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_stack.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n#stackPeek;     // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\n#stkSize = 0;   // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.#stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num) {\nconst node = new ListNode(num);\nnode.next = this.#stackPeek;\nthis.#stackPeek = node;\nthis.#stkSize++;\n}\n/* \u51fa\u6808 */\npop() {\nconst num = this.peek();\nthis.#stackPeek = this.#stackPeek.next;\nthis.#stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek() {\nif (!this.#stackPeek)\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#stackPeek;\nconst res = new Array(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate stackPeek: ListNode | null; // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nconst node = new ListNode(num);\nnode.next = this.stackPeek;\nthis.stackPeek = node;\nthis.stkSize++;\n}\n/* \u51fa\u6808 */\npop(): number {\nconst num = this.peek();\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nthis.stackPeek = this.stackPeek.next;\nthis.stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek(): number {\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.stackPeek;\nconst res = new Array<number>(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.c
    [class]{linkedListStack}-[func]{}\n
     linkedlist_stack.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack\n{\nprivate ListNode? stackPeek;  // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack()\n{\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num)\n{\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop()\n{\nif (stackPeek == null)\nthrow new Exception();\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek()\n{\nif (size() == 0 || stackPeek == null)\nthrow new Exception();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nif (stackPeek == null)\nreturn Array.Empty<int>();\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.Length - 1; i >= 0; i--)\n{\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate var _peek: ListNode? // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate var _size = 0 // \u6808\u7684\u957f\u5ea6\ninit() {}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nlet node = ListNode(x: num)\nnode.next = _peek\n_peek = node\n_size += 1\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n_peek = _peek?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn _peek!.val\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = _peek\nvar res = Array(repeating: 0, count: _size)\nfor i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_stack.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.stack_top = null;\nself.stk_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stk_size;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack_top.?.val;\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\nnode.next = self.stack_top;\nself.stack_top = node;\nself.stk_size += 1;\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.peek();\nself.stack_top = self.stack_top.?.next;\nself.stk_size -= 1;\nreturn num;\n} // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.stack_top;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[res.len - i - 1] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u4f7f\u7528\u300c\u6570\u7ec4\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u8003\u8651\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u5f53\u4f5c\u6808\u9876\u3002\u8fd9\u6837\u8bbe\u8ba1\u4e0b\uff0c\u300c\u5165\u6808\u300d\u4e0e\u300c\u51fa\u6808\u300d\u64cd\u4f5c\u5c31\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u300c\u6dfb\u52a0\u5143\u7d20\u300d\u4e0e\u300c\u5220\u9664\u5143\u7d20\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002
     ArrayStackpush()pop() 
     
     
     
    \u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u662f\u6e90\u6e90\u4e0d\u65ad\u7684\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528\u652f\u6301\u52a8\u6001\u6269\u5bb9\u7684\u300c\u5217\u8868\u300d\uff0c\u8fd9\u6837\u5c31\u65e0\u9700\u81ea\u884c\u5b9e\u73b0\u6570\u7ec4\u6269\u5bb9\u4e86\u3002\u4ee5\u4e0b\u662f\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_stack.java
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate ArrayList<Integer> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new ArrayList<>();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn stack.remove(size() - 1);\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn stack.get(size() - 1);\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic Object[] toArray() {\nreturn stack.toArray();\n}\n}\n
     array_stack.cpp
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate:\nvector<int> stack;\npublic:\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn stack.empty();\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nstack.push_back(num);\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint oldTop = top();\nstack.pop_back();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif(empty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stack.back();\n}\n/* \u8fd4\u56de Vector */\nvector<int> toVector() {\nreturn stack;\n}\n};\n
     array_stack.py
    class ArrayStack:\n\"\"\" \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 \"\"\"\ndef __init__(self) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__stack: List[int] = []\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nreturn len(self.__stack)\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__stack == []\ndef push(self, item: int) -> None:\n\"\"\" \u5165\u6808 \"\"\"\nself.__stack.append(item)\ndef pop(self) -> int:\n\"\"\" \u51fa\u6808 \"\"\"\nassert not self.is_empty(), \"\u6808\u4e3a\u7a7a\"\nreturn self.__stack.pop()\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u6808\u4e3a\u7a7a\"\nreturn self.__stack[-1]\ndef to_list(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nreturn self.__stack\n
     array_stack.go
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\ndata []int // \u6570\u636e\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\nreturn &arrayStack{\n// \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\ndata: make([]int, 0, 16),\n}\n}\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\nreturn len(s.data)\n}\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\nreturn s.size() == 0\n}\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n// \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\ns.data = append(s.data, v)\n}\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\nval := s.peek()\ns.data = s.data[:len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\nval := s.data[len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\nreturn s.data\n}\n
     array_stack.js
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n#stack;\nconstructor() {\nthis.#stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nthis.#stack.push(num);\n}\n/* \u51fa\u6808 */\npop() {\nif (this.empty())\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop() {\nif (this.empty())\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stack[this.#stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.#stack;\n}\n};\n
     array_stack.ts
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate stack: number[];\nconstructor() {\nthis.stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nthis.stack.push(num);\n}\n/* \u51fa\u6808 */\npop(): number | undefined {\nif (this.empty())\nthrow new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop(): number | undefined {\nif (this.empty())\nthrow new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack[this.stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.stack;\n}\n};\n
     array_stack.c
    [class]{arrayStack}-[func]{}\n
     array_stack.cs
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack\n{\nprivate List<int> stack;\npublic ArrayStack()\n{\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn stack.Count();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num)\n{\nstack.Add(num);\n}\n/* \u51fa\u6808 */\npublic int pop()\n{\nif (isEmpty())\nthrow new Exception();\nvar val = peek();\nstack.RemoveAt(size() - 1);\nreturn val;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek()\n{\nif (isEmpty())\nthrow new Exception();\nreturn stack[size() - 1];\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nreturn stack.ToArray();\n}\n}\n
     array_stack.swift
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate var stack: [Int]\ninit() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = []\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nstack.count\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nstack.isEmpty\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nstack.append(num)\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.removeLast()\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.last!\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nstack\n}\n}\n
     array_stack.zig
    // \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack: ?std.ArrayList(T) = null,     // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) void {\nif (self.stack == null) {\nself.stack = std.ArrayList(T).init(allocator);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.stack == null) return;\nself.stack.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stack.?.items.len;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack.?.items[self.size() - 1];\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\ntry self.stack.?.append(num);\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.stack.?.pop();\nreturn num;\n} // \u8fd4\u56de ArrayList\npub fn toList(self: *Self) std.ArrayList(T) {\nreturn self.stack.?;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":""},{"location":"chapter_stack_and_queue/stack/#_3","title":"\u652f\u6301\u64cd\u4f5c","text":"
    \u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\uff0c\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u7ecf\u8d85\u51fa\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002
    "},{"location":"chapter_stack_and_queue/stack/#_4","title":"\u65f6\u95f4\u6548\u7387","text":"
    \u5728\u6570\u7ec4\uff08\u5217\u8868\uff09\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u90fd\u662f\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u64cd\u4f5c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u6548\u7387\u5f88\u597d\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u5219\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u90a3\u4e48\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
     
    \u5728\u94fe\u8868\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u53d8\u6162\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u7ed3\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u800c\u6548\u7387\u4e0d\u5982\u6570\u7ec4\u3002\u8fdb\u4e00\u6b65\u5730\u601d\u8003\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u4e0d\u662f int \u800c\u662f\u7ed3\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u5c31\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u5347\u6548\u7387\u3002
     
    \u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff08\u4f8b\u5982 int , double \uff09\u65f6\uff0c\u5219\u7ed3\u8bba\u5982\u4e0b\uff1a
     
       
    • \u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u4f1a\u53d8\u6162\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64 \u603b\u4f53\u6548\u7387\u66f4\u9ad8\uff1b
      •  
    • \u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b \u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\uff1b
      •  
    "},{"location":"chapter_stack_and_queue/stack/#_5","title":"\u7a7a\u95f4\u6548\u7387","text":"
    \u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u7ed9\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u8fc7\u6211\u4eec\u7684\u9700\u6c42\u3002\u5e76\u4e14\u6269\u5bb9\u673a\u5236\u4e00\u822c\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u6bd4\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u6211\u4eec\u7684\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u6570\u7ec4\u5b9e\u73b0\u6808\u4f1a\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002
     
    \u5f53\u7136\uff0c\u7531\u4e8e\u7ed3\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64 \u94fe\u8868\u7ed3\u70b9\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u66f4\u5927\u3002
     
    \u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u7701\u5185\u5b58\uff0c\u9700\u8981 case-by-case \u5730\u5206\u6790\u3002
    "},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4. \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"
       
    • \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u5c06\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u300c\u540e\u9000\u300d\u64cd\u4f5c\u6765\u56de\u5230\u4e0a\u4e00\u9875\u9762\uff0c\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u5219\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
      •  
    • \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u5f53\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u5c31\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u6765\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\uff0c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u65f6\u51fa\u6808\u3002
      •  
    "},{"location":"chapter_stack_and_queue/summary/","title":"5.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u3002
      •  
    • \u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u66f4\u597d\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u6269\u5bb9\u65f6\u4f1a\u5bfc\u81f4\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u5bf9\u5730\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
      •  
    • \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u4f1a\u9020\u6210\u4e00\u5b9a\u7a7a\u95f4\u6d6a\u8d39\uff0c\u7136\u800c\u94fe\u8868\u7ed3\u70b9\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u5185\u5b58\u66f4\u5927\u3002
      •  
    • \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e24\u79cd\u5b9e\u73b0\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\uff0c\u4e0e\u4e0a\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u540c\u3002
      •  
    • \u53cc\u5411\u961f\u5217\u7684\u4e24\u7aef\u90fd\u53ef\u4ee5\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002
      •  
    "},{"location":"chapter_tree/avl_tree/","title":"7.4. \u00a0 AVL \u6811 *","text":"
    \u5728\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u7ae0\u8282\u4e2d\u63d0\u5230\uff0c\u5728\u8fdb\u884c\u591a\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002\u6b64\u65f6\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4f1a\u7531 \\(O(\\log n)\\) \u52a3\u5316\u81f3 \\(O(n)\\) \u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6267\u884c\u4e24\u6b65\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u8be5\u4e8c\u53c9\u641c\u7d22\u6811\u5c31\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002
     
     
     Fig. AVL \u6811\u5728\u5220\u9664\u7ed3\u70b9\u540e\u53d1\u751f\u9000\u5316 
     
    \u518d\u6bd4\u5982\uff0c\u5728\u4ee5\u4e0b\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u7ed3\u70b9\u540e\uff0c\u6811\u4e25\u91cd\u5411\u5de6\u504f\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u53d1\u751f\u52a3\u5316\u3002
     
     
     Fig. AVL \u6811\u5728\u63d2\u5165\u7ed3\u70b9\u540e\u53d1\u751f\u9000\u5316 
     
    G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u5176 1962 \u5e74\u53d1\u8868\u7684\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u4f7f\u5f97\u5728\u4e0d\u65ad\u6dfb\u52a0\u4e0e\u5220\u9664\u7ed3\u70b9\u540e\uff0cAVL \u6811\u4ecd\u7136\u4e0d\u4f1a\u53d1\u751f\u9000\u5316\uff0c\u8fdb\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u80fd\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002
     
    \u6362\u8a00\u4e4b\uff0c\u5728\u9891\u7e41\u589e\u5220\u67e5\u6539\u7684\u4f7f\u7528\u573a\u666f\u4e2d\uff0cAVL \u6811\u53ef\u59cb\u7ec8\u4fdd\u6301\u5f88\u9ad8\u7684\u6570\u636e\u589e\u5220\u67e5\u6539\u6548\u7387\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002
    "},{"location":"chapter_tree/avl_tree/#741-avl","title":"7.4.1. \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"
    \u300cAVL \u6811\u300d\u65e2\u662f\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u53c8\u662f\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u79cd\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u53c8\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u3002
    "},{"location":"chapter_tree/avl_tree/#_1","title":"\u7ed3\u70b9\u9ad8\u5ea6","text":"
    \u5728 AVL \u6811\u7684\u64cd\u4f5c\u4e2d\uff0c\u9700\u8981\u83b7\u53d6\u7ed3\u70b9\u300c\u9ad8\u5ea6 Height\u300d\uff0c\u6240\u4ee5\u7ed9 AVL \u6811\u7684\u7ed3\u70b9\u7c7b\u6dfb\u52a0 height \u53d8\u91cf\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\npublic int val;        // \u7ed3\u70b9\u503c\npublic int height;     // \u7ed3\u70b9\u9ad8\u5ea6\npublic TreeNode left;  // \u5de6\u5b50\u7ed3\u70b9\npublic TreeNode right; // \u53f3\u5b50\u7ed3\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nstruct TreeNode {\nint val{};          // \u7ed3\u70b9\u503c\nint height = 0;     // \u7ed3\u70b9\u9ad8\u5ea6\nTreeNode *left{};   // \u5de6\u5b50\u7ed3\u70b9\nTreeNode *right{};  // \u53f3\u5b50\u7ed3\u70b9\nTreeNode() = default;\nexplicit TreeNode(int x) : val(x){}\n};\n
     
    \"\"\" AVL \u6811\u7ed3\u70b9\u7c7b \"\"\"\nclass TreeNode:\ndef __init__(self, val: int):\nself.val: int = val                    # \u7ed3\u70b9\u503c\nself.height: int = 0                   # \u7ed3\u70b9\u9ad8\u5ea6\nself.left: Optional[TreeNode] = None   # \u5de6\u5b50\u7ed3\u70b9\u5f15\u7528\nself.right: Optional[TreeNode] = None  # \u53f3\u5b50\u7ed3\u70b9\u5f15\u7528\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\ntype TreeNode struct {\nVal    int       // \u7ed3\u70b9\u503c\nHeight int       // \u7ed3\u70b9\u9ad8\u5ea6\nLeft   *TreeNode // \u5de6\u5b50\u7ed3\u70b9\u5f15\u7528\nRight  *TreeNode // \u53f3\u5b50\u7ed3\u70b9\u5f15\u7528\n}\n
     
    class TreeNode {\nval; // \u7ed3\u70b9\u503c\nheight; //\u7ed3\u70b9\u9ad8\u5ea6\nleft; // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nconstructor(val, left, right, height) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n
     
    class TreeNode {\nval: number;            // \u7ed3\u70b9\u503c\nheight: number;         // \u7ed3\u70b9\u9ad8\u5ea6\nleft: TreeNode | null;  // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nright: TreeNode | null; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nconstructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height; this.left = left === undefined ? null : left; this.right = right === undefined ? null : right; }\n}\n
     
    \n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\npublic int val;          // \u7ed3\u70b9\u503c\npublic int height;       // \u7ed3\u70b9\u9ad8\u5ea6\npublic TreeNode? left;   // \u5de6\u5b50\u7ed3\u70b9\npublic TreeNode? right;  // \u53f3\u5b50\u7ed3\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar height: Int // \u7ed3\u70b9\u9ad8\u5ea6\nvar left: TreeNode? // \u5de6\u5b50\u7ed3\u70b9\nvar right: TreeNode? // \u53f3\u5b50\u7ed3\u70b9\ninit(x: Int) {\nval = x\nheight = 0\n}\n}\n
     
    \n
     
    \u300c\u7ed3\u70b9\u9ad8\u5ea6\u300d\u662f\u6700\u8fdc\u53f6\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u8d70\u8fc7\u7684\u300c\u8fb9\u300d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\uff0c\u53f6\u7ed3\u70b9\u7684\u9ad8\u5ea6\u4e3a 0 \uff0c\u7a7a\u7ed3\u70b9\u7684\u9ad8\u5ea6\u4e3a -1\u3002\u6211\u4eec\u5c01\u88c5\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u4e0e\u66f4\u65b0\u7ed3\u70b9\u7684\u9ad8\u5ea6\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
     avl_tree.cpp
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode* node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == nullptr ? -1 : node->height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode* node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode->height = max(height(node->left), height(node->right)) + 1;\n}\n
     avl_tree.py
    def height(self, node: Optional[TreeNode]) -> int:\n\"\"\" \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 \"\"\"\n# \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nif node is not None:\nreturn node.height\nreturn -1\ndef __update_height(self, node: Optional[TreeNode]):\n\"\"\" \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 \"\"\"\n# \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = max([self.height(node.left), self.height(node.right)]) + 1\n
     avl_tree.go
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nif node != nil {\nreturn node.Height\n}\nreturn -1\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\nlh := t.height(node.Left)\nrh := t.height(node.Right)\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif lh > rh {\nnode.Height = lh + 1\n} else {\nnode.Height = rh + 1\n}\n}\n
     avl_tree.js
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nheight(node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
     avl_tree.ts
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{height}\n[class]{aVLTree}-[func]{updateHeight}\n
     avl_tree.cs
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node)\n{\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node)\n{\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.Max(height(node.left), height(node.right)) + 1;\n}\n
     avl_tree.swift
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nnode == nil ? -1 : node!.height\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
     avl_tree.zig
    // \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n_ = self;\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn if (node == null) -1 else node.?.height;\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.?.height = std.math.max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_2","title":"\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50","text":"
    \u7ed3\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 Balance Factor\u300d\u662f \u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u9ad8\u5ea6\uff0c\u5e76\u5b9a\u4e49\u7a7a\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a 0 \u3002\u540c\u6837\u5730\uff0c\u6211\u4eec\u5c06\u83b7\u53d6\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u4ee5\u4fbf\u540e\u7eed\u4f7f\u7528\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
     avl_tree.cpp
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode* node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == nullptr) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
     avl_tree.py
    def balance_factor(self, node: Optional[TreeNode]) -> int:\n\"\"\" \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 \"\"\"\n# \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node is None:\nreturn 0\n# \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.left) - self.height(node.right)\n
     avl_tree.go
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node == nil {\nreturn 0\n}\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn t.height(node.Left) - t.height(node.Right)\n}\n
     avl_tree.js
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
     avl_tree.ts
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{balanceFactor}\n
     avl_tree.cs
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node)\n{\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
     avl_tree.swift
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nguard let node = node else { return 0 }\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node: node.left) - height(node: node.right)\n}\n
     avl_tree.zig
    // \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.?.left) - self.height(node.?.right);\n}\n
     
    Note
     
    \u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002
    "},{"location":"chapter_tree/avl_tree/#742-avl","title":"7.4.2. \u00a0 AVL \u6811\u65cb\u8f6c","text":"
    AVL \u6811\u7684\u72ec\u7279\u4e4b\u5904\u5728\u4e8e\u300c\u65cb\u8f6c Rotation\u300d\u7684\u64cd\u4f5c\uff0c\u5176\u53ef \u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u7ed3\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u8a00\u4e4b\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u53ef\u4ee5\u4f7f\u6811\u4fdd\u6301\u4e3a\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u3002
     
    \u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7684\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u7ed3\u70b9\u79f0\u4e3a\u300c\u5931\u8861\u7ed3\u70b9\u300d\u3002\u6839\u636e\u7ed3\u70b9\u7684\u5931\u8861\u60c5\u51b5\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a \u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u4e00\u8d77\u6765\u770b\u770b\u5b83\u4eec\u662f\u5982\u4f55\u64cd\u4f5c\u7684\u3002
    "},{"location":"chapter_tree/avl_tree/#case-1-","title":"Case 1 - \u53f3\u65cb","text":"
    \u5982\u4e0b\u56fe\u6240\u793a\uff08\u7ed3\u70b9\u4e0b\u65b9\u4e3a\u300c\u5e73\u8861\u56e0\u5b50\u300d\uff09\uff0c\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u7ed3\u70b9\u662f \u7ed3\u70b9 3\u3002\u6211\u4eec\u805a\u7126\u5728\u4ee5\u8be5\u5931\u8861\u7ed3\u70b9\u4e3a\u6839\u7ed3\u70b9\u7684\u5b50\u6811\u4e0a\uff0c\u5c06\u8be5\u7ed3\u70b9\u8bb0\u4e3a node \uff0c\u5c06\u5176\u5de6\u5b50\u7ed3\u70b9\u8bb0\u4e3a child \uff0c\u6267\u884c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u8be5\u5b50\u6811\u5df2\u7ecf\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4e3a\u4e8c\u53c9\u641c\u7d22\u6811\u3002
     <1><2><3><4> 
     
     
     
     
    \u8fdb\u800c\uff0c\u5982\u679c\u7ed3\u70b9 child \u672c\u8eab\u6709\u53f3\u5b50\u7ed3\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u53f3\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u5de6\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c 
     
    \u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u9700\u8981\u901a\u8fc7\u4fee\u6539\u7ed3\u70b9\u6307\u9488\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\nTreeNode child = node.left;\nTreeNode grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.cpp
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode* rightRotate(TreeNode* node) {\nTreeNode* child = node->left;\nTreeNode* grandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.py
    def __right_rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u53f3\u65cb\u64cd\u4f5c \"\"\"\nchild = node.left\ngrand_child = child.right\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node\nnode.left = grand_child\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n
     avl_tree.go
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\nchild := node.Left\ngrandChild := child.Right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.Right = node\nnode.Left = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.js
    /* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.ts
    /* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{rightRotate}\n
     avl_tree.cs
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node)\n{\nTreeNode? child = node.left;\nTreeNode? grandChild = child?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.swift
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.left\nlet grandChild = child?.right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild?.right = node\nnode?.left = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.zig
    // \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.left;\nvar grandChild = child.?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.?.right = node;\nnode.?.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#case-2-","title":"Case 2 - \u5de6\u65cb","text":"
    \u7c7b\u4f3c\u5730\uff0c\u5982\u679c\u5c06\u53d6\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u90a3\u4e48\u5219\u9700\u8981\u300c\u5de6\u65cb\u300d\u64cd\u4f5c\u3002
     
     
     Fig. \u5de6\u65cb\u64cd\u4f5c 
     
    \u540c\u7406\uff0c\u82e5\u7ed3\u70b9 child \u672c\u8eab\u6709\u5de6\u5b50\u7ed3\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u5de6\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u53f3\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c 
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u300c\u5de6\u65cb\u300d\u548c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u4e24\u8005\u5bf9\u5e94\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u6839\u636e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u5f88\u65b9\u4fbf\u5730\u4ece\u300c\u53f3\u65cb\u300d\u63a8\u5bfc\u51fa\u300c\u5de6\u65cb\u300d\u3002\u5177\u4f53\u5730\uff0c\u53ea\u9700\u5c06\u300c\u53f3\u65cb\u300d\u4ee3\u7801\u4e2d\u7684\u628a\u6240\u6709\u7684 left \u66ff\u6362\u4e3a right \u3001\u6240\u6709\u7684 right \u66ff\u6362\u4e3a left \uff0c\u5373\u53ef\u5f97\u5230\u300c\u5de6\u65cb\u300d\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\nTreeNode child = node.right;\nTreeNode grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.cpp
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode* leftRotate(TreeNode* node) {\nTreeNode* child = node->right;\nTreeNode* grandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.py
    def __left_rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u5de6\u65cb\u64cd\u4f5c \"\"\"\nchild = node.right\ngrand_child = child.left\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node\nnode.right = grand_child\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n
     avl_tree.go
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\nchild := node.Right\ngrandChild := child.Left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.Left = node\nnode.Right = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.js
    /* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.ts
    /* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{leftRotate}\n
     avl_tree.cs
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node)\n{\nTreeNode? child = node.right;\nTreeNode? grandChild = child?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.swift
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.right\nlet grandChild = child?.left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild?.left = node\nnode?.right = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.zig
    // \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.right;\nvar grandChild = child.?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.?.left = node;\nnode.?.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#case-3-","title":"Case 3 - \u5148\u5de6\u540e\u53f3","text":"
    \u5bf9\u4e8e\u4e0b\u56fe\u7684\u5931\u8861\u7ed3\u70b9 3 \uff0c\u5355\u4e00\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u6b64\u65f6\u9700\u8981\u300c\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u300d\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u5de6\u65cb\u300d\uff0c\u518d\u5bf9 node \u6267\u884c\u300c\u53f3\u65cb\u300d\u3002
     
     
     Fig. \u5148\u5de6\u65cb\u540e\u53f3\u65cb 
    "},{"location":"chapter_tree/avl_tree/#case-4-","title":"Case 4 - \u5148\u53f3\u540e\u5de6","text":"
    \u540c\u7406\uff0c\u53d6\u4ee5\u4e0a\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\uff0c\u5219\u9700\u8981\u300c\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u300d\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u53f3\u65cb\u300d\uff0c\u7136\u540e\u5bf9 node \u6267\u884c\u300c\u5de6\u65cb\u300d\u3002
     
     
     Fig. \u5148\u53f3\u65cb\u540e\u5de6\u65cb 
    "},{"location":"chapter_tree/avl_tree/#_3","title":"\u65cb\u8f6c\u7684\u9009\u62e9","text":"
    \u4e0b\u56fe\u63cf\u8ff0\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0 Cases \u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u91c7\u7528 \u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u540e\u5de6\u3001\u5148\u5de6\u540e\u53f3 \u7684\u65cb\u8f6c\u64cd\u4f5c\u3002
     
     
     Fig. AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 
     
    \u5177\u4f53\u5730\uff0c\u5728\u4ee3\u7801\u4e2d\u4f7f\u7528 \u5931\u8861\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u3001\u8f83\u9ad8\u4e00\u4fa7\u5b50\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u6765\u786e\u5b9a\u5931\u8861\u7ed3\u70b9\u5c5e\u4e8e\u4e0a\u56fe\u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002
     \u5931\u8861\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb 
    \u4e3a\u65b9\u4fbf\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6b64\u51fd\u6570\u6765\u65cb\u8f6c\u5404\u79cd\u5931\u8861\u60c5\u51b5\uff0c\u4f7f\u5931\u8861\u7ed3\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.cpp
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode* rotate(TreeNode* node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint _balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (_balanceFactor > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (_balanceFactor < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.py
    def __rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 \"\"\"\n# \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nbalance_factor = self.balance_factor(node)\n# \u5de6\u504f\u6811\nif balance_factor > 1:\nif self.balance_factor(node.left) >= 0:\n# \u53f3\u65cb\nreturn self.__right_rotate(node)\nelse:\n# \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = self.__left_rotate(node.left)\nreturn self.__right_rotate(node)\n# \u53f3\u504f\u6811\nelif balance_factor < -1:\nif self.balance_factor(node.right) <= 0:\n# \u5de6\u65cb\nreturn self.__left_rotate(node)\nelse:\n# \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = self.__right_rotate(node.right)\nreturn self.__left_rotate(node)\n# \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n
     avl_tree.go
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n// Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\nbf := t.balanceFactor(node)\n// \u5de6\u504f\u6811\nif bf > 1 {\nif t.balanceFactor(node.Left) >= 0 {\n// \u53f3\u65cb\nreturn t.rightRotate(node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.Left = t.leftRotate(node.Left)\nreturn t.rightRotate(node)\n}\n}\n// \u53f3\u504f\u6811\nif bf < -1 {\nif t.balanceFactor(node.Right) <= 0 {\n// \u5de6\u65cb\nreturn t.leftRotate(node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.Right = t.rightRotate(node.Right)\nreturn t.leftRotate(node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
     avl_tree.js
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.#rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.#leftRotate(node.left);\nreturn this.#rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.#leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.#rightRotate(node.right);\nreturn this.#leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.ts
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.leftRotate(node.left);\nreturn this.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.rightRotate(node.right);\nreturn this.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{rotate}\n
     avl_tree.cs
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node)\n{\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactorInt = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactorInt > 1)\n{\nif (balanceFactor(node.left) >= 0)\n{\n// \u53f3\u65cb\nreturn rightRotate(node);\n}\nelse\n{\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node?.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactorInt < -1)\n{\nif (balanceFactor(node.right) <= 0)\n{\n// \u5de6\u65cb\nreturn leftRotate(node);\n}\nelse\n{\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node?.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.swift
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balanceFactor = balanceFactor(node: node)\n// \u5de6\u504f\u6811\nif balanceFactor > 1 {\nif self.balanceFactor(node: node?.left) >= 0 {\n// \u53f3\u65cb\nreturn rightRotate(node: node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode?.left = leftRotate(node: node?.left)\nreturn rightRotate(node: node)\n}\n}\n// \u53f3\u504f\u6811\nif balanceFactor < -1 {\nif self.balanceFactor(node: node?.right) <= 0 {\n// \u5de6\u65cb\nreturn leftRotate(node: node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode?.right = rightRotate(node: node?.right)\nreturn leftRotate(node: node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
     avl_tree.zig
    // \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nvar balance_factor = self.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balance_factor > 1) {\nif (self.balanceFactor(node.?.left) >= 0) {\n// \u53f3\u65cb\nreturn self.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.?.left = self.leftRotate(node.?.left);\nreturn self.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balance_factor < -1) {\nif (self.balanceFactor(node.?.right) <= 0) {\n// \u5de6\u65cb\nreturn self.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.?.right = self.rightRotate(node.?.right);\nreturn self.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#743-avl","title":"7.4.3. \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#_4","title":"\u63d2\u5165\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u7684\u7ed3\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u4e3b\u4f53\u7c7b\u4f3c\u3002\u4e0d\u540c\u7684\u662f\uff0c\u5728\u63d2\u5165\u7ed3\u70b9\u540e\uff0c\u4ece\u8be5\u7ed3\u70b9\u5230\u6839\u7ed3\u70b9\u7684\u8def\u5f84\u4e0a\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u300c\u5931\u8861\u7ed3\u70b9\u300d\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8be5\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7ed3\u70b9\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode insert(int val) {\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.cpp
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode* insert(int val) {\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode* insertHelper(TreeNode* node, int val) {\nif (node == nullptr)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node->val)\nnode->left = insertHelper(node->left, val);\nelse if (val > node->val)\nnode->right = insertHelper(node->right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.py
    def insert(self, val) -> TreeNode:\n\"\"\" \u63d2\u5165\u7ed3\u70b9 \"\"\"\nself.__root = self.__insert_helper(self.__root, val)\nreturn self.__root\ndef __insert_helper(self, node: Optional[TreeNode], val: int) -> TreeNode:\n\"\"\" \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn TreeNode(val)\n# 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9\nif val < node.val:\nnode.left = self.__insert_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__insert_helper(node.right, val)\nelse:\n# \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
     avl_tree.go
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc (t *aVLTree) insert(val int) *TreeNode {\nt.root = t.insertHelper(t.root, val)\nreturn t.root\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn NewTreeNode(val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif val < node.Val {\nnode.Left = t.insertHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.insertHelper(node.Right, val)\n} else {\n// \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n
     avl_tree.js
    /* \u63d2\u5165\u7ed3\u70b9 */\ninsert(val) {\nthis.root = this.#insertHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val) node.left = this.#insertHelper(node.left, val);\nelse if (val > node.val) node.right = this.#insertHelper(node.right, val);\nelse return node; // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nthis.#updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.ts
    /* \u63d2\u5165\u7ed3\u70b9 */\ninsert(val: number): TreeNode {\nthis.root = this.insertHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val) {\nnode.left = this.insertHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.insertHelper(node.right, val);\n} else {\nreturn node; // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nthis.updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{insert}\n[class]{aVLTree}-[func]{insertHelper}\n
     avl_tree.cs
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode? insert(int val)\n{\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val)\n{\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.swift
    /* \u63d2\u5165\u7ed3\u70b9 */\n@discardableResult\nfunc insert(val: Int) -> TreeNode? {\nroot = insertHelper(node: root, val: val)\nreturn root\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn TreeNode(x: val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif val < node!.val {\nnode?.left = insertHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = insertHelper(node: node?.right, val: val)\n} else {\nreturn node // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nupdateHeight(node: node) // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n
     avl_tree.zig
    // \u63d2\u5165\u7ed3\u70b9\nfn insert(self: *Self, val: T) !?*inc.TreeNode(T) {\nself.root = try self.insertHelper(self.root, val);\nreturn self.root;\n}\n// \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) {\nvar tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\ntmp_node.init(val);\nreturn tmp_node;\n}\n// 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9\nif (val < node.?.val) {\nnode.?.left = try self.insertHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = try self.insertHelper(node.?.right, val);\n} else {\nreturn node;            // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nself.updateHeight(node);    // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_5","title":"\u5220\u9664\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u603b\u4f53\u76f8\u540c\u3002\u7c7b\u4f3c\u5730\uff0c\u5728\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u4e5f\u9700\u8981\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7ed3\u70b9\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode remove(int val) {\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode child = node.left != null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode temp = getInOrderNext(node.right);\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode getInOrderNext(TreeNode node) {\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left != null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.cpp
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode* remove(int val) {\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode* removeHelper(TreeNode* node, int val) {\nif (node == nullptr)\nreturn nullptr;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val)\nnode->left = removeHelper(node->left, val);\nelse if (val > node->val)\nnode->right = removeHelper(node->right, val);\nelse {\nif (node->left == nullptr || node->right == nullptr) {\nTreeNode* child = node->left != nullptr ? node->left : node->right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == nullptr) {\ndelete node;\nreturn nullptr;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\ndelete node;\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode* temp = getInOrderNext(node->right);\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode* getInOrderNext(TreeNode* node) {\nif (node == nullptr)\nreturn node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node->left != nullptr) {\nnode = node->left;\n}\nreturn node;\n}\n
     avl_tree.py
    def remove(self, val: int) -> Optional[TreeNode]:\n\"\"\" \u5220\u9664\u7ed3\u70b9 \"\"\"\nself.__root = self.__remove_helper(self.__root, val)\nreturn self.__root \ndef __remove_helper(self, node: Optional[TreeNode], val: int) -> Optional[TreeNode]:\n\"\"\" \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 \"\"\"\nif node is None:\nreturn None\n# 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif val < node.val:\nnode.left = self.__remove_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__remove_helper(node.right, val)\nelse:\nif node.left is None or node.right is None:\nchild = node.left or node.right\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child is None:\nreturn None\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse:\nnode = child\nelse:  # \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\ntemp = self.__get_inorder_next(node.right)\nnode.right = self.__remove_helper(node.right, temp.val)\nnode.val = temp.val\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\ndef __get_inorder_next(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 \"\"\"\nif node is None:\nreturn None\n# \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile node.left is not None:\nnode = node.left\nreturn node\n
     avl_tree.go
    /* \u5220\u9664\u7ed3\u70b9 */\nfunc (t *aVLTree) remove(val int) *TreeNode {\nroot := t.removeHelper(t.root, val)\nreturn root\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node.Val {\nnode.Left = t.removeHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.removeHelper(node.Right, val)\n} else {\nif node.Left == nil || node.Right == nil {\nchild := node.Left\nif node.Right != nil {\nchild = node.Right\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\ntemp := t.getInOrderNext(node.Right)\nnode.Right = t.removeHelper(node.Right, temp.Val)\nnode.Val = temp.Val\n}\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc (t *aVLTree) getInOrderNext(node *TreeNode) *TreeNode {\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nfor node.Left != nil {\nnode = node.Left\n}\nreturn node\n}\n
     avl_tree.js
    /* \u5220\u9664\u7ed3\u70b9 */\nremove(val) {\nthis.root = this.#removeHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) node.left = this.#removeHelper(node.left, val);\nelse if (val > node.val) node.right = this.#removeHelper(node.right, val);\nelse {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse node = child;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nconst temp = this.#getInOrderNext(node.right);\nnode.right = this.#removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.#updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\n#getInOrderNext(node) {\nif (node === null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left !== null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.ts
    /* \u5220\u9664\u7ed3\u70b9 */\nremove(val: number): TreeNode {\nthis.root = this.removeHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) {\nnode.left = this.removeHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.removeHelper(node.right, val);\n} else {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) {\nreturn null;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nconst temp = this.getInOrderNext(node.right);\nnode.right = this.removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\ngetInOrderNext(node: TreeNode): TreeNode {\nif (node === null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left !== null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{remove}\n[class]{aVLTree}-[func]{removeHelper}\n[class]{aVLTree}-[func]{getInOrderNext}\n
     avl_tree.cs
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode? remove(int val)\n{\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val)\n{\nif (node == null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse\n{\nif (node.left == null || node.right == null)\n{\nTreeNode? child = node.left != null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n}\nelse\n{\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode? temp = getInOrderNext(node.right);\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode? getInOrderNext(TreeNode? node)\n{\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left != null)\n{\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.swift
    /* \u5220\u9664\u7ed3\u70b9 */\n@discardableResult\nfunc remove(val: Int) -> TreeNode? {\nroot = removeHelper(node: root, val: val)\nreturn root\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node!.val {\nnode?.left = removeHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = removeHelper(node: node?.right, val: val)\n} else {\nif node?.left == nil || node?.right == nil {\nlet child = node?.left != nil ? node?.left : node?.right\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\nnode = child\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nlet temp = getInOrderNext(node: node?.right)\nnode?.right = removeHelper(node: node?.right, val: temp!.val)\nnode?.val = temp!.val\n}\n}\nupdateHeight(node: node) // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc getInOrderNext(node: TreeNode?) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile node?.left != nil {\nnode = node?.left\n}\nreturn node\n}\n
     avl_tree.zig
    // \u5220\u9664\u7ed3\u70b9\nfn remove(self: *Self, val: T) ?*inc.TreeNode(T) {\nself.root = self.removeHelper(self.root, val);\nreturn self.root;\n}\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) return null;\n// 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif (val < node.?.val) {\nnode.?.left = self.removeHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = self.removeHelper(node.?.right, val);\n} else {\nif (node.?.left == null or node.?.right == null) {\nvar child = if (node.?.left != null) node.?.left else node.?.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null) {\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n} else {\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nvar temp = self.getInOrderNext(node.?.right);\nnode.?.right = self.removeHelper(node.?.right, temp.?.val);\nnode.?.val = temp.?.val;\n}\n}\nself.updateHeight(node);    // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09\nfn getInOrderNext(self: *Self, node_: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n_ = self;\nvar node = node_;\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.?.left != null) {\nnode = node.?.left;\n}\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_6","title":"\u67e5\u627e\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u7684\u7ed3\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_tree/avl_tree/#744-avl","title":"7.4.4. \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"
       
    • \u7ec4\u7ec7\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u573a\u666f\uff1b
      •  
    • \u7528\u4e8e\u5efa\u7acb\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\uff1b
      •  
     
    \u4e3a\u4ec0\u4e48\u7ea2\u9ed1\u6811\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\uff1f
     
    \u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u56e0\u6b64\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u66f4\u5c11\uff0c\u7ed3\u70b9\u589e\u5220\u64cd\u4f5c\u76f8\u6bd4 AVL \u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002
    "},{"location":"chapter_tree/binary_search_tree/","title":"7.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"
    \u300c\u4e8c\u53c9\u641c\u7d22\u6811 Binary Search Tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a
     
       
    1. \u5bf9\u4e8e\u6839\u7ed3\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u7684\u503c \\(<\\) \u6839\u7ed3\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u7684\u503c\uff1b
      2.  
    2. \u4efb\u610f\u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u4e5f\u6ee1\u8db3\u6761\u4ef6 1. \uff1b
      3.  
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811 
    "},{"location":"chapter_tree/binary_search_tree/#731","title":"7.3.1. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_search_tree/#_1","title":"\u67e5\u627e\u7ed3\u70b9","text":"
    \u7ed9\u5b9a\u76ee\u6807\u7ed3\u70b9\u503c num \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u7ed3\u70b9 cur \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u7ed3\u70b9 root \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u7ed3\u70b9\u503c cur.val \u548c num \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb
     
       
    • \u82e5 cur.val < num \uff0c\u8bf4\u660e\u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.right \uff1b
      •  
    • \u82e5 cur.val > num \uff0c\u8bf4\u660e\u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.left \uff1b
      •  
    • \u82e5 cur.val = num \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u7ed3\u70b9\u5373\u53ef\uff1b
      •  
     <1><2><3><4> 
     
     
     
     
    \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u548c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u5982\u51fa\u4e00\u8f99\uff0c\u4e5f\u662f\u5728\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode search(int num) {\nTreeNode cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.cpp
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode* search(int num) {\nTreeNode* cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur->val > num) cur = cur->left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.py
    def search(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u67e5\u627e\u7ed3\u70b9 \"\"\"\ncur: Optional[TreeNode] = self.__root\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur is not None:\n# \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelif cur.val > num:\ncur = cur.left\n# \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse:\nbreak\nreturn cur\n
     binary_search_tree.go
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\nnode := bst.root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor node != nil {\nif node.Val < num {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nnode = node.Right\n} else if node.Val > num {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nnode = node.Left\n} else {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn node\n}\n
     binary_search_tree.js
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunction search(num) {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.ts
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunction search(num: number): TreeNode | null {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val < num) {\ncur = cur.right; // \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else if (cur.val > num) {\ncur = cur.left; // \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\nbreak; // \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{search}\n
     binary_search_tree.cs
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode? search(int num)\n{\nTreeNode? cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.swift
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunc search(num: Int) -> TreeNode? {\nvar cur = root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if cur!.val > num {\ncur = cur?.left\n}\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur\n}\n
     binary_search_tree.zig
    // \u67e5\u627e\u7ed3\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\nvar cur = self.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else if (cur.?.val > num) {\ncur = cur.?.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n} else {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
    "},{"location":"chapter_tree/binary_search_tree/#_2","title":"\u63d2\u5165\u7ed3\u70b9","text":"
    \u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 num \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u7ed3\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u5206\u4e3a\u4e24\u6b65\uff1a
     
       
    1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u7c7b\u4f3c\uff0c\u6211\u4eec\u4ece\u6839\u7ed3\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u7ed3\u70b9\u503c\u548c num \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u7ed3\u70b9\uff08\u904d\u5386\u5230 \\(\\text{null}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\uff1b
      2.  
    2. \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u7ed3\u70b9\uff1a\u521d\u59cb\u5316\u7ed3\u70b9 num \uff0c\u5c06\u8be5\u7ed3\u70b9\u653e\u5230 \\(\\text{null}\\) \u7684\u4f4d\u7f6e \uff1b
      3.  
     
    \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u7ed3\u70b9\uff0c\u5426\u5219\u5c06\u4f1a\u8fdd\u80cc\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\u82e5\u5f85\u63d2\u5165\u7ed3\u70b9\u5728\u6811\u4e2d\u5df2\u7ecf\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\nreturn node;\n}\n
     binary_search_tree.cpp
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode* insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr) return nullptr;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num) return nullptr;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur->left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode* node = new TreeNode(num);\nif (pre->val < num) pre->right = node;\nelse pre->left = node;\nreturn node;\n}\n
     binary_search_tree.py
    def insert(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u63d2\u5165\u7ed3\u70b9 \"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.__root is None:\nreturn None\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\ncur, pre = self.__root, None\nwhile cur is not None:\n# \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur.val == num:\nreturn None\npre = cur\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u63d2\u5165\u7ed3\u70b9 val\nnode = TreeNode(num)\nif pre.val < num:\npre.right = node\nelse:\npre.left = node\nreturn node\n
     binary_search_tree.go
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) *TreeNode {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5f85\u63d2\u5165\u7ed3\u70b9\u4e4b\u524d\u7684\u7ed3\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nreturn nil\n}\npre = cur\nif cur.Val < num {\ncur = cur.Right\n} else {\ncur = cur.Left\n}\n}\n// \u63d2\u5165\u7ed3\u70b9\nnode := NewTreeNode(num)\nif pre.Val < num {\npre.Right = node\n} else {\npre.Left = node\n}\nreturn cur\n}\n
     binary_search_tree.js
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunction insert(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return null;\nlet cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\nreturn node;\n}\n
     binary_search_tree.ts
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunction insert(num: number): TreeNode | null {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn null;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val === num) {\nreturn null; // \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = new TreeNode(num);\nif (pre!.val < num) {\npre!.right = node;\n} else {\npre!.left = node;\n}\nreturn node;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{insert}\n
     binary_search_tree.cs
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode? insert(int num)\n{\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode node = new TreeNode(num);\nif (pre != null)\n{\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\n}\nreturn node;\n}\n
     binary_search_tree.swift
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc insert(num: Int) -> TreeNode? {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn nil\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur!.val == num {\nreturn nil\n}\npre = cur\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = TreeNode(x: num)\nif pre!.val < num {\npre?.right = node\n} else {\npre?.left = node\n}\nreturn node\n}\n
     binary_search_tree.zig
    // \u63d2\u5165\u7ed3\u70b9\nfn insert(self: *Self, num: T) !?*inc.TreeNode(T) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return null;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.?.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nvar node = try self.mem_allocator.create(inc.TreeNode(T));\nnode.init(num);\nif (pre.?.val < num) {\npre.?.right = node;\n} else {\npre.?.left = node;\n}\nreturn node;\n}\n
     
    \u4e3a\u4e86\u63d2\u5165\u7ed3\u70b9\uff0c\u9700\u8981\u501f\u52a9 \u8f85\u52a9\u7ed3\u70b9 pre \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u7ed3\u70b9\uff0c\u8fd9\u6837\u5728\u904d\u5386\u5230 \\(\\text{null}\\) \u65f6\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u7ed3\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u7ed3\u70b9\u63d2\u5165\u64cd\u4f5c\u3002
     
    \u4e0e\u67e5\u627e\u7ed3\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u7ed3\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
    "},{"location":"chapter_tree/binary_search_tree/#_3","title":"\u5220\u9664\u7ed3\u70b9","text":"
    \u4e0e\u63d2\u5165\u7ed3\u70b9\u4e00\u6837\uff0c\u6211\u4eec\u9700\u8981\u5728\u5220\u9664\u64cd\u4f5c\u540e\u7ef4\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u7ed3\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u3002\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5728\u4e8c\u53c9\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u83b7\u53d6\u5f85\u5220\u9664\u7ed3\u70b9\u3002\u63a5\u4e0b\u6765\uff0c\u6839\u636e\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf\uff0c\u5220\u9664\u64cd\u4f5c\u9700\u8981\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 0\\) \u65f6\uff0c\u8868\u660e\u5f85\u5220\u9664\u7ed3\u70b9\u662f\u53f6\u7ed3\u70b9\uff0c\u76f4\u63a5\u5220\u9664\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u7ed3\u70b9\uff08\u5ea6\u4e3a 0\uff09 
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u7ed3\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u7ed3\u70b9\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u7ed3\u70b9\uff08\u5ea6\u4e3a 1\uff09 
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 2\\) \u65f6\uff0c\u5220\u9664\u64cd\u4f5c\u5206\u4e3a\u4e09\u6b65\uff1a
     
       
    1. \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\u5728 \u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 \u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff0c\u8bb0\u4e3a nex \uff1b
      2.  
    2. \u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex \uff1b
      3.  
    3. \u4f7f\u7528 nex \u66ff\u6362\u5f85\u5220\u9664\u7ed3\u70b9\uff1b
      4.  
     <1><2><3><4> 
     
     
     
     
    \u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u4e5f\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u7ed3\u70b9 \\(O(\\log n)\\) \uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u7ed3\u70b9 \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nTreeNode child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left == cur) pre.left = child;\nelse pre.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode nex = getInOrderNext(cur.right);\nint tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode getInOrderNext(TreeNode root) {\nif (root == null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left != null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.cpp
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode* remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr) return nullptr;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur->left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == nullptr) return nullptr;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur->left == nullptr || cur->right == nullptr) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u7ed3\u70b9\nTreeNode* child = cur->left != nullptr ? cur->left : cur->right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre->left == cur) pre->left = child;\nelse pre->right = child;\n// \u91ca\u653e\u5185\u5b58\ndelete cur;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode* nex = getInOrderNext(cur->right);\nint tmp = nex->val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex->val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur->val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode* getInOrderNext(TreeNode* root) {\nif (root == nullptr) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root->left != nullptr) {\nroot = root->left;\n}\nreturn root;\n}\n
     binary_search_tree.py
    def remove(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u5220\u9664\u7ed3\u70b9 \"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.__root is None:\nreturn None\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\ncur, pre = self.__root, None\nwhile cur is not None:\n# \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur.val == num:\nbreak\npre = cur\nif cur.val < num:  # \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur.right\nelse:  # \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur.left\n# \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur is None:\nreturn None\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif cur.left is None or cur.right is None:\n# \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nchild = cur.left or cur.right\n# \u5220\u9664\u7ed3\u70b9 cur\nif pre.left == cur:\npre.left = child\nelse:\npre.right = child\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse:\n# \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nnex: TreeNode = self.get_inorder_next(cur.right)\ntmp: int = nex.val\n# \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nself.remove(nex.val)\n# \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp\nreturn cur\ndef get_inorder_next(self, root: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 \"\"\"\nif root is None:\nreturn root\n# \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile root.left is not None:\nroot = root.left\nreturn root\n
     binary_search_tree.go
    /* \u5220\u9664\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) *TreeNode {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5f85\u5220\u9664\u7ed3\u70b9\u4e4b\u524d\u7684\u7ed3\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nbreak\n}\npre = cur\nif cur.Val < num {\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\ncur = cur.Right\n} else {\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\ncur = cur.Left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u4e3a 0 \u6216 1\nif cur.Left == nil || cur.Right == nil {\nvar child *TreeNode = nil\n// \u53d6\u51fa\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\nif cur.Left != nil {\nchild = cur.Left\n} else {\nchild = cur.Right\n}\n// \u5c06\u5b50\u7ed3\u70b9\u66ff\u6362\u4e3a\u5f85\u5220\u9664\u7ed3\u70b9\nif pre.Left == cur {\npre.Left = child\n} else {\npre.Right = child\n}\n// \u5b50\u7ed3\u70b9\u6570\u4e3a 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u7ed3\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nnext := bst.getInOrderNext(cur)\ntemp := next.Val\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 next\nbst.remove(next.Val)\n// \u5c06 next \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.Val = temp\n}\nreturn cur\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc (bst *binarySearchTree) getInOrderNext(node *TreeNode) *TreeNode {\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nfor node.Left != nil {\nnode = node.Left\n}\nreturn node\n}\n
     binary_search_tree.js
    /* \u5220\u9664\u7ed3\u70b9 */\nfunction remove(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return null;\nlet cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left === cur) pre.left = child;\nelse pre.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet nex = getInOrderNext(cur.right);\nlet tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunction getInOrderNext(root) {\nif (root === null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left !== null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.ts
    /* \u5220\u9664\u7ed3\u70b9 */\nfunction remove(num: number): TreeNode | null {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn null;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) {\nbreak;\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) {\nreturn null;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre!.left === cur) {\npre!.left = child;\n} else {\npre!.right = child;\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet next = getInOrderNext(cur.right);\nlet tmp = next!.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(next!.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunction getInOrderNext(root: TreeNode | null): TreeNode | null {\nif (root === null) {\nreturn null;\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left !== null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{remove}\n[class]{binarySearchTree}-[func]{getInOrderNext}\n
     binary_search_tree.cs
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode? remove(int num)\n{\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null || pre == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null)\n{\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nTreeNode? child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left == cur)\n{\npre.left = child;\n}\nelse\n{\npre.right = child;\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse\n{\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode? nex = getInOrderNext(cur.right);\nif (nex != null)\n{\nint tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode? getInOrderNext(TreeNode? root)\n{\nif (root == null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left != null)\n{\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.swift
    /* \u5220\u9664\u7ed3\u70b9 */\n@discardableResult\nfunc remove(num: Int) -> TreeNode? {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn nil\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur!.val == num {\nbreak\n}\npre = cur\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif cur?.left == nil || cur?.right == nil {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur?.left != nil ? cur?.left : cur?.right\n// \u5220\u9664\u7ed3\u70b9 cur\nif pre?.left === cur {\npre?.left = child\n} else {\npre?.right = child\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet nex = getInOrderNext(root: cur?.right)\nlet tmp = nex!.val\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(num: nex!.val)\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur?.val = tmp\n}\nreturn cur\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc getInOrderNext(root: TreeNode?) -> TreeNode? {\nvar root = root\nif root == nil {\nreturn root\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile root?.left != nil {\nroot = root?.left\n}\nreturn root\n}\n
     binary_search_tree.zig
    // \u5220\u9664\u7ed3\u70b9\nfn remove(self: *Self, num: T) ?*inc.TreeNode(T) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return null;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.?.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.?.left == null or cur.?.right == null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nvar child = if (cur.?.left != null) cur.?.left else cur.?.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.?.left == cur) {\npre.?.left = child;\n} else {\npre.?.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nvar nex = self.getInOrderNext(cur.?.right);\nvar tmp = nex.?.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\n_ = self.remove(nex.?.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.?.val = tmp;\n}\nreturn cur;\n}\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09\nfn getInOrderNext(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n_ = self;\nvar node_tmp = node;\nif (node_tmp == null) return null;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node_tmp.?.left != null) {\nnode_tmp = node_tmp.?.left;\n}\nreturn node_tmp;\n}\n
    "},{"location":"chapter_tree/binary_search_tree/#_4","title":"\u6392\u5e8f","text":"
    \u6211\u4eec\u77e5\u9053\uff0c\u300c\u4e2d\u5e8f\u904d\u5386\u300d\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u9075\u5faa\u201c\u5de6\u5b50\u7ed3\u70b9 \\(<\\) \u6839\u7ed3\u70b9 \\(<\\) \u53f3\u5b50\u7ed3\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u7ed3\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u6761\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002
     
    \u501f\u52a9\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u800c\u65e0\u9700\u989d\u5916\u6392\u5e8f\uff0c\u975e\u5e38\u9ad8\u6548\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 
    "},{"location":"chapter_tree/binary_search_tree/#732","title":"7.3.2. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"
    \u5047\u8bbe\u7ed9\u5b9a \\(n\\) \u4e2a\u6570\u5b57\uff0c\u6700\u5e38\u7528\u7684\u5b58\u50a8\u65b9\u5f0f\u662f\u300c\u6570\u7ec4\u300d\uff0c\u90a3\u4e48\u5bf9\u4e8e\u8fd9\u4e32\u4e71\u5e8f\u7684\u6570\u5b57\uff0c\u5e38\u89c1\u64cd\u4f5c\u7684\u6548\u7387\u4e3a\uff1a
     
       
    • \u67e5\u627e\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u65e0\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u6570\u7ec4\u6765\u786e\u5b9a\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u63d2\u5165\u5143\u7d20\uff1a\u53ea\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u5148\u67e5\u627e\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u518d\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u8be5\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20\uff1a\u9700\u8981\u904d\u5386\u6570\u7ec4\u6765\u786e\u5b9a\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
     
    \u4e3a\u4e86\u5f97\u5230\u5148\u9a8c\u4fe1\u606f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u9884\u5148\u5c06\u6570\u7ec4\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230\u4e00\u4e2a\u300c\u6392\u5e8f\u6570\u7ec4\u300d\uff0c\u6b64\u65f6\u64cd\u4f5c\u6548\u7387\u4e3a\uff1a
     
       
    • \u67e5\u627e\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u5e73\u5747\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff1b
      •  
    • \u63d2\u5165\u5143\u7d20\uff1a\u5148\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u518d\u63d2\u5165\u5230\u6307\u5b9a\u4f4d\u7f6e\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u5148\u67e5\u627e\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u518d\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u8be5\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20\uff1a\u6570\u7ec4\u5934\u90e8\u548c\u5c3e\u90e8\u5143\u7d20\u5373\u662f\u6700\u5c0f\u548c\u6700\u5927\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
      •  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u65e0\u5e8f\u6570\u7ec4\u548c\u6709\u5e8f\u6570\u7ec4\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u504f\u79d1\u201d\u7684\uff0c\u5373\u6709\u7684\u5feb\u6709\u7684\u6162\uff1b\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5728\u6570\u636e\u91cf \\(n\\) \u5f88\u5927\u65f6\u6709\u5de8\u5927\u4f18\u52bf\u3002
     \u65e0\u5e8f\u6570\u7ec4 \u6709\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u6307\u5b9a\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \\(O(\\log n)\\)"},{"location":"chapter_tree/binary_search_tree/#733","title":"7.3.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u9000\u5316","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u662f\u201c\u5de6\u53f3\u5e73\u8861\u201d\u7684\uff08\u8be6\u89c1\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u7ae0\u8282\uff09\uff0c\u6b64\u65f6\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u7ed3\u70b9\u3002
     
    \u5982\u679c\u6211\u4eec\u52a8\u6001\u5730\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u6b64\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u9000\u5316\u4e4b \\(O(n)\\) \u3002
     
    Note
     
    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5982\u4f55\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\uff0c\u4e5f\u662f\u4e00\u4e2a\u9700\u8981\u91cd\u8981\u8003\u8651\u7684\u95ee\u9898\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\u4e0e\u9000\u5316 
    "},{"location":"chapter_tree/binary_search_tree/#734","title":"7.3.4. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"
       
    • \u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u9ad8\u6548\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
      •  
    • \u5404\u79cd\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4fdd\u6301\u5176\u5df2\u6392\u5e8f\u3002
      •  
    "},{"location":"chapter_tree/binary_tree/","title":"7.1. \u00a0 \u4e8c\u53c9\u6811","text":"
    \u300c\u4e8c\u53c9\u6811 Binary Tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u7c7b\u4f3c\u4e8e\u94fe\u8868\uff0c\u4e8c\u53c9\u6811\u4e5f\u662f\u4ee5\u7ed3\u70b9\u4e3a\u5355\u4f4d\u5b58\u50a8\u7684\uff0c\u7ed3\u70b9\u5305\u542b\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u7ed3\u70b9\u503c\nTreeNode left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;          // \u7ed3\u70b9\u503c\nTreeNode *left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode *right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n
     
    \"\"\" \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b \"\"\"\nclass TreeNode:\ndef __init__(self, val: int):\nself.val: int = val                   # \u7ed3\u70b9\u503c\nself.left: Optional[TreeNode] = None  # \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nself.right: Optional[TreeNode] = None # \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal   int\nLeft  *TreeNode\nRight *TreeNode\n}\n/* \u7ed3\u70b9\u521d\u59cb\u5316\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\nreturn &TreeNode{\nLeft:  nil,\nRight: nil,\nVal:   v,\n}\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nfunction TreeNode(val, left, right) {\nthis.val = (val === undefined ? 0 : val); // \u7ed3\u70b9\u503c\nthis.left = (left === undefined ? null : left); // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nthis.right = (right === undefined ? null : right); // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nval: number;\nleft: TreeNode | null;\nright: TreeNode | null;\nconstructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.left = left === undefined ? null : left; // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nthis.right = right === undefined ? null : right; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n}\n}\n
     
    \n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nint val;          // \u7ed3\u70b9\u503c\nTreeNode? left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode? right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar left: TreeNode? // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nvar right: TreeNode? // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\ninit(x: Int) {\nval = x\n}\n}\n
     
    \n
     
    \u7ed3\u70b9\u7684\u4e24\u4e2a\u6307\u9488\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u7ed3\u70b9 Left Child Node\u300d\u548c\u300c\u53f3\u5b50\u7ed3\u70b9 Right Child Node\u300d\uff0c\u5e76\u4e14\u79f0\u8be5\u7ed3\u70b9\u4e3a\u4e24\u4e2a\u5b50\u7ed3\u70b9\u7684\u300c\u7236\u7ed3\u70b9 Parent Node\u300d\u3002\u7ed9\u5b9a\u4e8c\u53c9\u6811\u67d0\u7ed3\u70b9\uff0c\u5c06\u5de6\u5b50\u7ed3\u70b9\u4ee5\u4e0b\u7684\u6811\u79f0\u4e3a\u8be5\u7ed3\u70b9\u7684\u300c\u5de6\u5b50\u6811 Left Subtree\u300d\uff0c\u53f3\u5b50\u6811\u540c\u7406\u3002
     
    \u9664\u4e86\u53f6\u7ed3\u70b9\u5916\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u6709\u5b50\u7ed3\u70b9\u548c\u5b50\u6811\u3002\u4f8b\u5982\uff0c\u82e5\u5c06\u4e0b\u56fe\u7684\u300c\u7ed3\u70b9 2\u300d\u770b\u4f5c\u7236\u7ed3\u70b9\uff0c\u90a3\u4e48\u5176\u5de6\u5b50\u7ed3\u70b9\u548c\u53f3\u5b50\u7ed3\u70b9\u5206\u522b\u4e3a\u300c\u7ed3\u70b9 4\u300d\u548c\u300c\u7ed3\u70b9 5\u300d\uff0c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5206\u522b\u4e3a\u300c\u7ed3\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u7ed3\u70b9\u5f62\u6210\u7684\u6811\u300d\u548c\u300c\u7ed3\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u7ed3\u70b9\u5f62\u6210\u7684\u6811\u300d\u3002
     
     
     Fig. \u7236\u7ed3\u70b9\u3001\u5b50\u7ed3\u70b9\u3001\u5b50\u6811 
    "},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1. \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"
    \u4e8c\u53c9\u6811\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5efa\u8bae\u5c3d\u91cf\u7406\u89e3\u5e76\u8bb0\u4f4f\u3002\u540e\u7eed\u53ef\u80fd\u9057\u5fd8\uff0c\u53ef\u4ee5\u5728\u9700\u8981\u4f7f\u7528\u65f6\u56de\u6765\u67e5\u770b\u786e\u8ba4\u3002
     
       
    • \u300c\u6839\u7ed3\u70b9 Root Node\u300d\uff1a\u4e8c\u53c9\u6811\u6700\u9876\u5c42\u7684\u7ed3\u70b9\uff0c\u5176\u6ca1\u6709\u7236\u7ed3\u70b9\uff1b
      •  
    • \u300c\u53f6\u7ed3\u70b9 Leaf Node\u300d\uff1a\u6ca1\u6709\u5b50\u7ed3\u70b9\u7684\u7ed3\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u90fd\u6307\u5411 \\(\\text{null}\\) \uff1b
      •  
    • \u7ed3\u70b9\u6240\u5904\u300c\u5c42 Level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u4f9d\u6b21\u589e\u52a0\uff0c\u6839\u7ed3\u70b9\u6240\u5904\u5c42\u4e3a 1 \uff1b
      •  
    • \u7ed3\u70b9\u300c\u5ea6 Degree\u300d\uff1a\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf\u3002\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u8303\u56f4\u662f 0, 1, 2 \uff1b
      •  
    • \u300c\u8fb9 Edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u7ed3\u70b9\u7684\u8fb9\uff0c\u5373\u7ed3\u70b9\u6307\u9488\uff1b
      •  
    • \u4e8c\u53c9\u6811\u300c\u9ad8\u5ea6\u300d\uff1a\u4e8c\u53c9\u6811\u4e2d\u6839\u7ed3\u70b9\u5230\u6700\u8fdc\u53f6\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
    • \u7ed3\u70b9\u300c\u6df1\u5ea6 Depth\u300d \uff1a\u6839\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
    • \u7ed3\u70b9\u300c\u9ad8\u5ea6 Height\u300d\uff1a\u6700\u8fdc\u53f6\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed 
     
    \u9ad8\u5ea6\u4e0e\u6df1\u5ea6\u7684\u5b9a\u4e49
     
    \u503c\u5f97\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u300c\u9ad8\u5ea6\u300d\u548c\u300c\u6df1\u5ea6\u300d\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u800c\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u7ed3\u70b9\u7684\u6570\u91cf\u201d\uff0c\u6b64\u65f6\u9ad8\u5ea6\u6216\u6df1\u5ea6\u90fd\u9700\u8981 + 1 \u3002
    "},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2. \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":"
    \u521d\u59cb\u5316\u4e8c\u53c9\u6811\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5148\u521d\u59cb\u5316\u7ed3\u70b9\uff0c\u518d\u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree.java
    // \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.cpp
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
     binary_tree.py
    \"\"\" \u521d\u59cb\u5316\u4e8c\u53c9\u6811 \"\"\"\n# \u521d\u59cb\u5316\u7ed3\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
     binary_tree.go
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
     binary_tree.js
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.ts
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.c
    \n
     binary_tree.cs
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.swift
    // \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
     binary_tree.zig
    \n
     
    \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u90fd\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u5b9e\u73b0\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree.java
    TreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.cpp
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1->left = n2;\n
     binary_tree.py
    \"\"\" \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 \"\"\"\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2\n
     binary_tree.go
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u7ed3\u70b9 P\nn1.Left = n2\n
     binary_tree.js
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.ts
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.c
    \n
     binary_tree.cs
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.swift
    let P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2\n
     binary_tree.zig
    \n
     
    Note
     
    \u63d2\u5165\u7ed3\u70b9\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u5220\u9664\u7ed3\u70b9\u5f80\u5f80\u610f\u5473\u7740\u5220\u9664\u4e86\u8be5\u7ed3\u70b9\u7684\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u90fd\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u5b9e\u73b0\u6709\u610f\u4e49\u7684\u64cd\u4f5c\u3002
    "},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3. \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#_1","title":"\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 Perfect Binary Tree\u300d\u7684\u6240\u6709\u5c42\u7684\u7ed3\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u6240\u6709\u7ed3\u70b9\u7684\u5ea6 = 2 \uff1b\u82e5\u6811\u9ad8\u5ea6 \\(= h\\) \uff0c\u5219\u7ed3\u70b9\u603b\u6570 \\(= 2^{h+1} - 1\\) \uff0c\u5448\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u7740\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u3002
     
    Tip
     
    \u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\uff0c\u8bf7\u6ce8\u610f\u4e0e\u5b8c\u6ee1\u4e8c\u53c9\u6811\u533a\u5206\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_2","title":"\u5b8c\u5168\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u5168\u4e8c\u53c9\u6811 Complete Binary Tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u7ed3\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u7ed3\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002
     
    \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u5982\u679c\u6309\u7167\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u6765\u5b58\u50a8\uff0c\u90a3\u4e48\u7a7a\u7ed3\u70b9 null \u4e00\u5b9a\u5168\u90e8\u51fa\u73b0\u5728\u5e8f\u5217\u7684\u5c3e\u90e8\uff0c\u56e0\u6b64\u6211\u4eec\u5c31\u53ef\u4ee5\u4e0d\u7528\u5b58\u50a8\u8fd9\u4e9b null \u4e86\u3002
     
     
     Fig. \u5b8c\u5168\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_3","title":"\u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 Full Binary Tree\u300d\u9664\u4e86\u53f6\u7ed3\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u7ed3\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u5b8c\u6ee1\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_4","title":"\u5e73\u8861\u4e8c\u53c9\u6811","text":"
    \u300c\u5e73\u8861\u4e8c\u53c9\u6811 Balanced Binary Tree\u300d\u4e2d\u4efb\u610f\u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c \\(\\leq 1\\) \u3002
     
     
     Fig. \u5e73\u8861\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4. \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"
    \u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u7684\u7ed3\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff1b\u800c\u5f53\u6240\u6709\u7ed3\u70b9\u90fd\u504f\u5411\u4e00\u8fb9\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u300c\u94fe\u8868\u300d\u3002
     
       
    • \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u201c\u6700\u4f73\u72b6\u6001\u201d\uff0c\u53ef\u4ee5\u5b8c\u5168\u53d1\u6325\u51fa\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\uff1b
      •  
    • \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \uff1b
      •  
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u4e0e\u6700\u4e8c\u53c9\u6811\u7684\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u5dee\u60c5\u51b5 
     
    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5728\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u7ed3\u70b9\u6570\u91cf\u3001\u7ed3\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u6216\u6781\u5c0f\u503c\u3002
     \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u7ed3\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u53f6\u7ed3\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u7ed3\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u6811\u7684\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \u65f6\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree/#715","title":"7.1.5. \u00a0 \u4e8c\u53c9\u6811\u8868\u793a\u65b9\u5f0f *","text":"
    \u6211\u4eec\u4e00\u822c\u4f7f\u7528\u4e8c\u53c9\u6811\u7684\u300c\u94fe\u8868\u8868\u793a\u300d\uff0c\u5373\u5b58\u50a8\u5355\u4f4d\u4e3a\u7ed3\u70b9 TreeNode \uff0c\u7ed3\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\uff08\u5f15\u7528\uff09\u76f8\u8fde\u63a5\u3002\u672c\u6587\u524d\u8ff0\u793a\u4f8b\u4ee3\u7801\u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u5728\u94fe\u8868\u8868\u793a\u4e0b\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002
     
    \u90a3\u80fd\u5426\u53ef\u4ee5\u7528\u300c\u6570\u7ec4\u8868\u793a\u300d\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002\u5148\u6765\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff0c\u5c06\u7ed3\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u7f16\u53f7\uff08\u4ece 0 \u5f00\u59cb\uff09\uff0c\u90a3\u4e48\u53ef\u4ee5\u63a8\u5bfc\u5f97\u51fa\u7236\u7ed3\u70b9\u7d22\u5f15\u4e0e\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u300c\u6620\u5c04\u516c\u5f0f\u300d\uff1a\u8bbe\u7ed3\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u7ed3\u70b9\u7684\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \u3001\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002
     
    \u672c\u8d28\u4e0a\uff0c\u6620\u5c04\u516c\u5f0f\u7684\u4f5c\u7528\u5c31\u662f\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u5bf9\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u7684\u4efb\u610f\u7ed3\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u4f7f\u7528\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b50\u7ed3\u70b9\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\uff08\u5373\u6570\u7ec4\uff09\u6765\u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u7136\u800c\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u53ea\u662f\u4e2a\u4f8b\uff0c\u4e8c\u53c9\u6811\u4e2d\u95f4\u5c42\u5f80\u5f80\u5b58\u5728\u8bb8\u591a\u7a7a\u7ed3\u70b9\uff08\u5373 null \uff09\uff0c\u800c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b\u7a7a\u7ed3\u70b9\uff0c\u5e76\u4e14\u6211\u4eec\u65e0\u6cd5\u5355\u51ed\u5e8f\u5217\u6765\u731c\u6d4b\u7a7a\u7ed3\u70b9\u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u5373\u7406\u8bba\u4e0a\u5b58\u5728\u8bb8\u591a\u79cd\u4e8c\u53c9\u6811\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002\u663e\u7136\uff0c\u8fd9\u79cd\u60c5\u51b5\u65e0\u6cd5\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u4e8c\u53c9\u6811\u3002
     
     
     Fig. \u7ed9\u5b9a\u6570\u7ec4\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 
     
    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8003\u8651\u6309\u7167\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5f62\u5f0f\u6765\u8868\u793a\u6240\u6709\u4e8c\u53c9\u6811\uff0c\u5373\u5728\u5e8f\u5217\u4e2d\u4f7f\u7528\u7279\u6b8a\u7b26\u53f7\u6765\u663e\u5f0f\u5730\u8868\u793a\u201c\u7a7a\u4f4d\u201d\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5e8f\u5217\uff08\u6570\u7ec4\uff09\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4e3a\u4e86\u7b26\u5408\u6570\u636e\u7c7b\u578b\u4e3a int \uff0c\u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\n// \u8be5\u65b9\u6cd5\u7684\u4f7f\u7528\u524d\u63d0\u662f\u6ca1\u6709\u7ed3\u70b9\u7684\u503c = INT_MAX\nvector<int> tree = { 1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15 };\n
     
    \"\"\" \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a \"\"\"\n# \u76f4\u63a5\u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
     
    \n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n
     
    \n
     
     
     Fig. \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u56de\u987e\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u7684\u5b9a\u4e49\uff0c\u5176\u53ea\u6709\u6700\u5e95\u5c42\u6709\u7a7a\u7ed3\u70b9\uff0c\u5e76\u4e14\u6700\u5e95\u5c42\u7684\u7ed3\u70b9\u5c3d\u91cf\u9760\u5de6\uff0c\u56e0\u800c\u6240\u6709\u7a7a\u7ed3\u70b9\u90fd\u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002\u56e0\u4e3a\u6211\u4eec\u5148\u9a8c\u5730\u786e\u5b9a\u4e86\u7a7a\u4f4d\u7684\u4f4d\u7f6e\uff0c\u6240\u4ee5\u5728\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u201c\u7a7a\u4f4d\u201d\u3002\u56e0\u6b64\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002
     
     
     Fig. \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u6570\u7ec4\u8868\u793a\u6709\u4e24\u4e2a\u4f18\u70b9\uff1a \u4e00\u662f\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u8282\u7701\u7a7a\u95f4\uff1b\u4e8c\u662f\u53ef\u4ee5\u968f\u673a\u8bbf\u95ee\u7ed3\u70b9\u3002\u7136\u800c\uff0c\u5f53\u4e8c\u53c9\u6811\u4e2d\u7684\u201c\u7a7a\u4f4d\u201d\u5f88\u591a\u65f6\uff0c\u6570\u7ec4\u4e2d\u53ea\u5305\u542b\u5f88\u5c11\u7ed3\u70b9\u7684\u6570\u636e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u5f88\u4f4e\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2. \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"
    \u4ece\u7269\u7406\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u904d\u5386\u65b9\u5f0f\u4e5f\u662f\u901a\u8fc7\u6307\u9488\uff08\u5373\u5f15\u7528\uff09\u9010\u4e2a\u904d\u5386\u7ed3\u70b9\u3002\u540c\u65f6\uff0c\u6811\u8fd8\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u5bfc\u81f4\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u4f7f\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002
     
    \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u5f0f\u6709\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1. \u00a0 \u5c42\u5e8f\u904d\u5386","text":"
    \u300c\u5c42\u5e8f\u904d\u5386 Level-Order Traversal\u300d\u4ece\u9876\u81f3\u5e95\u3001\u4e00\u5c42\u4e00\u5c42\u5730\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u5c42\u4e2d\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u7ed3\u70b9\u3002
     
    \u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u662f\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Traversal\u300d\uff0c\u5176\u4f53\u73b0\u7740\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5c42\u8fdb\u904d\u5386\u65b9\u5f0f\u3002
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 
    "},{"location":"chapter_tree/binary_tree_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u4e00\u822c\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u7684\u89c4\u5219\u662f\u201c\u5148\u8fdb\u5148\u51fa\u201d\uff0c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u89c4\u5219\u662f \u201d\u4e00\u5c42\u5c42\u5e73\u63a8\u201c \uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree_bfs.java
    /* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nQueue<TreeNode> queue = new LinkedList<>() {{ add(root); }};\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<Integer> list = new ArrayList<>();\nwhile (!queue.isEmpty()) {\nTreeNode node = queue.poll();  // \u961f\u5217\u51fa\u961f\nlist.add(node.val);            // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null)\nqueue.offer(node.left);    // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right != null)\nqueue.offer(node.right);   // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.cpp
    /* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode* root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue<TreeNode*> queue;\nqueue.push(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvector<int> vec;\nwhile (!queue.empty()) {\nTreeNode* node = queue.front();\nqueue.pop();                 // \u961f\u5217\u51fa\u961f\nvec.push_back(node->val);    // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node->left != nullptr)\nqueue.push(node->left);  // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node->right != nullptr)\nqueue.push(node->right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn vec;\n}\n
     binary_tree_bfs.py
    def level_order(root: Optional[TreeNode]) -> List[int]:\n\"\"\" \u5c42\u5e8f\u904d\u5386 \"\"\"\n# \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue: Deque[TreeNode] = collections.deque()\nqueue.append(root)\n# \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nres: List[int] = []\nwhile queue:\nnode: TreeNode = queue.popleft() # \u961f\u5217\u51fa\u961f\nres.append(node.val)             # \u4fdd\u5b58\u7ed3\u70b9\u503c\nif node.left is not None:\nqueue.append(node.left)      # \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif node.right is not None:\nqueue.append(node.right)     # \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\nreturn res\n
     binary_tree_bfs.go
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []int {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue := list.New()\nqueue.PushBack(root)\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nnums := make([]int, 0)\nfor queue.Len() > 0 {\n// \u961f\u5217\u51fa\u961f\nnode := queue.Remove(queue.Front()).(*TreeNode)\n// \u4fdd\u5b58\u7ed3\u70b9\u503c\nnums = append(nums, node.Val)\nif node.Left != nil {\n// \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nqueue.PushBack(node.Left)\n}\nif node.Right != nil {\n// \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\nqueue.PushBack(node.Right)\n}\n}\nreturn nums\n}\n
     binary_tree_bfs.js
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list = [];\nwhile (queue.length) {\nlet node = queue.shift();   // \u961f\u5217\u51fa\u961f\nlist.push(node.val);        // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left)\nqueue.push(node.left);  // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right)\nqueue.push(node.right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.ts
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list: number[] = [];\nwhile (queue.length) {\nlet node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left) {\nqueue.push(node.left); // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif (node.right) {\nqueue.push(node.right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\n}\nreturn list;\n}\n
     binary_tree_bfs.c
    [class]{}-[func]{levelOrder}\n
     binary_tree_bfs.cs
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode root)\n{\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nQueue<TreeNode> queue = new();\nqueue.Enqueue(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> list = new();\nwhile (queue.Count != 0)\n{\nTreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\nlist.Add(node.val);              // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null)\nqueue.Enqueue(node.left);    // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right != null)\nqueue.Enqueue(node.right);   // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.swift
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nvar queue: [TreeNode] = [root]\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list: [Int] = []\nwhile !queue.isEmpty {\nlet node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\nlist.append(node.val) // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif let left = node.left {\nqueue.append(left) // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif let right = node.right {\nqueue.append(right) // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\n}\nreturn list\n}\n
     binary_tree_bfs.zig
    // \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst L = std.TailQueue(*inc.TreeNode(T));\nvar queue = L{};\nvar root_node = try mem_allocator.create(L.Node);\nroot_node.data = root;\nqueue.append(root_node); // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list = std.ArrayList(T).init(std.heap.page_allocator);\nwhile (queue.len > 0) {\nvar queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\nvar node = queue_node.data;\ntry list.append(node.val);              // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.left.?;\nqueue.append(tmp_node);             // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif (node.right != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.right.?;\nqueue.append(tmp_node);             // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}        }\nreturn list;\n}\n
    "},{"location":"chapter_tree/binary_tree_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u7ed3\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7ed3\u70b9\u6570\u91cf\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5f53\u4e3a\u6ee1\u4e8c\u53c9\u6811\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\(\\frac{n + 1}{2}\\) \u4e2a\u7ed3\u70b9\uff0c\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2. \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"
    \u76f8\u5bf9\u5730\uff0c\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u5176\u4f53\u73b0\u7740\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5de6\u4fa7\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7684\u793a\u610f\u56fe\uff0c\u53f3\u4e0a\u65b9\u662f\u5bf9\u5e94\u7684\u9012\u5f52\u5b9e\u73b0\u4ee3\u7801\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u4e2a\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u8d70\u7684\u8fc7\u7a0b\u4e2d\uff0c\u5728\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386 
     \u4f4d\u7f6e \u542b\u4e49 \u6b64\u5904\u8bbf\u95ee\u7ed3\u70b9\u65f6\u5bf9\u5e94 \u6a59\u8272\u5706\u5708\u5904 \u521a\u8fdb\u5165\u6b64\u7ed3\u70b9\uff0c\u5373\u5c06\u8bbf\u95ee\u8be5\u7ed3\u70b9\u7684\u5de6\u5b50\u6811 \u524d\u5e8f\u904d\u5386 Pre-Order Traversal \u84dd\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\uff0c\u5373\u5c06\u8bbf\u95ee\u53f3\u5b50\u6811 \u4e2d\u5e8f\u904d\u5386 In-Order Traversal \u7d2b\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff0c\u5373\u5c06\u8fd4\u56de \u540e\u5e8f\u904d\u5386 Post-Order Traversal"},{"location":"chapter_tree/binary_tree_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree_dfs.java
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.add(root.val);\n}\n
     binary_tree_dfs.cpp
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nvec.push_back(root->val);\npreOrder(root->left);\npreOrder(root->right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left);\nvec.push_back(root->val);\ninOrder(root->right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root->left);\npostOrder(root->right);\nvec.push_back(root->val);\n}\n
     binary_tree_dfs.py
    def pre_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u524d\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nres.append(root.val)\npre_order(root=root.left)\npre_order(root=root.right)\ndef in_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u4e2d\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\nin_order(root=root.left)\nres.append(root.val)\nin_order(root=root.right)\ndef post_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u540e\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npost_order(root=root.left)\npost_order(root=root.right)\nres.append(root.val)\n
     binary_tree_dfs.go
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nnums = append(nums, node.Val)\npreOrder(node.Left)\npreOrder(node.Right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.Left)\nnums = append(nums, node.Val)\ninOrder(node.Right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(node.Left)\npostOrder(node.Right)\nnums = append(nums, node.Val)\n}\n
     binary_tree_dfs.js
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
     binary_tree_dfs.ts
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
     binary_tree_dfs.c
    [class]{}-[func]{preOrder}\n[class]{}-[func]{inOrder}\n[class]{}-[func]{postOrder}\n
     binary_tree_dfs.cs
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.Add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.Add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.Add(root.val);\n}\n
     binary_tree_dfs.swift
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.append(root.val)\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root: root.left)\nlist.append(root.val)\ninOrder(root: root.right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root: root.left)\npostOrder(root: root.right)\nlist.append(root.val)\n}\n
     binary_tree_dfs.zig
    // \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\ntry list.append(root.?.val);\ntry preOrder(T, root.?.left);\ntry preOrder(T, root.?.right);\n}\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ntry inOrder(T, root.?.left);\ntry list.append(root.?.val);\ntry inOrder(T, root.?.right);\n}\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\ntry postOrder(T, root.?.left);\ntry postOrder(T, root.?.right);\ntry list.append(root.?.val);\n}\n
     
    Note
     
    \u4f7f\u7528\u5faa\u73af\u4e00\u6837\u53ef\u4ee5\u5b9e\u73b0\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\uff0c\u4f46\u4ee3\u7801\u76f8\u5bf9\u7e41\u7410\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u7ed3\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7ed3\u70b9\u6570\u91cf\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5f53\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
    "},{"location":"chapter_tree/summary/","title":"7.5. \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#_1","title":"\u4e8c\u53c9\u6811","text":"
       
    • \u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u5305\u542b\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\uff0c\u5206\u522b\u6307\u5411\u5de6\u5b50\u7ed3\u70b9\u548c\u53f3\u5b50\u7ed3\u70b9\u3002
      •  
    • \u9009\u5b9a\u4e8c\u53c9\u6811\u4e2d\u67d0\u7ed3\u70b9\uff0c\u5c06\u5176\u5de6\uff08\u53f3\uff09\u5b50\u7ed3\u70b9\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002
      •  
    • \u4e8c\u53c9\u6811\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5305\u62ec\u6839\u7ed3\u70b9\u3001\u53f6\u7ed3\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u3001\u6df1\u5ea6\u7b49\u3002
      •  
    • \u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u7ed3\u70b9\u63d2\u5165\u3001\u7ed3\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002
      •  
    • \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u5305\u62ec\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u3001\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u72b6\u6001\uff0c\u94fe\u8868\u5219\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002
      •  
    • \u4e8c\u53c9\u6811\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u8868\u793a\uff0c\u5177\u4f53\u505a\u6cd5\u662f\u5c06\u7ed3\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u57fa\u4e8e\u7236\u7ed3\u70b9\u548c\u5b50\u7ed3\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5b9e\u73b0\u6307\u9488\u3002
      •  
    "},{"location":"chapter_tree/summary/#_2","title":"\u4e8c\u53c9\u6811\u904d\u5386","text":"
       
    • \u4e8c\u53c9\u6811\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5c42\u8fdb\u5f0f\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\u3002
      •  
    • \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f53\u73b0\u7740\u201c\u8d70\u5230\u5934\u3001\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_tree/summary/#_3","title":"\u4e8c\u53c9\u641c\u7d22\u6811","text":"
       
    • \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u540e\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \uff0c\u56e0\u6b64\u5982\u4f55\u907f\u514d\u9000\u5316\u662f\u975e\u5e38\u91cd\u8981\u7684\u8bfe\u9898\u3002
      •  
    • AVL \u6811\u53c8\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5176\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u5f97\u5728\u4e0d\u65ad\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u4ecd\u7136\u53ef\u4ee5\u4fdd\u6301\u4e8c\u53c9\u6811\u7684\u5e73\u8861\uff08\u4e0d\u9000\u5316\uff09\u3002
      •  
    • AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u7ed3\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u6062\u590d\u5e73\u8861\u3002
      •  
    "}]}
\ No newline at end of file
+{"config":{"lang":["ja"],"separator":"[\\s\\-\uff0c\u3002]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"
      \u300a Hello \u7b97\u6cd5 \u300b  
     \u52a8\u753b\u56fe\u89e3\u3001\u80fd\u8fd0\u884c\u3001\u53ef\u63d0\u95ee\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5feb\u901f\u5165\u95e8\u6559\u7a0b 
     
     
     @Krahets 
      \u63a8\u8350\u8bed  
    Quote
     
    \u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d
     
    \u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388
     
    Quote
     
    \u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d
     
    \u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6
      \u300c\u5168\u4e66\u52a8\u753b\u56fe\u89e3\u300d  
     \u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u7535\u8111\u3001\u5e73\u677f\u3001\u624b\u673a\u5168\u7ec8\u7aef\u9605\u8bfb 
     
     
     \"A picture is worth a thousand words.\" 
     
     \u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d 
      \u300c\u4ee3\u7801\u4e00\u952e\u8fd0\u884c\u300d  
     \u63d0\u4f9b\u5404\u4e2a\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u7b80\u6d01\u5b9e\u73b0\u4e0e\u6d4b\u8bd5\u6837\u4f8b\uff0c\u7686\u53ef\u76f4\u63a5\u8fd0\u884c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00 
     
     
     \"Talk is cheap. Show me the code.\" 
     
     \u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d 
      \u300c\u53ef\u8ba8\u8bba\u4e0e\u63d0\u95ee\u300d  
     \u9f13\u52b1\u5c0f\u4f19\u4f34\u4eec\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u6210\u957f\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d 
     
     
     \u201c\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d 
     
     \u4e00\u8d77\u52a0\u6cb9\uff01 
      \u5e8f  
    \u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u300a\u5251\u6307 Offer\u300b\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u5f88\u591a\u5c0f\u4f19\u4f34\u7684\u559c\u7231\u4e0e\u652f\u6301\u3002\u5728\u6b64\u671f\u95f4\uff0c\u6211\u4e5f\u56de\u590d\u4e86\u8bb8\u591a\u8bfb\u8005\u7684\u8bc4\u8bba\u95ee\u9898\uff0c\u9047\u5230\u6700\u591a\u7684\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u5b66\u4e60\u7b97\u6cd5\u201d\u3002\u6211\u6e10\u6e10\u4e5f\u5bf9\u8fd9\u4e2a\u95ee\u9898\u597d\u5947\u4e86\u8d77\u6765\u3002
     
    \u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u5e94\u8be5\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u5f0f\uff0c\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\u3002\u7136\u800c\uff0c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5730\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u53d7\u632b\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e66\u7c4d\u4e5f\u662f\u5e38\u7528\u65b9\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u5e94\u4ed8\u7b14\u9762\u8bd5\u5df2\u7ecf\u5360\u7528\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u539a\u91cd\u7684\u4e66\u672c\u4e5f\u56e0\u6b64\u6210\u4e3a\u5de8\u5927\u7684\u6311\u6218\u3002
     
    \u5982\u679c\u4f60\u4e5f\u6709\u4e0a\u8ff0\u70e6\u607c\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u4e8e\u8be5\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u867d\u7136\u4e0d\u4e00\u5b9a\u6b63\u786e\uff0c\u4f46\u81f3\u5c11\u4ee3\u8868\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u8fd9\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u5f97\u5fc3\u5e94\u624b\u5730\u5237\u9898\u4e0e\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u642d\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002
      \u4f5c\u8005\u7b80\u4ecb  
    \u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u5176 LeetBook\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 22 \u4e07\u672c\u3002
      \u81f4\u8c22  
    \u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u7684\u7fa4\u7b56\u7fa4\u529b\u4e0b\u9010\u6b65\u6210\u957f\uff0c\u611f\u8c22\u6bcf\u4e00\u4f4d\u64b0\u7a3f\u4eba\uff0c\u662f\u4ed6\u4eec\u7684\u65e0\u79c1\u5949\u732e\u8ba9\u8fd9\u672c\u4e66\u53d8\u5f97\u66f4\u597d\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\uff09\uff1a
     
     
     
    \u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 justin\u2010tse, krahets, nuomi1, Reanon, sjinzh \u5b8c\u6210\uff0c\u611f\u8c22\u4ed6\u4eec\u7684\u8f9b\u52e4\u4ed8\u51fa\uff01
     justin-tseJS / TS krahetsJava / Python nuomi1Swift ReanonGo / C sjinzhRust / Zig"},{"location":"chapter_appendix/contribution/","title":"12.2. \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"
    \u5f00\u6e90\u7684\u9b45\u529b
     
    \u7eb8\u8d28\u4e66\u7c4d\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u9700\u8981\u6570\u5e74\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002\u4f46\u5728\u672c\u5f00\u6e90 HTML \u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002
     
    \u7531\u4e8e\u4f5c\u8005\u6c34\u5e73\u6709\u9650\uff0c\u4e66\u4e2d\u5185\u5bb9\u96be\u514d\u758f\u6f0f\u8c2c\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u53d1\u73b0\u7b14\u8bef\u3001\u65e0\u6548\u94fe\u63a5\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u3001\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u60a8\u5e2e\u5fd9\u4fee\u6b63\uff0c\u4ee5\u5e2e\u52a9\u5176\u4ed6\u8bfb\u8005\u83b7\u53d6\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u5185\u5bb9\u3002\u6240\u6709\u64b0\u7a3f\u4eba\u5c06\u88ab\u5c55\u793a\u5728\u4ed3\u5e93\u4e0e\u7f51\u7ad9\u4e3b\u9875\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\uff01
    "},{"location":"chapter_appendix/contribution/#1221","title":"12.2.1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"
    \u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u4e00\u4e2a\u300c\u7f16\u8f91\u300d\u56fe\u6807\uff0c\u4f60\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u5b57\u6216\u4ee3\u7801\uff1a
     
       
    1. \u70b9\u51fb\u7f16\u8f91\u6309\u94ae\uff0c\u5982\u679c\u9047\u5230\u63d0\u793a\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\uff0c\u8bf7\u901a\u8fc7\uff1b
      2.  
    2. \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u5e76\u68c0\u67e5\u5185\u5bb9\u6b63\u786e\u6027\uff0c\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7edf\u4e00\uff1b
      3.  
    3. \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u66f4\u6539\u8bf4\u660e\uff0c\u7136\u540e\u5355\u51fb\u201cPropose file change\u201d\u6309\u94ae\uff1b\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u5373\u53ef\u3002
      4.  
     
     
     Fig. \u9875\u9762\u7f16\u8f91\u6309\u952e 
     
    \u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u56fe\u7247\u95ee\u9898\uff0c\u6211\u4f1a\u7b2c\u4e00\u65f6\u95f4\u91cd\u65b0\u753b\u56fe\u5e76\u66ff\u6362\u56fe\u7247\u3002
    "},{"location":"chapter_appendix/contribution/#1222","title":"12.2.2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"
    \u5982\u679c\u60a8\u60f3\u8981\u53c2\u4e0e\u672c\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u7ffb\u8bd1\u4ee3\u7801\u81f3\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u62d3\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd Pull Request \u5de5\u4f5c\u6d41\u7a0b\uff1a
     
       
    1. \u767b\u5f55 GitHub \uff0c\u5e76 Fork \u672c\u4ed3\u5e93 \u81f3\u4e2a\u4eba\u8d26\u53f7\uff1b
      2.  
    2. \u8fdb\u5165 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 git clone \u514b\u9686\u8be5\u4ed3\u5e93\u81f3\u672c\u5730\uff1b
      3.  
    3. \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u901a\u8fc7\u8fd0\u884c\u6d4b\u8bd5\u6765\u9a8c\u8bc1\u4ee3\u7801\u6b63\u786e\u6027\uff1b
      4.  
    4. \u5c06\u672c\u5730\u66f4\u6539 Commit \uff0c\u5e76 Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\uff1b
      5.  
    5. \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u5373\u53ef\uff1b
      6.  
    "},{"location":"chapter_appendix/contribution/#1223-docker","title":"12.2.3. \u00a0 Docker \u90e8\u7f72","text":"
    \u4f60\u53ef\u4ee5\u4f7f\u7528 Docker \u6765\u90e8\u7f72\u672c\u9879\u76ee\u3002\u7a0d\u7b49\u7247\u523b\uff0c\u5373\u53ef\u4f7f\u7528\u6d4f\u89c8\u5668\u6253\u5f00 http://localhost:8000 \u8bbf\u95ee\u672c\u9879\u76ee\u3002
     
    git clone https://github.com/krahets/hello-algo.git\ncd hello-algo\ndocker-compose up -d\n
     
    \u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\u3002
     
    docker-compose down\n
    "},{"location":"chapter_appendix/installation/","title":"12.1. \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1211-vscode","title":"12.1.1. \u00a0 \u5b89\u88c5 VSCode","text":"
    \u672c\u4e66\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730 IDE \uff0c\u4e0b\u8f7d\u5e76\u5b89\u88c5 VSCode \u3002
    "},{"location":"chapter_appendix/installation/#1212-java","title":"12.1.2. \u00a0 Java \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 java \uff0c\u5b89\u88c5 Java Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1213-cc","title":"12.1.3. \u00a0 C/C++ \u73af\u5883","text":"
       
    1. Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff0cMacOS \u81ea\u5e26 Clang \u65e0\u9700\u5b89\u88c5\u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c++ \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1214-python","title":"12.1.4. \u00a0 Python \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 python \uff0c\u5b89\u88c5 Python Extension Pack \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1215-go","title":"12.1.5. \u00a0 Go \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 go \uff0c\u5b89\u88c5 Go \u3002
      3.  
    3. \u5feb\u6377\u952e Ctrl + Shift + P \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 Go: Install/Update Tools \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
      4.  
    "},{"location":"chapter_appendix/installation/#1216-javascript","title":"12.1.6. \u00a0 JavaScript \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 javascript \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1217-c","title":"12.1.7. \u00a0 C# \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 6.0 \uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c# \uff0c\u5b89\u88c5 c# \u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1218-swift","title":"12.1.8. \u00a0 Swift \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift\uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 swift\uff0c\u5b89\u88c5 Swift for Visual Studio Code\u3002
      3.  
    "},{"location":"chapter_appendix/installation/#1219-rust","title":"12.1.9. \u00a0 Rust \u73af\u5883","text":"
       
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust\uff1b
      2.  
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 rust\uff0c\u5b89\u88c5 rust-analyzer\u3002
      3.  
    "},{"location":"chapter_array_and_linkedlist/array/","title":"4.1. \u00a0 \u6570\u7ec4","text":"
    \u300c\u6570\u7ec4 Array\u300d\u662f\u4e00\u79cd\u5c06 \u76f8\u540c\u7c7b\u578b\u5143\u7d20 \u5b58\u50a8\u5728 \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u7684\u6570\u636e\u7ed3\u6784\uff0c\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u5143\u7d20\u7684\u300c\u7d22\u5f15 Index\u300d\u3002
     
     
     Fig. \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f 
     
    Note
     
    \u89c2\u5bdf\u4e0a\u56fe\uff0c\u6211\u4eec\u53d1\u73b0 \u6570\u7ec4\u9996\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \u3002\u4f60\u53ef\u80fd\u4f1a\u60f3\uff0c\u8fd9\u5e76\u4e0d\u7b26\u5408\u65e5\u5e38\u4e60\u60ef\uff0c\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(1\\) \u5462\uff0c\u8fd9\u4e0d\u662f\u66f4\u52a0\u81ea\u7136\u5417\uff1f\u6211\u8ba4\u540c\u4f60\u7684\u60f3\u6cd5\uff0c\u4f46\u8bf7\u5148\u8bb0\u4f4f\u8fd9\u4e2a\u8bbe\u5b9a\uff0c\u540e\u9762\u8bb2\u5185\u5b58\u5730\u5740\u8ba1\u7b97\u65f6\uff0c\u6211\u4f1a\u5c1d\u8bd5\u89e3\u7b54\u8fd9\u4e2a\u95ee\u9898\u3002
     
    \u6570\u7ec4\u521d\u59cb\u5316\u3002\u4e00\u822c\u4f1a\u7528\u5230\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u4e24\u79cd\u5199\u6cd5\uff0c\u53ef\u6839\u636e\u9700\u6c42\u9009\u53d6\u3002\u5728\u4e0d\u7ed9\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u4e00\u822c\u6240\u6709\u5143\u7d20\u4f1a\u88ab\u521d\u59cb\u5316\u4e3a\u9ed8\u8ba4\u503c \\(0\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
     array.cpp
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
     array.py
    \"\"\" \u521d\u59cb\u5316\u6570\u7ec4 \"\"\"\narr: List[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: List[int] = [1, 3, 2, 5, 4]  \n
     array.go
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
     array.js
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
     array.ts
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
     array.c
    int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
     array.cs
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
     array.swift
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
     array.zig
    // \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n
    "},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1. \u00a0 \u6570\u7ec4\u4f18\u70b9","text":"
    \u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u6570\u7ec4\u4e2d\uff0c\u8ba1\u7b97\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u3001\u548c\u4e00\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5229\u7528\u4ee5\u4e0b\u516c\u5f0f\u53ef\u4ee5\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u6b64\u5143\u7d20\u3002
     
     
     Fig. \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 
     
    # \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n
     
    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u5143\u7d20\u7d22\u5f15\u4ece 0 \u5f00\u59cb\u7f16\u53f7\uff1f \u6839\u636e\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u8868\u793a\u7684\u662f\u5185\u5b58\u5730\u5740\u504f\u79fb\u91cf\uff0c\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u90a3\u4e48\u7d22\u5f15\u662f \\(0\\) \u4e5f\u5c31\u5f88\u81ea\u7136\u4e86\u3002
     
    \u8bbf\u95ee\u5143\u7d20\u7684\u9ad8\u6548\u6027\u5e26\u6765\u4e86\u8bb8\u591a\u4fbf\u5229\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u83b7\u53d6\u4e00\u4e2a\u6570\u7ec4\u4e2d\u7684\u5143\u7d20\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = ThreadLocalRandom.current().\nnextInt(0, nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.cpp
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int* nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.py
    def random_access(nums: List[int]) -> int:\n\"\"\" \u968f\u673a\u8bbf\u95ee\u5143\u7d20 \"\"\"\n# \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandom_index = random.randint(0, len(nums) - 1)\n# \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandom_num = nums[random_index]\nreturn random_num\n
     array.go
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandomIndex := rand.Intn(len(nums))\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandomNum = nums[randomIndex]\nreturn\n}\n
     array.js
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
     array.ts
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
     array.c
    [class]{}-[func]{randomAccess}\n
     array.cs
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums)\n{\nRandom random = new();\n// \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = random.Next(nums.Length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
     array.swift
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n// \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet randomIndex = nums.indices.randomElement()!\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet randomNum = nums[randomIndex]\nreturn randomNum\n}\n
     array.zig
    // \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n// \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\nvar randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nvar randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2. \u00a0 \u6570\u7ec4\u7f3a\u70b9","text":"
    \u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7531\u4e8e\u7cfb\u7edf\u65e0\u6cd5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u56e0\u6b64\u6570\u7ec4\u957f\u5ea6\u65e0\u6cd5\u6269\u5c55\u3002\u800c\u82e5\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u65b0\u5efa\u4e00\u4e2a\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u662f\u975e\u5e38\u8017\u65f6\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.cpp
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint* extend(int* nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint* res = new int[size + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] nums;\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.py
    def extend(nums: List[int], enlarge: int) -> List[int]:\n\"\"\" \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 \"\"\"\n# \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres = [0] * (len(nums) + enlarge)\n# \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in range(len(nums)):\nres[i] = nums[i]\n# \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n
     array.go
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres := make([]int, len(nums)+enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i, num := range nums {\nres[i] = num\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
     array.js
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.ts
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.c
    [class]{}-[func]{extend}\n
     array.cs
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge)\n{\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.Length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     array.swift
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = Array(repeating: 0, count: nums.count + enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in nums.indices {\nres[i] = nums[i]\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
     array.zig
    // \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = try mem_allocator.alloc(i32, nums.len + enlarge);\nstd.mem.set(i32, res, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(i32, res, nums);\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
     
    \u6570\u7ec4\u4e2d\u63d2\u5165\u6216\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\u4e0b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u653e\u4efb\u4f55\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4e0d\u5f97\u4e0d\u5c06\u6b64\u7d22\u5f15\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002
     
     
     Fig. \u6570\u7ec4\u63d2\u5165\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#Swift array.java
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.cpp
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int* nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.py
    def insert(nums: List[int], num: int, index: int) -> None:\n\"\"\" \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num \"\"\"\n# \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in range(len(nums) - 1, index, -1):\nnums[i] = nums[i - 1]\n# \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n
     array.go
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i := len(nums) - 1; i > index; i-- {\nnums[i] = nums[i-1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
     array.js
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.ts
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.c
    [class]{}-[func]{insert}\n
     array.cs
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index)\n{\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.Length - 1; i > index; i--)\n{\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
     array.swift
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: nums.count - 1, next: { $0 > index + 1 ? $0 - 1 : nil }) {\nnums[i] = nums[i - 1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
     
    \u5220\u9664\u5143\u7d20\u4e5f\u662f\u7c7b\u4f3c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5220\u9664\u5143\u7d20\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6211\u4eec\u65e0\u9700\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002
     
     
     Fig. \u6570\u7ec4\u5220\u9664\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.cpp
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int* nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.py
    def remove(nums: List[int], index: int) -> None:\n\"\"\" \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 \"\"\"\n# \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in range(index, len(nums) - 1):\nnums[i] = nums[i + 1]\n
     array.go
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i := index; i < len(nums)-1; i++ {\nnums[i] = nums[i+1]\n}\n}\n
     array.js
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums, index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.ts
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
     array.c
    [class]{}-[func]{removeItem}\n
     array.cs
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index)\n{\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.Length - 1; i++)\n{\nnums[i] = nums[i + 1];\n}\n}\n
     array.swift
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\nlet count = nums.count\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: index, next: { $0 < count - 1 - 1 ? $0 + 1 : nil }) {\nnums[i] = nums[i + 1]\n}\n}\n
     array.zig
    // \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar i = index;\nwhile (i < nums.len - 1) : (i += 1) {\nnums[i] = nums[i + 1];\n}\n}\n
     
    \u603b\u7ed3\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\uff1a
     
       
    • \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(N)\\) \uff0c\u5176\u4e2d \\(N\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
      •  
    • \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u88ab\u4e22\u5931\u3002
      •  
    • \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u4e00\u822c\u4f1a\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u6211\u4eec\u4e0d\u5173\u5fc3\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u540c\u65f6\u4e5f\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
      •  
    "},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3. \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":"
    \u6570\u7ec4\u904d\u5386\u3002\u4ee5\u4e0b\u4ecb\u7ecd\u4e24\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int num : nums) {\ncount++;\n}\n}\n
     array.cpp
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int* nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
     array.py
    def traverse(nums: List[int]) -> None:\n\"\"\" \u904d\u5386\u6570\u7ec4 \"\"\"\ncount = 0\n# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor num in nums:\ncount += 1\n
     array.go
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\ncount := 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\ncount++\n}\ncount = 0\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor range nums {\ncount++\n}\n}\n
     array.js
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
     array.ts
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
     array.c
    [class]{}-[func]{traverse}\n
     array.cs
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums)\n{\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nforeach (int num in nums)\n{\ncount++;\n}\n}\n
     array.swift
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\nvar count = 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor _ in nums.indices {\ncount += 1\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor _ in nums {\ncount += 1\n}\n}\n
     array.zig
    // \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\nvar count: i32 = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nvar i: i32 = 0;\nwhile (i < nums.len) : (i += 1) {\ncount += 1;\n}\ncount = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (nums) |_| {\ncount += 1;\n}\n}\n
     
    \u6570\u7ec4\u67e5\u627e\u3002\u901a\u8fc7\u904d\u5386\u6570\u7ec4\uff0c\u67e5\u627e\u6570\u7ec4\u5185\u7684\u6307\u5b9a\u5143\u7d20\uff0c\u5e76\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array.java
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.cpp
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int* nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.py
    def find(nums: List[int], target: int) -> int:\n\"\"\" \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 \"\"\"\nfor i in range(len(nums)):\nif nums[i] == target:\nreturn i\nreturn -1\n
     array.go
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\nindex = -1\nfor i := 0; i < len(nums); i++ {\nif nums[i] == target {\nindex = i\nbreak\n}\n}\nreturn\n}\n
     array.js
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
     array.ts
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] === target) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     array.c
    [class]{}-[func]{find}\n
     array.cs
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target)\n{\nfor (int i = 0; i < nums.Length; i++)\n{\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
     array.swift
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\nfor i in nums.indices {\nif nums[i] == target {\nreturn i\n}\n}\nreturn -1\n}\n
     array.zig
    // \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\nfor (nums) |num, i| {\nif (num == target) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#414","title":"4.1.4. \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"
    \u968f\u673a\u8bbf\u95ee\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u6837\u672c\u7684\u968f\u673a\u62bd\u53d6\u3002
     
    \u4e8c\u5206\u67e5\u627e\u3002\u4f8b\u5982\u524d\u6587\u67e5\u5b57\u5178\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u4e2d\u7684\u6240\u6709\u5b57\u6309\u7167\u62fc\u97f3\u987a\u5e8f\u5b58\u50a8\u5728\u6570\u7ec4\u4e2d\uff0c\u7136\u540e\u4f7f\u7528\u4e0e\u65e5\u5e38\u67e5\u7eb8\u8d28\u5b57\u5178\u76f8\u540c\u7684\u201c\u7ffb\u5f00\u4e2d\u95f4\uff0c\u6392\u9664\u4e00\u534a\u201d\u7684\u65b9\u5f0f\uff0c\u6765\u5b9e\u73b0\u4e00\u4e2a\u67e5\u7535\u5b50\u5b57\u5178\u7684\u7b97\u6cd5\u3002
     
    \u6df1\u5ea6\u5b66\u4e60\u3002\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002
    "},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2. \u00a0 \u94fe\u8868","text":"
    \u5f15\u8a00
     
    \u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u6392\u9664\u5df2\u5360\u7528\u7684\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u5f80\u5f80\u662f\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u7684\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u5185\u5b58\u7a7a\u95f4\u8fde\u7eed\uff0c\u5f53\u6211\u4eec\u9700\u8981\u7533\u8bf7\u4e00\u4e2a\u5f88\u5927\u7684\u6570\u7ec4\u65f6\uff0c\u7cfb\u7edf\u4e0d\u4e00\u5b9a\u5b58\u5728\u8fd9\u4e48\u5927\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u3002\u800c\u94fe\u8868\u5219\u66f4\u52a0\u7075\u6d3b\uff0c\u4e0d\u9700\u8981\u5185\u5b58\u662f\u8fde\u7eed\u7684\uff0c\u53ea\u8981\u5269\u4f59\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u591f\u7528\u5373\u53ef\u3002
     
    \u300c\u94fe\u8868 Linked List\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u5355\u72ec\u7684\u5bf9\u8c61\uff0c\u5404\u4e2a\u5143\u7d20\uff08\u4e00\u822c\u79f0\u4e3a\u7ed3\u70b9\uff09\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u8fde\u63a5\u3002\u7531\u4e8e\u7ed3\u70b9\u4e2d\u8bb0\u5f55\u4e86\u8fde\u63a5\u5173\u7cfb\uff0c\u56e0\u6b64\u94fe\u8868\u7684\u5b58\u50a8\u65b9\u5f0f\u76f8\u6bd4\u4e8e\u6570\u7ec4\u66f4\u52a0\u7075\u6d3b\uff0c\u7cfb\u7edf\u4e0d\u5fc5\u4fdd\u8bc1\u5185\u5b58\u5730\u5740\u7684\u8fde\u7eed\u6027\u3002
     
    \u94fe\u8868\u7684\u300c\u7ed3\u70b9 Node\u300d\u5305\u542b\u4e24\u9879\u6570\u636e\uff0c\u4e00\u662f\u7ed3\u70b9\u300c\u503c Value\u300d\uff0c\u4e8c\u662f\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u300c\u6307\u9488 Pointer\u300d\uff08\u6216\u79f0\u300c\u5f15\u7528 Reference\u300d\uff09\u3002
     
     
     Fig. \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u7ed3\u70b9\u503c\nListNode *next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
     
    \"\"\" \u94fe\u8868\u7ed3\u70b9\u7c7b \"\"\" \nclass ListNode:\ndef __init__(self, val: int):\nself.val: int = val                  # \u7ed3\u70b9\u503c\nself.next: Optional[ListNode] = None # \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\nVal  int       // \u7ed3\u70b9\u503c\nNext *ListNode // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\nreturn &ListNode{\nVal:  val,\nNext: nil,\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nconstructor(val, next) {\nthis.val = (val === undefined ? 0 : val);       // \u7ed3\u70b9\u503c\nthis.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nconstructor(val?: number, next?: ListNode | null) {\nthis.val = val === undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u7ed3\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\n// typedef \u4f5c\u7528\u662f\u4e3a\u4e00\u79cd\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\u4e00\u4e2a\u65b0\u540d\u5b57\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570\uff0c\u521d\u59cb\u5316\u4e00\u4e2a\u65b0\u7ed3\u70b9 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nreturn node;\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode\n{\nint val;         // \u7ed3\u70b9\u503c\nListNode next;   // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\nListNode(int x) => val = x;  //\u6784\u9020\u51fd\u6570\n}\n
     
    /* \u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
     
    // \u94fe\u8868\u7ed3\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u7ed3\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\n}\n};\n}\n
     
    \u5c3e\u7ed3\u70b9\u6307\u5411\u4ec0\u4e48\uff1f \u6211\u4eec\u4e00\u822c\u5c06\u94fe\u8868\u7684\u6700\u540e\u4e00\u4e2a\u7ed3\u70b9\u79f0\u4e3a\u300c\u5c3e\u7ed3\u70b9\u300d\uff0c\u5176\u6307\u5411\u7684\u662f\u300c\u7a7a\u300d\uff0c\u5728 Java / C++ / Python \u4e2d\u5206\u522b\u8bb0\u4e3a null / nullptr / None \u3002\u5728\u4e0d\u5f15\u8d77\u6b67\u4e49\u4e0b\uff0c\u672c\u4e66\u90fd\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u3002
     
    \u94fe\u8868\u521d\u59cb\u5316\u65b9\u6cd5\u3002\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u5f15\u7528\u6307\u5411\u5173\u7cfb\u3002\u5b8c\u6210\u540e\uff0c\u5373\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u9996\u4e2a\u7ed3\u70b9\uff08\u5373\u5934\u7ed3\u70b9\uff09\u51fa\u53d1\uff0c\u8bbf\u95ee\u5176\u4f59\u6240\u6709\u7684\u7ed3\u70b9\u3002
     
    Tip
     
    \u6211\u4eec\u901a\u5e38\u5c06\u5934\u7ed3\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u4f8b\u5982\u5934\u7ed3\u70b9 head \u548c\u94fe\u8868 head \u5b9e\u9645\u4e0a\u662f\u540c\u4e49\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.cpp
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
     linked_list.py
    \"\"\" \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 \"\"\"\n# \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
     linked_list.go
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
     linked_list.js
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.ts
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.c
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
     linked_list.cs
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
     linked_list.swift
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
     linked_list.zig
    // \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u7ed3\u70b9 \nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1. \u00a0 \u94fe\u8868\u4f18\u70b9","text":"
    \u5728\u94fe\u8868\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u7684\u64cd\u4f5c\u6548\u7387\u9ad8\u3002\u6bd4\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u5728\u94fe\u8868\u4e2d\u95f4\u7684\u4e24\u4e2a\u7ed3\u70b9 A , B \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u7ed3\u70b9 P \uff0c\u6211\u4eec\u53ea\u9700\u8981\u6539\u53d8\u4e24\u4e2a\u7ed3\u70b9\u6307\u9488\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u76f8\u6bd4\u6570\u7ec4\u7684\u63d2\u5165\u64cd\u4f5c\u9ad8\u6548\u5f88\u591a\u3002
     
     
     Fig. \u94fe\u8868\u63d2\u5165\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.cpp
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode* n0, ListNode* P) {\nListNode* n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
     linked_list.py
    def insert(n0: ListNode, P: ListNode) -> None:\n\"\"\" \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P \"\"\"\nn1 = n0.next\nP.next = n1\nn0.next = P\n
     linked_list.go
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\nn1 := n0.Next\nP.Next = n1\nn0.Next = P\n}\n
     linked_list.js
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunction insert(n0, P) {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.ts
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.c
    [class]{}-[func]{insertNode}\n
     linked_list.cs
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nvoid insert(ListNode n0, ListNode P)\n{\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
     linked_list.swift
    /* \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\nlet n1 = n0.next\nP.next = n1\nn0.next = P\n}\n
     linked_list.zig
    // \u5728\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u7ed3\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\nvar n1 = n0.?.next;\nP.?.next = n1;\nn0.?.next = P;\n}\n
     
    \u5728\u94fe\u8868\u4e2d\u5220\u9664\u7ed3\u70b9\u4e5f\u5f88\u65b9\u4fbf\uff0c\u53ea\u9700\u8981\u6539\u53d8\u4e00\u4e2a\u7ed3\u70b9\u6307\u9488\u5373\u53ef\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u867d\u7136\u5728\u5b8c\u6210\u5220\u9664\u540e\u7ed3\u70b9 P \u4ecd\u7136\u6307\u5411 n1 \uff0c\u4f46\u5b9e\u9645\u4e0a P \u5df2\u7ecf\u4e0d\u5c5e\u4e8e\u6b64\u94fe\u8868\u4e86\uff0c\u56e0\u4e3a\u904d\u5386\u6b64\u94fe\u8868\u662f\u65e0\u6cd5\u8bbf\u95ee\u5230 P \u7684\u3002
     
     
     Fig. \u94fe\u8868\u5220\u9664\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.cpp
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode* n0) {\nif (n0->next == nullptr)\nreturn;\n// n0 -> P -> n1\nListNode* P = n0->next;\nListNode* n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\ndelete P;\n}\n
     linked_list.py
    def remove(n0: ListNode) -> None:\n\"\"\" \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 \"\"\"\nif not n0.next:\nreturn\n# n0 -> P -> n1\nP = n0.next\nn1 = P.next\nn0.next = n1\n
     linked_list.go
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc removeNode(n0 *ListNode) {\nif n0.Next == nil {\nreturn\n}\n// n0 -> P -> n1\nP := n0.Next\nn1 := P.Next\nn0.Next = n1\n}\n
     linked_list.js
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction remove(n0) {\nif (!n0.next)\nreturn;\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.ts
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction remove(n0: ListNode): void {\nif (!n0.next) {\nreturn;\n}\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.c
    [class]{}-[func]{removeNode}\n
     linked_list.cs
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nvoid remove(ListNode n0)\n{\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
     linked_list.swift
    /* \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc remove(n0: ListNode) {\nif n0.next == nil {\nreturn\n}\n// n0 -> P -> n1\nlet P = n0.next\nlet n1 = P?.next\nn0.next = n1\nP?.next = nil\n}\n
     linked_list.zig
    // \u5220\u9664\u94fe\u8868\u7684\u7ed3\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u7ed3\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\nif (n0.?.next == null) return;\n// n0 -> P -> n1\nvar P = n0.?.next;\nvar n1 = P.?.next;\nn0.?.next = n1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#422","title":"4.2.2. \u00a0 \u94fe\u8868\u7f3a\u70b9","text":"
    \u94fe\u8868\u8bbf\u95ee\u7ed3\u70b9\u6548\u7387\u4f4e\u3002\u4e0a\u8282\u63d0\u5230\uff0c\u6570\u7ec4\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u4efb\u610f\u5143\u7d20\uff0c\u4f46\u94fe\u8868\u65e0\u6cd5\u76f4\u63a5\u8bbf\u95ee\u4efb\u610f\u7ed3\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u8ba1\u7b97\u673a\u9700\u8981\u4ece\u5934\u7ed3\u70b9\u51fa\u53d1\uff0c\u4e00\u4e2a\u4e00\u4e2a\u5730\u5411\u540e\u904d\u5386\u5230\u76ee\u6807\u7ed3\u70b9\u3002\u4f8b\u5982\uff0c\u5018\u82e5\u60f3\u8981\u8bbf\u95ee\u94fe\u8868\u7d22\u5f15\u4e3a index \uff08\u5373\u7b2c index + 1 \u4e2a\uff09\u7684\u7ed3\u70b9\uff0c\u90a3\u4e48\u9700\u8981 index \u6b21\u8bbf\u95ee\u64cd\u4f5c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.cpp
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode* access(ListNode* head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == nullptr)\nreturn nullptr;\nhead = head->next;\n}\nreturn head;\n}\n
     linked_list.py
    def access(head: ListNode, index: int) -> Optional[ListNode]:\n\"\"\" \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 \"\"\"\nfor _ in range(index):\nif not head:\nreturn None\nhead = head.next\nreturn head\n
     linked_list.go
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\nfor i := 0; i < index; i++ {\nif head == nil {\nreturn nil\n}\nhead = head.Next\n}\nreturn head\n}\n
     linked_list.js
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunction access(head, index) {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.ts
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.c
    [class]{}-[func]{access}\n
     linked_list.cs
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nListNode? access(ListNode head, int index)\n{\nfor (int i = 0; i < index; i++)\n{\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
     linked_list.swift
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\nvar head: ListNode? = head\nfor _ in 0 ..< index {\nif head == nil {\nreturn nil\n}\nhead = head?.next\n}\nreturn head\n}\n
     linked_list.zig
    // \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u7ed3\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\nvar head = node;\nvar i: i32 = 0;\nwhile (i < index) : (i += 1) {\nhead = head.?.next;\nif (head == null) return null;\n}\nreturn head;\n}\n
     
    \u94fe\u8868\u7684\u5185\u5b58\u5360\u7528\u591a\u3002\u94fe\u8868\u4ee5\u7ed3\u70b9\u4e3a\u5355\u4f4d\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u9664\u4e86\u4fdd\u5b58\u503c\u5916\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u8fd9\u610f\u5473\u7740\u540c\u6837\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u9700\u8981\u5360\u7528\u66f4\u591a\u5185\u5b58\u7a7a\u95f4\u3002
    "},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3. \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u904d\u5386\u94fe\u8868\u67e5\u627e\u3002\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u94fe\u8868\u5185\u503c\u4e3a target \u7684\u7ed3\u70b9\uff0c\u8f93\u51fa\u7ed3\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linked_list.java
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.cpp
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode* head, int target) {\nint index = 0;\nwhile (head != nullptr) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.py
    def find(head: ListNode, target: int) -> int:\n\"\"\" \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 \"\"\"\nindex = 0\nwhile head:\nif head.val == target:\nreturn index\nhead = head.next\nindex += 1\nreturn -1\n
     linked_list.go
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc findNode(head *ListNode, target int) int {\nindex := 0\nfor head != nil {\nif head.Val == target {\nreturn index\n}\nhead = head.Next\nindex++\n}\nreturn -1\n}\n
     linked_list.js
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction find(head, target) {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
     linked_list.ts
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
     linked_list.c
    [class]{}-[func]{findNode}\n
     linked_list.cs
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nint find(ListNode head, int target)\n{\nint index = 0;\nwhile (head != null)\n{\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
     linked_list.swift
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\nvar head: ListNode? = head\nvar index = 0\nwhile head != nil {\nif head?.val == target {\nreturn index\n}\nhead = head?.next\nindex += 1\n}\nreturn -1\n}\n
     linked_list.zig
    // \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u7ed3\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\nvar head = node;\nvar index: i32 = 0;\nwhile (head != null) {\nif (head.?.val == target) return index;\nhead = head.?.next;\nindex += 1;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4. \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"
    \u5355\u5411\u94fe\u8868\u3002\u5373\u4e0a\u8ff0\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u7ed3\u70b9\u6709\u300c\u503c\u300d\u548c\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u300c\u6307\u9488\uff08\u5f15\u7528\uff09\u300d\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u7ed3\u70b9\u79f0\u4e3a\u5934\u7ed3\u70b9\uff0c\u5c3e\u7ed3\u70b9\u6307\u5411 null \u3002
     
    \u73af\u5f62\u94fe\u8868\u3002\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u7ed3\u70b9\u6307\u5411\u5934\u7ed3\u70b9\uff08\u5373\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u7ed3\u70b9\u770b\u4f5c\u662f\u5934\u7ed3\u70b9\u3002
     
    \u53cc\u5411\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u4ec5\u8bb0\u5f55\u4e86\u4e00\u4e2a\u65b9\u5411\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\uff0c\u5728\u53cc\u5411\u94fe\u8868\u7684\u7ed3\u70b9\u5b9a\u4e49\u4e2d\uff0c\u540c\u65f6\u6709\u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\uff08\u540e\u7ee7\u7ed3\u70b9\uff09\u548c\u4e0a\u4e00\u7ed3\u70b9\uff08\u524d\u9a71\u7ed3\u70b9\uff09\u7684\u300c\u6307\u9488\uff08\u5f15\u7528\uff09\u300d\u3002\u53cc\u5411\u94fe\u8868\u76f8\u5bf9\u4e8e\u5355\u5411\u94fe\u8868\u66f4\u52a0\u7075\u6d3b\uff0c\u5373\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u7ed3\u70b9\u503c\nListNode *next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode *prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
     
    \"\"\" \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b \"\"\" \nclass ListNode:\ndef __init__(self, val: int):\nself.val: int = val                   # \u7ed3\u70b9\u503c\nself.next: Optional[ListNode] = None  # \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nself.prev: Optional[ListNode] = None  # \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\nVal  int             // \u7ed3\u70b9\u503c\nNext *DoublyListNode // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nPrev *DoublyListNode // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\nreturn &DoublyListNode{\nVal:  val,\nNext: nil,\nPrev: nil,\n}\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nprev;\nconstructor(val, next) {\nthis.val = val  ===  undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nprev: ListNode | null;\nconstructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\nthis.val = val  ===  undefined ? 0 : val;        // \u7ed3\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
     
    \n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u7ed3\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) => val = x;  // \u6784\u9020\u51fd\u6570\n}\n
     
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nvar prev: ListNode? // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
     
    // \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u7ed3\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nprev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n
     
     
     Fig. \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b 
    "},{"location":"chapter_array_and_linkedlist/list/","title":"4.3. \u00a0 \u5217\u8868","text":"
    \u7531\u4e8e\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u6570\u7ec4\u7684\u5b9e\u7528\u6027\u5927\u5927\u964d\u4f4e\u3002\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e8b\u5148\u5e76\u4e0d\u77e5\u9053\u4f1a\u8f93\u5165\u591a\u5c11\u6570\u636e\uff0c\u8fd9\u5c31\u4e3a\u6570\u7ec4\u957f\u5ea6\u7684\u9009\u62e9\u5e26\u6765\u4e86\u5f88\u5927\u56f0\u96be\u3002\u957f\u5ea6\u9009\u5c0f\u4e86\uff0c\u9700\u8981\u5728\u6dfb\u52a0\u6570\u636e\u4e2d\u9891\u7e41\u5730\u6269\u5bb9\u6570\u7ec4\uff1b\u957f\u5ea6\u9009\u5927\u4e86\uff0c\u53c8\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
     
    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u88ab\u79f0\u4e3a\u300c\u5217\u8868 List\u300d\u7684\u6570\u636e\u7ed3\u6784\u3002\u5217\u8868\u53ef\u4ee5\u88ab\u7406\u89e3\u4e3a\u957f\u5ea6\u53ef\u53d8\u7684\u6570\u7ec4\uff0c\u56e0\u6b64\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u7ec4 Dynamic Array\u300d\u3002\u5217\u8868\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\uff0c\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u4f18\u70b9\uff0c\u540c\u65f6\u8fd8\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u5b9e\u65f6\u6269\u5bb9\u3002\u5728\u5217\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u4e0d\u7528\u62c5\u5fc3\u8d85\u8fc7\u5bb9\u91cf\u9650\u5236\u3002
    "},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1. \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u521d\u59cb\u5316\u5217\u8868\u3002\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u5230\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> list1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> list = new ArrayList<>(Arrays.asList(numbers));\n
     list.cpp
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 list\n// \u65e0\u521d\u59cb\u503c\nvector<int> list1;\n// \u6709\u521d\u59cb\u503c\nvector<int> list = { 1, 3, 2, 5, 4 };\n
     list.py
    \"\"\" \u521d\u59cb\u5316\u5217\u8868 \"\"\"\n# \u65e0\u521d\u59cb\u503c\nlist1: List[int] = []\n# \u6709\u521d\u59cb\u503c\nlist: List[int] = [1, 3, 2, 5, 4]\n
     list_test.go
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlist1 := []int\n// \u6709\u521d\u59cb\u503c\nlist := []int{1, 3, 2, 5, 4}\n
     list.js
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1 = [];\n// \u6709\u521d\u59cb\u503c\nconst list = [1, 3, 2, 5, 4];\n
     list.ts
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst list: number[] = [1, 3, 2, 5, 4];\n
     list.c
    \n
     list.cs
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = new ();\n// \u6709\u521d\u59cb\u503c\nint[] numbers = new int[] { 1, 3, 2, 5, 4 };\nList<int> list = numbers.ToList();\n
     list.swift
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet list1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar list = [1, 3, 2, 5, 4]\n
     list.zig
    // \u521d\u59cb\u5316\u5217\u8868\nvar list = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list.deinit();\ntry list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n
     
    \u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\u3002\u5217\u8868\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.cpp
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.py
    \"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\nnum: int = list[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\"\"\" \u66f4\u65b0\u5143\u7d20 \"\"\"\nlist[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list_test.go
    /* \u8bbf\u95ee\u5143\u7d20 */\nnum := list[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.js
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.ts
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.c
    \n
     list.cs
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.swift
    /* \u8bbf\u95ee\u5143\u7d20 */\nlet num = list[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
     list.zig
    // \u8bbf\u95ee\u5143\u7d20\nvar num = list.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n// \u66f4\u65b0\u5143\u7d20\nlist.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n
     
    \u5728\u5217\u8868\u4e2d\u6dfb\u52a0\u3001\u63d2\u5165\u3001\u5220\u9664\u5143\u7d20\u3002\u76f8\u5bf9\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u662f\u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u4e00\u6837\u4f4e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.cpp
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push_back(1);\nlist.push_back(3);\nlist.push_back(2);\nlist.push_back(5);\nlist.push_back(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(list.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.erase(list.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.py
    \"\"\" \u6e05\u7a7a\u5217\u8868 \"\"\"\nlist.clear()\n\"\"\" \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 \"\"\"\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n\"\"\" \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 \"\"\"\nlist.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\"\"\" \u5220\u9664\u5143\u7d20 \"\"\"\nlist.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list_test.go
    /* \u6e05\u7a7a\u5217\u8868 */\nlist = nil\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist = append(list, 1)\nlist = append(list, 3)\nlist = append(list, 2)\nlist = append(list, 5)\nlist = append(list, 4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist = append(list[:3], append([]int{6}, list[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist = append(list[:3], list[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.js
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
     list.ts
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
     list.c
    \n
     list.cs
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.Clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.Add(1);\nlist.Add(3);\nlist.Add(2);\nlist.Add(5);\nlist.Add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.Insert(3, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.RemoveAt(3);\n
     list.swift
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.removeAll()\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     list.zig
    // \u6e05\u7a7a\u5217\u8868\nlist.clearRetainingCapacity();\n// \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry list.append(1);\ntry list.append(3);\ntry list.append(2);\ntry list.append(5);\ntry list.append(4);\n// \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry list.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n// \u5220\u9664\u5143\u7d20\n_ = list.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
     
    \u904d\u5386\u5217\u8868\u3002\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u4f7f\u7528\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528 for-each \u76f4\u63a5\u904d\u5386\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
     list.cpp
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
     list.py
    \"\"\" \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 \"\"\"\ncount: int = 0\nfor i in range(len(list)):\ncount += 1\n\"\"\" \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 \"\"\"\ncount: int = 0\nfor n in list:\ncount += 1\n
     list_test.go
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(list); i++ {\ncount++\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor range list {\ncount++\n}\n
     list.js
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
     list.ts
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
     list.c
    \n
     list.cs
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.Count(); i++)\n{\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int n in list)\n{\ncount++;\n}\n
     list.swift
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor _ in list.indices {\ncount += 1\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _ in list {\ncount += 1\n}\n
     list.zig
    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < list.items.len) : (i += 1) {\ncount += 1;\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (list.items) |_| {\ncount += 1;\n}\n
     
    \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\u3002\u518d\u521b\u5efa\u4e00\u4e2a\u65b0\u5217\u8868 list1 \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u4e2d\u4e00\u4e2a\u5217\u8868\u62fc\u63a5\u5230\u53e6\u4e00\u4e2a\u7684\u5c3e\u90e8\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.cpp
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> list1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\nlist.insert(list.end(), list1.begin(), list1.end());\n
     list.py
    \"\"\" \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 \"\"\"\nlist1: List[int] = [6, 8, 7, 10, 9]\nlist += list1  # \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list_test.go
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlist1 := []int{6, 8, 7, 10, 9}\nlist = append(list, list1...)  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.js
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1 = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.ts
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1: number[] = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.c
    \n
     list.cs
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = new() { 6, 8, 7, 10, 9 };\nlist.AddRange(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.swift
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet list1 = [6, 8, 7, 10, 9]\nlist.append(contentsOf: list1) // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     list.zig
    // \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar list1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list1.deinit();\ntry list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry list.insertSlice(list.items.len, list1.items); // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
     
    \u6392\u5e8f\u5217\u8868\u3002\u6392\u5e8f\u4e5f\u662f\u5e38\u7528\u7684\u65b9\u6cd5\u4e4b\u4e00\uff0c\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u5bdf\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u53cc\u6307\u9488\u300d\u7b97\u6cd5\u4e86\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig list.java
    /* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(list);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.cpp
    /* \u6392\u5e8f\u5217\u8868 */\nsort(list.begin(), list.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.py
    \"\"\" \u6392\u5e8f\u5217\u8868 \"\"\"\nlist.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list_test.go
    /* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(list)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.js
    /* \u6392\u5e8f\u5217\u8868 */  list.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.ts
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.c
    \n
     list.cs
    /* \u6392\u5e8f\u5217\u8868 */\nlist.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.swift
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
     list.zig
    // \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32));\n
    "},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2. \u00a0 \u5217\u8868\u7b80\u6613\u5b9e\u73b0 *","text":"
    \u4e3a\u4e86\u5e2e\u52a9\u52a0\u6df1\u5bf9\u5217\u8868\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5728\u6b64\u63d0\u4f9b\u4e00\u4e2a\u5217\u8868\u7684\u7b80\u6613\u7248\u672c\u7684\u5b9e\u73b0\u3002\u9700\u8981\u5173\u6ce8\u4e09\u4e2a\u6838\u5fc3\u70b9\uff1a
     
       
    • \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u7684\u521d\u59cb\u5bb9\u91cf initialCapacity \u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
      •  
    • \u6570\u91cf\u8bb0\u5f55\uff1a\u9700\u8981\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf size \uff0c\u7528\u6765\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u6709\u591a\u5c11\u4e2a\u5143\u7d20\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u4e0e\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u7684\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002
      •  
    • \u6269\u5bb9\u673a\u5236\uff1a\u63d2\u5165\u5143\u7d20\u6709\u53ef\u80fd\u5bfc\u81f4\u8d85\u51fa\u5217\u8868\u5bb9\u91cf\uff0c\u6b64\u65f6\u9700\u8981\u6269\u5bb9\u5217\u8868\uff0c\u65b9\u6cd5\u662f\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\u6765\u66ff\u6362\u5f53\u524d\u6570\u7ec4\u3002\u9700\u8981\u7ed9\u5b9a\u4e00\u4e2a\u6269\u5bb9\u500d\u6570 extendRatio \uff0c\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002
      •  
     
    \u672c\u793a\u4f8b\u662f\u4e3a\u4e86\u5e2e\u52a9\u8bfb\u8005\u5bf9\u5982\u4f55\u5b9e\u73b0\u5217\u8868\u4ea7\u751f\u76f4\u89c2\u7684\u8ba4\u8bc6\u3002\u5b9e\u9645\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u5217\u8868\u7684\u5b9e\u73b0\u8fdc\u6bd4\u4ee5\u4e0b\u4ee3\u7801\u590d\u6742\u4e14\u6807\u51c6\uff0c\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u5b66\u4e60\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_list.java
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int capacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int size = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[capacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size() {\nreturn size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\nnums[size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = Arrays.copyOf(nums, capacity() * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\ncapacity = nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\nint size = size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[size];\nfor (int i = 0; i < size; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
     my_list.cpp
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate:\nint* nums;                // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nint numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint extendRatio = 2;      // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\nnums = new int[numsCapacity];\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~MyList() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\nnums[size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size() - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size() - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nint newCapacity = capacity() * extendRatio;\nint* tmp = nums;\nnums = new int[newCapacity];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size(); i++) {\nnums[i] = tmp[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] tmp;\nnumsCapacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> vec(size());\nfor (int i = 0; i < size(); i++) {\nvec[i] = nums[i];\n}\nreturn vec;\n}\n};\n
     my_list.py
    class MyList:\n\"\"\" \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__capacity: int = 10                       # \u5217\u8868\u5bb9\u91cf\nself.__nums: List[int] = [0] * self.__capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nself.__size: int = 0                            # \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nself.__extend_ratio: int = 2                    # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 \"\"\"\nreturn self.__size\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u5217\u8868\u5bb9\u91cf \"\"\"\nreturn self.__capacity\ndef get(self, index: int) -> int:\n\"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\n# \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nreturn self.__nums[index]\ndef set(self, num: int, index: int) -> None:\n\"\"\" \u66f4\u65b0\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nself.__nums[index] = num\ndef add(self, num: int) -> None:\n\"\"\" \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 \"\"\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size() == self.capacity():\nself.extend_capacity()\nself.__nums[self.__size] = num\nself.__size += 1\ndef insert(self, num: int, index: int) -> None:\n\"\"\" \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.__size == self.capacity():\nself.extend_capacity()\n# \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in range(self.__size - 1, index - 1, -1):\nself.__nums[j + 1] = self.__nums[j]\nself.__nums[index] = num\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size += 1\ndef remove(self, index: int) -> int:\n\"\"\" \u5220\u9664\u5143\u7d20 \"\"\"\nassert index >= 0 and index < self.__size, \"\u7d22\u5f15\u8d8a\u754c\"\nnum = self.__nums[index]\n# \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in range(index, self.__size - 1):\nself.__nums[j] = self.__nums[j + 1]\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size -= 1\n# \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\ndef extend_capacity(self) -> None:\n\"\"\" \u5217\u8868\u6269\u5bb9 \"\"\"\n# \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a self.__size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nself.__nums = self.__nums + [0] * self.capacity() * (self.__extend_ratio - 1)\n# \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.__capacity = len(self.__nums)\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 \"\"\"\nreturn self.__nums[:self.__size]\n
     my_list.go
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\ntype myList struct {\nnumsCapacity int\nnums         []int\nnumsSize     int\nextendRatio  int\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newMyList() *myList {\nreturn &myList{\nnumsCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\nnums:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio:  2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\nreturn l.numsSize\n}\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\nreturn l.numsCapacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn l.nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nl.nums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\nl.nums[l.numsSize] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j := l.numsSize - 1; j >= index; j-- {\nl.nums[j+1] = l.nums[j]\n}\nl.nums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnum := l.nums[index]\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j := index; j < l.numsSize-1; j++ {\nl.nums[j] = l.nums[j+1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize--\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a self.__size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nl.nums = append(l.nums, make([]int, l.numsCapacity*(l.extendRatio-1))...)\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nl.numsCapacity = len(l.nums)\n}\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nreturn l.nums[:l.numsSize]\n}\n
     my_list.js
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\n#nums = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n#capacity = 10; // \u5217\u8868\u5bb9\u91cf\n#size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n#extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#nums = new Array(this.#capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nsize() {\nreturn this.#size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\ncapacity() {\nreturn this.#capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nget(index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.#nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nset(index, num) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.#nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nadd(num) {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.#nums[this.#size] = num;\nthis.#size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\ninsert(index, num) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this.#size - 1; j >= index; j--) {\nthis.#nums[j + 1] = this.#nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#nums[index] = num;\nthis.#size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nremove(index) {\nif (index < 0 || index >= this.#size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.#nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this.#size - 1; j++) {\nthis.#nums[j] = this.#nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nextendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.#nums = this.#nums.concat(\nnew Array(this.capacity() * (this.#extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis.#capacity = this.#nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\ntoArray() {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
     my_list.ts
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate nums: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\nprivate _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.nums = new Array(this._capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic size(): number {\nreturn this._size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic capacity(): number {\nreturn this._capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic get(index: number): number {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic set(index: number, num: number): void {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic add(num: number): void {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this._size === this._capacity)\nthis.extendCapacity();\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.nums[this._size] = num;\nthis._size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic insert(index: number, num: number): void {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this._size === this._capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this._size - 1; j >= index; j--) {\nthis.nums[j + 1] = this.nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.nums[index] = num;\nthis._size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic remove(index: number): number {\nif (index < 0 || index >= this._size)\nthrow new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this._size - 1; j++) {\nthis.nums[j] = this.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis._size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic extendCapacity(): void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.nums = this.nums.concat(\nnew Array(this.capacity() * (this.extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis._capacity = this.nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic toArray(): number[] {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
     my_list.c
    [class]{myList}-[func]{}\n
     my_list.cs
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList\n{\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList()\n{\nnums = new int[numsCapacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size()\n{\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity()\n{\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index)\n{\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num)\n{\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\nnums[numsSize] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = numsSize - 1; j >= index; j--)\n{\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index)\n{\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < numsSize - 1; j++)\n{\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity()\n{\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a numsCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nSystem.Array.Resize(ref nums, numsCapacity * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nnumsCapacity = nums.Length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray()\n{\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[numsSize];\nfor (int i = 0; i < numsSize; i++)\n{\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
     my_list.swift
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate var nums: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\nprivate var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nnums = Array(repeating: 0, count: _capacity)\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nfunc size() -> Int {\n_size\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc capacity() -> Int {\n_capacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc get(index: Int) -> Int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc set(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc add(num: Int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\nnums[_size] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc insert(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\nnums[j + 1] = nums[j]\n}\nnums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5220\u9664\u5143\u7d20 */\n@discardableResult\nfunc remove(index: Int) -> Int {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nlet num = nums[index]\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in index ..< (_size - 1) {\nnums[j] = nums[j + 1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size -= 1\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = nums + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = nums.count\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nfunc toArray() -> [Int] {\nvar nums = Array(repeating: 0, count: _size)\nfor i in 0 ..< _size {\nnums[i] = get(index: i)\n}\nreturn nums\n}\n}\n
     my_list.zig
    // \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\nfn MyList(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnums_capacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\nnum_size: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextend_ratio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.nums = try self.mem_allocator.alloc(T, self.nums_capacity);\nstd.mem.set(T, self.nums, @as(T, 0));\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\npub fn size(self: *Self) usize {\nreturn self.num_size;\n}\n// \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.nums_capacity;\n}\n// \u8bbf\u95ee\u5143\u7d20\npub fn get(self: *Self, index: usize) T {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn self.nums[index];\n}  // \u66f4\u65b0\u5143\u7d20\npub fn set(self: *Self, index: usize, num: T) void {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nself.nums[index] = num;\n}  // \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\npub fn add(self: *Self, num: T) !void {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\nself.nums[self.size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size += 1;\n}  // \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\npub fn insert(self: *Self, index: usize, num: T) !void {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\n// \u7d22\u5f15 i \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar j = self.size() - 1;\nwhile (j >= index) : (j -= 1) {\nself.nums[j + 1] = self.nums[j];\n}\nself.nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size += 1;\n}\n// \u5220\u9664\u5143\u7d20\npub fn remove(self: *Self, index: usize) T {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nvar num = self.nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar j = index;\nwhile (j < self.size() - 1) : (j += 1) {\nself.nums[j] = self.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.num_size -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n// \u5217\u8868\u6269\u5bb9\npub fn extendCapacity(self: *Self) !void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extend_ratio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nvar newCapacity = self.capacity() * self.extend_ratio;\nvar extend = try self.mem_allocator.alloc(T, newCapacity);\nstd.mem.set(T, extend, @as(T, 0));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(T, extend, self.nums);\nself.nums = extend;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.nums_capacity = newCapacity;\n}\n// \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar nums = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, nums, @as(T, 0));\nfor (nums) |*num, i| {\nnum.* = self.get(i);\n}\nreturn nums;\n}\n};\n}\n
    "},{"location":"chapter_array_and_linkedlist/summary/","title":"4.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff0c\u5373\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u4f18\u70b9\u4e0e\u7f3a\u70b9\u5448\u73b0\u51fa\u6b64\u6d88\u5f7c\u957f\u7684\u5173\u7cfb\u3002
      •  
    • \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5185\u5b58\u7a7a\u95f4\u5360\u7528\u5c0f\uff1b\u4f46\u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
      •  
    • \u94fe\u8868\u53ef\u901a\u8fc7\u66f4\u6539\u6307\u9488\u5b9e\u73b0\u9ad8\u6548\u7684\u7ed3\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u5730\u4fee\u6539\u957f\u5ea6\uff1b\u4f46\u7ed3\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u6709\u5355\u5411\u94fe\u8868\u3001\u5faa\u73af\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002
      •  
    • \u5217\u8868\u53c8\u79f0\u52a8\u6001\u6570\u7ec4\uff0c\u662f\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4fdd\u5b58\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u6539\u53d8\u957f\u5ea6\u3002\u5217\u8868\u7684\u51fa\u73b0\u5927\u5927\u63d0\u5347\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
      •  
    • \u4e0b\u8868\u603b\u7ed3\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u9879\u7279\u6027\u3002
      •  
     \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u79bb\u6563\u5185\u5b58\u7a7a\u95f4 \u6570\u636e\u7ed3\u6784\u957f\u5ea6 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u957f\u5ea6\u53ef\u53d8 \u5185\u5b58\u4f7f\u7528\u7387 \u5360\u7528\u5185\u5b58\u5c11\u3001\u7f13\u5b58\u5c40\u90e8\u6027\u597d \u5360\u7528\u5185\u5b58\u591a \u4f18\u52bf\u64cd\u4f5c \u968f\u673a\u8bbf\u95ee \u63d2\u5165\u3001\u5220\u9664 
    \u7f13\u5b58\u5c40\u90e8\u6027\u7684\u7b80\u5355\u89e3\u91ca
     
    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6570\u636e\u8bfb\u5199\u901f\u5ea6\u6392\u5e8f\u662f\u201c\u786c\u76d8 < \u5185\u5b58 < CPU \u7f13\u5b58\u201d\u3002\u5f53\u6211\u4eec\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u5b83\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u8ba1\u7b97\u673a\u53ea\u80fd\u6328\u4e2a\u5730\u7f13\u5b58\u5404\u4e2a\u7ed3\u70b9\uff0c\u8fd9\u6837\u7684\u591a\u6b21\u201c\u642c\u8fd0\u201d\u964d\u4f4e\u4e86\u6574\u4f53\u6548\u7387\u3002
     
       
    • \u4e0b\u8868\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u79cd\u64cd\u4f5c\u6548\u7387\u3002
      •  
     \u64cd\u4f5c \u6570\u7ec4 \u94fe\u8868 \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(N)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(N)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(N)\\) \\(O(1)\\)"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1. \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1. \u00a0 \u7b97\u6cd5\u8bc4\u4ef7\u7ef4\u5ea6","text":"
    \u5728\u5f00\u59cb\u5b66\u4e60\u7b97\u6cd5\u4e4b\u524d\uff0c\u6211\u4eec\u9996\u5148\u8981\u60f3\u6e05\u695a\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u662f\u4ec0\u4e48\uff0c\u6216\u8005\u8bf4\uff0c\u5982\u4f55\u6765\u8bc4\u5224\u7b97\u6cd5\u7684\u597d\u4e0e\u574f\u3002\u6574\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u8bbe\u8ba1\u7b97\u6cd5\u65f6\u8ffd\u6c42\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002
     
       
    1. \u627e\u5230\u95ee\u9898\u89e3\u6cd5\u3002\u7b97\u6cd5\u9700\u8981\u80fd\u591f\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u4e0b\uff0c\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
      2.  
    2. \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\u3002\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u800c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u6548\u7387\u5c3d\u53ef\u80fd\u7684\u9ad8\u3002
      3.  
     
    \u6362\u8a00\u4e4b\uff0c\u5728\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5219\u662f\u4e3b\u8981\u8bc4\u4ef7\u7ef4\u5ea6\uff0c\u5305\u62ec\uff1a
     
       
    • \u65f6\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u7684\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
      •  
    • \u7a7a\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u3002
      •  
     
    \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8ffd\u6c42\u201c\u8fd0\u884c\u901f\u5ea6\u5feb\u3001\u5360\u7528\u5185\u5b58\u5c11\u201d\uff0c\u800c\u5982\u4f55\u53bb\u8bc4\u4ef7\u7b97\u6cd5\u6548\u7387\u5219\u662f\u975e\u5e38\u91cd\u8981\u7684\u95ee\u9898\uff0c\u56e0\u4e3a\u53ea\u6709\u77e5\u9053\u5982\u4f55\u8bc4\u4ef7\u7b97\u6cd5\uff0c\u624d\u80fd\u53bb\u505a\u7b97\u6cd5\u4e4b\u95f4\u7684\u5bf9\u6bd4\u5206\u6790\uff0c\u4ee5\u53ca\u4f18\u5316\u7b97\u6cd5\u8bbe\u8ba1\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2. \u00a0 \u6548\u7387\u8bc4\u4f30\u65b9\u6cd5","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#_1","title":"\u5b9e\u9645\u6d4b\u8bd5","text":"
    \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 A \u548c \u7b97\u6cd5 B \uff0c\u90fd\u80fd\u591f\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u4e24\u4e2a\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002\u6211\u4eec\u80fd\u591f\u60f3\u5230\u7684\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\uff0c\u5c31\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u628a\u4e24\u4e2a\u7b97\u6cd5\u90fd\u5b8c\u6574\u8dd1\u4e00\u904d\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u662f\u4e5f\u5b58\u5728\u5f88\u5927\u7684\u786c\u4f24\u3002
     
    \u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u5230\u7b97\u6cd5\u7684\u6027\u80fd\u8868\u73b0\u3002\u4f8b\u5982\uff0c\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 A \u6bd4\u7b97\u6cd5 B \u8fd0\u884c\u65f6\u95f4\u66f4\u77ed\uff1b\u4f46\u6362\u5230\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u4f1a\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u5c55\u5f00\u6d4b\u8bd5\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002
     
    \u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u5927\u5c0f\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u5448\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u8868\u73b0\u3002\u6bd4\u5982\uff0c\u6709\u53ef\u80fd\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 A \u8fd0\u884c\u65f6\u95f4\u77ed\u4e8e\u7b97\u6cd5 B \uff0c\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u622a\u7136\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u82e5\u60f3\u8981\u8fbe\u5230\u5177\u6709\u8bf4\u670d\u529b\u7684\u5bf9\u6bd4\u7ed3\u679c\uff0c\u90a3\u4e48\u9700\u8981\u8f93\u5165\u5404\u79cd\u4f53\u91cf\u6570\u636e\uff0c\u8fd9\u6837\u7684\u6d4b\u8bd5\u9700\u8981\u5360\u7528\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#_2","title":"\u7406\u8bba\u4f30\u7b97","text":"
    \u65e2\u7136\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u5f88\u5927\u7684\u5c40\u9650\u6027\uff0c\u90a3\u4e48\u6211\u4eec\u662f\u5426\u53ef\u4ee5\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\uff0c\u5c31\u83b7\u77e5\u7b97\u6cd5\u7684\u6548\u7387\u6c34\u5e73\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\uff0c\u6211\u4eec\u5c06\u6b64\u4f30\u7b97\u65b9\u6cd5\u79f0\u4e3a\u300c\u590d\u6742\u5ea6\u5206\u6790 Complexity Analysis\u300d\u6216\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 Asymptotic Complexity Analysis\u300d\u3002
     
    \u590d\u6742\u5ea6\u5206\u6790\u8bc4\u4f30\u7684\u662f\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u53e5\u8bdd\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\uff1a
     
       
    • \u201c\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u201d\u53ef\u5206\u4e3a\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u548c\u201c\u5360\u7528\u7a7a\u95f4\u201d\uff0c\u8fdb\u800c\u53ef\u5c06\u590d\u6742\u5ea6\u5206\u4e3a\u300c\u65f6\u95f4\u590d\u6742\u5ea6 Time Complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u3002
      •  
    • \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u201d\u4ee3\u8868\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u6709\u5173\uff0c\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff1b
      •  
    • \u201c\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u4e0d\u5173\u5fc3\u7b97\u6cd5\u5177\u4f53\u4f7f\u7528\u4e86\u591a\u5c11\u65f6\u95f4\u6216\u5360\u7528\u4e86\u591a\u5c11\u7a7a\u95f4\uff0c\u800c\u662f\u7ed9\u51fa\u4e00\u79cd\u201c\u8d8b\u52bf\u6027\u5206\u6790\u201d\uff1b
      •  
     
    \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\u3002\u4e00\u662f\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002\u4e8c\u662f\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u53ef\u4ee5\u53cd\u6620\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002
     
    \u5982\u679c\u611f\u89c9\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u7684\u6982\u5ff5\u4e00\u77e5\u534a\u89e3\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u540e\u7eed\u7ae0\u8282\u4f1a\u5c55\u5f00\u4ecb\u7ecd\u3002
    "},{"location":"chapter_computational_complexity/performance_evaluation/#213","title":"2.1.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790\u91cd\u8981\u6027","text":"
    \u590d\u6742\u5ea6\u5206\u6790\u7ed9\u51fa\u4e00\u628a\u8bc4\u4ef7\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u544a\u8bc9\u6211\u4eec\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u9700\u8981\u591a\u5c11\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u4e5f\u8ba9\u6211\u4eec\u53ef\u4ee5\u5f00\u5c55\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u5bf9\u6bd4\u3002
     
    \u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u51fa\u53d1\uff0c\u5176\u5e76\u4e0d\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u7ae0\u5185\u5bb9\u3002\u4f46\u662f\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u8005\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u9700\u8981\u5206\u6790\u5b83\u7684\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002\u56e0\u6b64\uff0c\u5728\u5c55\u5f00\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5efa\u8bae\u8bfb\u8005\u5148\u5bf9\u590d\u6742\u5ea6\u5efa\u7acb\u8d77\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u5e76\u4e14\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u6848\u4f8b\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/","title":"2.3. \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"
    \u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u7edf\u8ba1 \u7b97\u6cd5\u4f7f\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5f88\u7c7b\u4f3c\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/#231","title":"2.3.1. \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"
    \u7b97\u6cd5\u8fd0\u884c\u4e2d\uff0c\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u6709\u4ee5\u4e0b\u51e0\u79cd\uff1a
     
       
    • \u300c\u8f93\u5165\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\uff1b
      •  
    • \u300c\u6682\u5b58\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\uff1b
      •  
    • \u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\uff1b
      •  
     
    Tip
     
    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u8303\u56f4\u662f\u300c\u6682\u5b58\u7a7a\u95f4\u300d+\u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u3002
     
    \u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a
     
       
    • \u300c\u6682\u5b58\u6570\u636e\u300d\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5404\u79cd \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61 \u7b49\u3002
      •  
    • \u300c\u6808\u5e27\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u90fd\u4f1a\u5728\u6808\u7684\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u65f6\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
      •  
    • \u300c\u6307\u4ee4\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u4e00\u822c\u5ffd\u7565\u4e0d\u8ba1\u3002
      •  
     
     
     Fig. \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nfinal int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7ed3\u6784\u4f53 */\nstruct Node {\nint val;\nNode *next;\nNode(int x) : val(x), next(nullptr) {}\n};\n/* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    \"\"\" \u7c7b \"\"\"\nclass Node:\ndef __init__(self, x: int):\nself.val: int = x                 # \u7ed3\u70b9\u503c\nself.next: Optional[Node] = None  # \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n\"\"\" \u51fd\u6570 \"\"\"\ndef function() -> int:\n# do something...\nreturn 0\ndef algorithm(n) -> int:  # \u8f93\u5165\u6570\u636e\nA: int = 0            # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\nb: int = 0            # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnode = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc: int = function()   # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn A + b + c      # \u8f93\u51fa\u6570\u636e\n
     
    /* \u7ed3\u6784\u4f53 */\ntype node struct {\nval  int\nnext *node\n}\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\nreturn &node{val: val}\n}\n/* \u51fd\u6570 */\nfunc function() int {\n// do something...\nreturn 0\n}\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\nconst a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nb := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnewNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nval;\nnext;\nconstructor(val) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc() {\n// do something\nreturn 0;\n}\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\nconst a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nval: number;\nnext: Node | null;\nconstructor(val?: number) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u7ed3\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n// do something\nreturn 0;\n}\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\nconst a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n
     
    \n
     
    /* \u7c7b */\nclass Node\n{\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function()\n{\n// do something...\nreturn 0;\n}\nint algorithm(int n)          // \u8f93\u5165\u6570\u636e\n{\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
     
    /* \u7c7b */\nclass Node {\nvar val: Int\nvar next: Node?\ninit(x: Int) {\nval = x\n}\n}\n/* \u51fd\u6570 */\nfunc function() -> Int {\n// do something...\nreturn 0\n}\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\nlet a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nvar b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/space_complexity/#232","title":"2.3.2. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"
    \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u548c\u65f6\u95f4\u590d\u6742\u5ea6\u603b\u4f53\u7c7b\u4f3c\uff0c\u53ea\u662f\u4ece\u7edf\u8ba1\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u201d\u53d8\u4e3a\u7edf\u8ba1\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u4e00\u822c\u53ea\u5173\u6ce8\u300c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u300d\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u4e2a\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u4fdd\u8bc1\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002
     
    \u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\uff0c\u5206\u522b\u4e3a\u8f93\u5165\u6570\u636e\u7684\u6700\u5dee\u5206\u5e03\u3001\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u6700\u5dee\u65f6\u95f4\u70b9\u3002
     
       
    • \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\u3002\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u662f\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 nums \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
      •  
    • \u4ee5\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\u3002\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 nums \u65f6\uff0c\u7a0b\u5e8f\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\nint[] nums = new int[n]; // O(n)\n}\n
     
    void algorithm(int n) {\nint a = 0;               // O(1)\nvector<int> b(10000);    // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
     
    def algorithm(n: int) -> None:\na: int = 0                     # O(1)\nb: List[int] = [0] * 10000     # O(1)\nif n > 10:\nnums: List[int] = [0] * n  # O(n)\n
     
    func algorithm(n int) {\na := 0                      // O(1)\nb := make([]int, 10000)     // O(1)\nvar nums []int\nif n > 10 {\nnums := make([]int, n)  // O(n)\n}\nfmt.Println(a, b, nums)\n}\n
     
    function algorithm(n) {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
     
    function algorithm(n: number): void {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
     
    \n
     
    void algorithm(int n)\n{\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\n{\nint[] nums = new int[n]; // O(n)\n}\n}\n
     
    func algorithm(n: Int) {\nlet a = 0 // O(1)\nlet b = Array(repeating: 0, count: 10000) // O(1)\nif n > 10 {\nlet nums = Array(repeating: 0, count: n) // O(n)\n}\n}\n
     
    \n
     
    \u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u4f8b\u5982\u51fd\u6570 loop()\uff0c\u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 function() \uff0c\u6bcf\u8f6e\u4e2d\u7684 function() \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002\u800c\u9012\u5f52\u51fd\u6570 recur() \u5728\u8fd0\u884c\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 recur() \uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
     
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
     
    def function() -> int:\n# do something\nreturn 0\n\"\"\" \u5faa\u73af O(1) \"\"\"\ndef loop(n: int) -> None:\nfor _ in range(n):\nfunction()\n\"\"\" \u9012\u5f52 O(n) \"\"\"\ndef recur(n: int) -> int:\nif n == 1: return\nreturn recur(n - 1)\n
     
    func function() int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\nfor i := 0; i < n; i++ {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\nif n == 1 {\nreturn\n}\nrecur(n - 1)\n}\n
     
    function constFunc() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
     
    function constFunc(): number {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
     
    \n
     
    int function()\n{\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nint recur(int n)\n{\nif (n == 1) return 1;\nreturn recur(n - 1);\n}\n
     
    @discardableResult\nfunc function() -> Int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\nfor _ in 0 ..< n {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\nif n == 1 {\nreturn\n}\nrecur(n: n - 1)\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/space_complexity/#233","title":"2.3.3. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09
     \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\] 
     
     Fig. \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b 
     
    Tip
     
    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u5148\u805a\u7126\u5728\u7406\u89e3\u7a7a\u95f4\u590d\u6742\u5ea6\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u4e0a\u3002
    "},{"location":"chapter_computational_complexity/space_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"
    \u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002
     
    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u5373\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
     space_complexity.cpp
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nvector<int> nums(10000);\nListNode node(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
     space_complexity.py
    def constant(n: int) -> None:\n\"\"\" \u5e38\u6570\u9636 \"\"\"\n# \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\na: int = 0\nnums: List[int] = [0] * 10000\nnode = ListNode(0)\n# \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nc: int = 0\n# \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nfunction()\n
     space_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0\nb := 0\nnums := make([]int, 10000)\nListNode := newNode(0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nvar c int\nfor i := 0; i < n; i++ {\nc = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i := 0; i < n; i++ {\nfunction()\n}\nfmt.Println(a, b, nums, c, ListNode)\n}\n
     space_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
     space_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceConstant}\n
     space_complexity.cs
    /* \u5e38\u6570\u9636 */\nvoid constant(int n)\n{\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nint a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++)\n{\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n
     space_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nlet a = 0\nvar b = 0\nlet nums = Array(repeating: 0, count: 10000)\nlet node = ListNode(x: 0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nlet c = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nfunction()\n}\n}\n
     space_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a: i32 = 0;\nvar b: i32 = 0;\nvar nums = [_]i32{0}**10000;\nvar node = inc.ListNode(i32){.val = 0};\nvar i: i32 = 0;\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nwhile (i < n) : (i += 1) {\nvar c: i32 = 0;\n_ = c;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\ni = 0;\nwhile (i < n) : (i += 1) {\n_ = function();\n}\n_ = a;\n_ = b;\n_ = nums;\n_ = node;\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"
    \u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nnodes.add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<Integer, String> map = new HashMap<>();\nfor (int i = 0; i < n; i++) {\nmap.put(i, String.valueOf(i));\n}\n}\n
     space_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvector<int> nums(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvector<ListNode> nodes;\nfor (int i = 0; i < n; i++) {\nnodes.push_back(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nunordered_map<int, string> map;\nfor (int i = 0; i < n; i++) {\nmap[i] = to_string(i);\n}\n}\n
     space_complexity.py
    def linear(n: int) -> None:\n\"\"\" \u7ebf\u6027\u9636 \"\"\"\n# \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nnums: List[int] = [0] * n\n# \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nmapp: Dict = {}\nfor i in range(n):\nmapp[i] = str(i)\n
     space_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n_ = make([]int, n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes []*node\nfor i := 0; i < n; i++ {\nnodes = append(nodes, newNode(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nm := make(map[int]string, n)\nfor i := 0; i < n; i++ {\nm[i] = strconv.Itoa(i)\n}\n}\n
     space_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
     space_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes: ListNode[] = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceLinear}\n
     space_complexity.cs
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n)\n{\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new();\nfor (int i = 0; i < n; i++)\n{\nnodes.Add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nDictionary<int, String> map = new();\nfor (int i = 0; i < n; i++)\n{\nmap.Add(i, i.ToString());\n}\n}\n
     space_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet nums = Array(repeating: 0, count: n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet nodes = (0 ..< n).map { ListNode(x: $0) }\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
     space_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvar nums = [_]i32{0}**n;\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ntry nodes.append(i);\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\ndefer map.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\nconst string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\ndefer std.heap.page_allocator.free(string);\ntry map.put(i, string);\n}\n_ = nums;\n}\n
     
    \u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nSystem.out.println(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\ncout << \"\u9012\u5f52 n = \" << n << endl;\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.py
    def linear_recur(n: int) -> None:\n\"\"\" \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 \"\"\"\nprint(\"\u9012\u5f52 n =\", n)\nif n == 1: return\nlinear_recur(n - 1)\n
     space_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\nfmt.Println(\"\u9012\u5f52 n =\", n)\nif n == 1 {\nreturn\n}\nspaceLinearRecur(n - 1)\n}\n
     space_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.c
    [class]{}-[func]{spaceLinearRecur}\n
     space_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n)\n{\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     space_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\nprint(\"\u9012\u5f52 n = \\(n)\")\nif n == 1 {\nreturn\n}\nlinearRecur(n: n - 1)\n}\n
     space_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\nstd.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
     
     
     Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"
    \u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\u7684\u77e9\u9635\u3001\u56fe\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[][] numMatrix = new int[n][n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<Integer>> numList = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<Integer> tmp = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n
     space_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvector<vector<int>> numMatrix;\nfor (int i = 0; i < n; i++) {\nvector<int> tmp;\nfor (int j = 0; j < n; j++) {\ntmp.push_back(0);\n}\nnumMatrix.push_back(tmp);\n}\n}\n
     space_complexity.py
    def quadratic(n: int) -> None:\n\"\"\" \u5e73\u65b9\u9636 \"\"\"\n# \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nnum_matrix: List[List[int]] = [[0] * n for _ in range(n)]\n
     space_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nnumMatrix := make([][]int, n)\nfor i := 0; i < n; i++ {\nnumMatrix[i] = make([]int, n)\n}\n}\n
     space_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n).fill(null).map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
     space_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
     space_complexity.c
    [class]{}-[func]{spaceQuadratic}\n
     space_complexity.cs
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n)\n{\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[,] numMatrix = new int[n, n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = new();\nfor (int i = 0; i < n; i++)\n{\nList<int> tmp = new();\nfor (int j = 0; j < n; j++)\n{\ntmp.Add(0);\n}\nnumList.Add(tmp);\n}\n}\n
     space_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
     space_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvar nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\nvar tmp = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer tmp.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ntry tmp.append(0);\n}\ntry nodes.append(tmp);\n}\n}\n
     
    \u5728\u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \uff0c\u5e76\u4e14\u6bcf\u4e2a\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n, n-1, n-2, ..., 2, 1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nint[] nums = new int[n];\nSystem.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nvector<int> nums(n);\ncout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.py
    def quadratic_recur(n: int) -> int:\n\"\"\" \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 \"\"\"\nif n <= 0: return 0\n# \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nnums: List[int] = [0] * n\nreturn quadratic_recur(n - 1)\n
     space_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\nif n <= 0 {\nreturn 0\n}\nnums := make([]int, n)\nfmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\nreturn spaceQuadraticRecur(n - 1)\n}\n
     space_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.c
    [class]{}-[func]{spaceQuadraticRecur}\n
     space_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n)\n{\nif (n <= 0) return 0;\nint[] nums = new int[n];\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\nreturn quadraticRecur(n - 1);\n}\n
     space_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\nif n <= 0 {\nreturn 0\n}\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = Array(repeating: 0, count: n)\nprint(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\nreturn quadraticRecur(n: n - 1)\n}\n
     space_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\nif (n <= 0) return 0;\nvar nums = [_]i32{0}**n;\nstd.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\nreturn quadraticRecur(n - 1);\n}\n
     
     
     Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"
    \u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u7684\u7ed3\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u4f7f\u7528 \\(O(2^n)\\) \u7a7a\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig space_complexity.java
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode* buildTree(int n) {\nif (n == 0) return nullptr;\nTreeNode* root = new TreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.py
    def build_tree(n: int) -> Optional[TreeNode]:\n\"\"\" \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 \"\"\"\nif n == 0: return None\nroot = TreeNode(0)\nroot.left = build_tree(n - 1)\nroot.right = build_tree(n - 1)\nreturn root\n
     space_complexity.go
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\nif n == 0 {\nreturn nil\n}\nroot := newTreeNode(0)\nroot.left = buildTree(n - 1)\nroot.right = buildTree(n - 1)\nreturn root\n}\n
     space_complexity.js
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.c
    [class]{}-[func]{buildTree}\n
     space_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n)\n{\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
     space_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\nif n == 0 {\nreturn nil\n}\nlet root = TreeNode(x: 0)\nroot.left = buildTree(n: n - 1)\nroot.right = buildTree(n: n - 1)\nreturn root\n}\n
     space_complexity.zig
    // \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\nif (n == 0) return null;\nconst root = try mem_allocator.create(inc.TreeNode(i32));\nroot.init(0);\nroot.left = try buildTree(mem_allocator, n - 1);\nroot.right = try buildTree(mem_allocator, n - 1);\nreturn root;\n}\n
     
     
     Fig. \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/space_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"
    \u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3001\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u7b49\u3002
     
    \u4f8b\u5982\u300c\u5f52\u5e76\u6392\u5e8f\u300d\uff0c\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u53ef\u4ee5\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     
    \u518d\u4f8b\u5982\u300c\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\u300d\uff0c\u8f93\u5165\u4efb\u610f\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n) = O(\\log n)\\) \u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/","title":"2.4. \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u591f\u8fbe\u5230\u6700\u4f18\uff0c\u800c\u5b9e\u9645\u4e0a\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u662f\u975e\u5e38\u56f0\u96be\u7684\u3002
     
    \u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5f80\u5f80\u662f\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\u7684\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u628a\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u300c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u300d\uff1b\u53cd\u4e4b\uff0c\u79f0\u4e4b\u4e3a\u300c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u300d\u3002\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002
     
    \u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u90fd\u662f\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\u7684\uff0c\u53ea\u8981\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u8981\u592a\u79bb\u8c31\u3001\u80fd\u63a5\u53d7\u5c31\u884c\uff0c\u56e0\u6b64\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u6700\u4e3a\u5e38\u7528\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#241","title":"2.4.1. \u00a0 \u793a\u4f8b\u9898\u76ee *","text":"
    \u4ee5 LeetCode \u5168\u7ad9\u7b2c\u4e00\u9898 \u4e24\u6570\u4e4b\u548c \u4e3a\u4f8b\u3002
     
    \u4e24\u6570\u4e4b\u548c
     
    \u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u6574\u6570\u76ee\u6807\u503c target \uff0c\u8bf7\u4f60\u5728\u8be5\u6570\u7ec4\u4e2d\u627e\u51fa\u201c\u548c\u201d\u4e3a\u76ee\u6807\u503c target \u7684\u90a3\u4e24\u4e2a\u6574\u6570\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u4e0b\u6807\u3002
     
    \u4f60\u53ef\u4ee5\u5047\u8bbe\u6bcf\u79cd\u8f93\u5165\u53ea\u4f1a\u5bf9\u5e94\u4e00\u4e2a\u7b54\u6848\u3002\u4f46\u662f\uff0c\u6570\u7ec4\u4e2d\u540c\u4e00\u4e2a\u5143\u7d20\u5728\u7b54\u6848\u91cc\u4e0d\u80fd\u91cd\u590d\u51fa\u73b0\u3002
     
    \u4f60\u53ef\u4ee5\u6309\u4efb\u610f\u987a\u5e8f\u8fd4\u56de\u7b54\u6848\u3002
     
    \u300c\u66b4\u529b\u679a\u4e3e\u300d\u548c\u300c\u8f85\u52a9\u54c8\u5e0c\u8868\u300d\u5206\u522b\u5bf9\u5e94 \u7a7a\u95f4\u6700\u4f18 \u548c \u65f6\u95f4\u6700\u4f18 \u7684\u4e24\u79cd\u89e3\u6cd5\u3002\u672c\u7740\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\u7684\u539f\u5219\uff0c\u540e\u8005\u662f\u672c\u9898\u7684\u6700\u4f73\u89e3\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#_1","title":"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e","text":"
    \u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u6240\u6709\u53ef\u80fd\u6027\u3002\u901a\u8fc7\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a target \uff0c\u82e5\u662f\u5219\u8fd4\u56de\u5b83\u4fe9\u7684\u7d22\u5f15\uff08\u5373\u4e0b\u6807\uff09\u5373\u53ef\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig leetcode_two_sum.java
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.cpp
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int>& nums, int target) {\nint size = nums.size();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn { i, j };\n}\n}\nreturn {};\n}\n
     leetcode_two_sum.py
    def two_sum_brute_force(nums: List[int], target: int) -> List[int]:\n\"\"\" \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e \"\"\"\n# \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in range(len(nums) - 1):\nfor j in range(i + 1, len(nums)):\nif nums[i] + nums[j] == target:\nreturn [i, j]\nreturn []\n
     leetcode_two_sum.go
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\nsize := len(nums)\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i := 0; i < size-1; i++ {\nfor j := i + 1; i < size; j++ {\nif nums[i]+nums[j] == target {\nreturn []int{i, j}\n}\n}\n}\nreturn nil\n}\n
     leetcode_two_sum.js
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
     leetcode_two_sum.ts
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n};\n
     leetcode_two_sum.c
    [class]{}-[func]{twoSumBruteForce}\n
     leetcode_two_sum.cs
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target)\n{\nint size = nums.Length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++)\n{\nfor (int j = i + 1; j < size; j++)\n{\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.swift
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in nums.indices.dropLast() {\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[i] + nums[j] == target {\nreturn [i, j]\n}\n}\n}\nreturn [0]\n}\n
     leetcode_two_sum.zig
    // \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\nvar size: usize = nums.len;\nvar i: usize = 0;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nwhile (i < size - 1) : (i += 1) {\nvar j = i + 1;\nwhile (j < size) : (j += 1) {\nif (nums[i] + nums[j] == target) {\nreturn [_]i32{@intCast(i32, i), @intCast(i32, j)};\n}\n}\n}\nreturn null;\n}\n
     
    \u8be5\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5c5e\u4e8e\u65f6\u95f4\u6362\u7a7a\u95f4\u3002\u672c\u65b9\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u8f83\u9ad8\uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002
    "},{"location":"chapter_computational_complexity/space_time_tradeoff/#_2","title":"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868","text":"
    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0ckey \u4e3a\u6570\u7ec4\u5143\u7d20\u3001value \u4e3a\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5e76\u6267\u884c\uff1a
     
       
    1. \u5224\u65ad\u6570\u5b57 target - num \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\u5219\u76f4\u63a5\u8fd4\u56de\u8be5\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1b
      2.  
    2. \u5c06\u5143\u7d20 num \u548c\u5176\u7d22\u5f15\u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\uff1b
      3.  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig leetcode_two_sum.java
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nMap<Integer, Integer> dic = new HashMap<>();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn new int[] { dic.get(target - nums[i]), i };\n}\ndic.put(nums[i], i);\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.cpp
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int>& nums, int target) {\nint size = nums.size();\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nunordered_map<int, int> dic;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.find(target - nums[i]) != dic.end()) {\nreturn { dic[target - nums[i]], i };\n}\ndic.emplace(nums[i], i);\n}\nreturn {};\n}\n
     leetcode_two_sum.py
    def two_sum_hash_table(nums: List[int], target: int) -> List[int]:\n\"\"\" \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 \"\"\"\n# \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\ndic = {}\n# \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in range(len(nums)):\nif target - nums[i] in dic:\nreturn [dic[target - nums[i]], i]\ndic[nums[i]] = i\nreturn []\n
     leetcode_two_sum.go
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nhashTable := map[int]int{}\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor idx, val := range nums {\nif preIdx, ok := hashTable[target-val]; ok {\nreturn []int{preIdx, idx}\n}\nhashTable[val] = idx\n}\nreturn nil\n}\n
     leetcode_two_sum.js
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m = {};\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nif (m[nums[i]] !== undefined) {\nreturn [m[nums[i]], i];\n} else {\nm[target - nums[i]] = i;\n}\n}\nreturn [];\n}\n
     leetcode_two_sum.ts
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m: Map<number, number> = new Map();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nlet index = m.get(nums[i]);\nif (index !== undefined) {\nreturn [index, i];\n} else {\nm.set(target - nums[i], i);\n}\n}\nreturn [];\n};\n
     leetcode_two_sum.c
    [class]{}-[func]{twoSumHashTable}\n
     leetcode_two_sum.cs
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target)\n{\nint size = nums.Length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nDictionary<int, int> dic = new();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++)\n{\nif (dic.ContainsKey(target - nums[i]))\n{\nreturn new int[] { dic[target - nums[i]], i };\n}\ndic.Add(nums[i], i);\n}\nreturn new int[0];\n}\n
     leetcode_two_sum.swift
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic: [Int: Int] = [:]\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in nums.indices {\nif let j = dic[target - nums[i]] {\nreturn [j, i]\n}\ndic[nums[i]] = i\n}\nreturn [0]\n}\n
     leetcode_two_sum.zig
    // \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\nvar size: usize = nums.len;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\ndefer dic.deinit();\nvar i: usize = 0;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nwhile (i < size) : (i += 1) {\nif (dic.contains(target - nums[i])) {\nreturn [_]i32{dic.get(target - nums[i]).?, @intCast(i32, i)};\n}\ntry dic.put(nums[i], @intCast(i32, i));\n}\nreturn null;\n}\n
     
    \u8be5\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff0c\u4f53\u73b0\u7a7a\u95f4\u6362\u65f6\u95f4\u3002\u672c\u65b9\u6cd5\u867d\u7136\u5f15\u5165\u4e86\u989d\u5916\u7a7a\u95f4\u4f7f\u7528\uff0c\u4f46\u65f6\u95f4\u548c\u7a7a\u95f4\u4f7f\u7528\u6574\u4f53\u66f4\u52a0\u5747\u8861\uff0c\u56e0\u6b64\u4e3a\u672c\u9898\u6700\u4f18\u89e3\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/summary/","title":"2.5. \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#_1","title":"\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"
       
    • \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u7b97\u6cd5\u6027\u80fd\u7684\u4e24\u4e2a\u91cd\u8981\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3002
      •  
    • \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\uff0c\u5e76\u4e14\u975e\u5e38\u8017\u8d39\u8ba1\u7b97\u8d44\u6e90\u3002
      •  
    • \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u5927\u5c0f\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\u3002
      •  
    "},{"location":"chapter_computational_complexity/summary/#_2","title":"\u65f6\u95f4\u590d\u6742\u5ea6","text":"
       
    • \u65f6\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u6bd4\u5982\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u6027\u3002
      •  
    • \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5373\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u5176\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u4e8e\u6b63\u65e0\u7a77\u65f6\uff0c\\(T(n)\\) \u5904\u4e8e\u4f55\u79cd\u589e\u957f\u7ea7\u522b\u3002
      •  
    • \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\uff0c\u518d\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
      •  
    • \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n \\log n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) , \\(O(n!)\\) \u3002
      •  
    • \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u540e\u8005\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u9700\u8981\u6ee1\u8db3\u82db\u523b\u7684\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
      •  
    • \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u53cd\u6620\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u6700\u8d34\u5408\u5b9e\u9645\u4f7f\u7528\u60c5\u51b5\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\uff0c\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002
      •  
    "},{"location":"chapter_computational_complexity/summary/#_3","title":"\u7a7a\u95f4\u590d\u6742\u5ea6","text":"
       
    • \u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5b9a\u4e49\u7c7b\u4f3c\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u7b97\u6cd5\u5360\u7528\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002
      •  
    • \u7b97\u6cd5\u8fd0\u884c\u4e2d\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u8ba1\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6307\u4ee4\u7a7a\u95f4\u3001\u6570\u636e\u7a7a\u95f4\u3001\u6808\u5e27\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u4e00\u822c\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u624d\u4f1a\u5f71\u54cd\u5230\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
      •  
    • \u6211\u4eec\u4e00\u822c\u53ea\u5173\u5fc3\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u95f4\u70b9\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
      •  
    • \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) \u3002
      •  
    "},{"location":"chapter_computational_complexity/time_complexity/","title":"2.2. \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/time_complexity/#221","title":"2.2.1. \u00a0 \u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4","text":"
    \u8fd0\u884c\u65f6\u95f4\u80fd\u591f\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u4f53\u73b0\u51fa\u7b97\u6cd5\u7684\u6548\u7387\u6c34\u5e73\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981 \u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4 \uff0c\u8be5\u5982\u4f55\u505a\u5462\uff1f
     
       
    1. \u9996\u5148\u9700\u8981 \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0 \uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u90fd\u4f1a\u5f71\u54cd\u5230\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
      2.  
    2. \u8bc4\u4f30 \u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u7684\u6240\u9700\u8fd0\u884c\u65f6\u95f4 \uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c + \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c * \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c\u9700\u8981 5 ns \u7b49\u3002
      3.  
    3. \u6839\u636e\u4ee3\u7801 \u7edf\u8ba1\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf \uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u5373\u53ef\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002
      4.  
     
    \u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e3a \\(6n + 12\\) ns \u3002
     \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nSystem.out.println(0);     // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\ncout << 0 << endl;         // 5 ns\n}\n}\n
     
    # \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int) -> None:\na = 2      # 1 ns\na = a + 1  # 1 ns\na = a * 2  # 10 ns\n# \u5faa\u73af n \u6b21\nfor _ in range(n):  # 1 ns\nprint(0)        # 5 ns\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\na := 2      // 1 ns\na = a + 1   // 1 ns\na = a * 2   // 10 ns\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {    // 1 ns\nfmt.Println(a)          // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\nvar a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\nvar a: number = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintf(\"%d\", 0);            // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n)\n{\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++)\n{  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nConsole.WriteLine(0);     // 5 ns\n}\n}\n
     
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\nvar a = 2 // 1 ns\na = a + 1 // 1 ns\na = a * 2 // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // 1 ns\nprint(0) // 5 ns\n}\n}\n
     
    \n
     
    \u4f46\u5b9e\u9645\u4e0a\uff0c \u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u6bd5\u7adf\u7b97\u6cd5\u9700\u8981\u8dd1\u5728\u5404\u5f0f\u5404\u6837\u7684\u5e73\u53f0\u4e4b\u4e0a\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u4e00\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u4e3a\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#222","title":"2.2.2. \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"
    \u300c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u300d\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u505a\u6cd5\uff0c\u5176\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f \u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf \u3002
     
    \u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u4f8b\u5b50\u6765\u7406\u89e3\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 A , B , C \u3002
     
       
    • \u7b97\u6cd5 A \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
      •  
    • \u7b97\u6cd5 B \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u6210\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u300c\u7ebf\u6027\u9636\u300d\u3002
      •  
    • \u7b97\u6cd5 C \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u4ecd\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c A \u76f8\u540c\uff0c\u4ecd\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nSystem.out.println(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nSystem.out.println(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nSystem.out.println(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\ncout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\ncout << 0 << endl;\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\ncout << 0 << endl;\n}\n}\n
     
    # \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int) -> None:\nprint(0)\n# \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int) -> None:\nfor _ in range(n):\nprint(0)\n# \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int) -> None:\nfor _ in range(1000000):\nprint(0)\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\nfmt.Println(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\nfor i := 0; i < n; i++ {\nfmt.Println(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\nfor i := 0; i < 1000000; i++ {\nfmt.Println(0)\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nprintf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nprintf(\"%d\", 0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprintf(\"%d\", 0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n)\n{\nConsole.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nConsole.WriteLine(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n)\n{\nfor (int i = 0; i < 1000000; i++)\n{\nConsole.WriteLine(0);\n}\n}\n
     
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\nprint(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\nfor _ in 0 ..< n {\nprint(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\nfor _ in 0 ..< 1000000 {\nprint(0)\n}\n}\n
     
    \n
     
     
     Fig. \u7b97\u6cd5 A, B, C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf 
     
    \u76f8\u6bd4\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7684\u505a\u6cd5\u6709\u4ec0\u4e48\u597d\u5904\u5462\uff1f\u4ee5\u53ca\u6709\u4ec0\u4e48\u4e0d\u8db3\uff1f
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u7b97\u6cd5 B \u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u662f\u7ebf\u6027\u7684\uff0c\u5728 \\(n > 1\\) \u65f6\u6162\u4e8e\u7b97\u6cd5 A \uff0c\u5728 \\(n > 1000000\\) \u65f6\u6162\u4e8e\u7b97\u6cd5 C \u3002\u5b9e\u8d28\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u300c\u7ebf\u6027\u9636\u300d\u7684\u7b97\u6cd5\uff0c\u8fd9\u4e5f\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u52a0\u7b80\u4fbf\u3002\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7edf\u8ba1\u300c\u8ba1\u7b97\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\u300d\u7b80\u5316\u4e3a\u7edf\u8ba1\u300c\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf\u300d\uff0c\u8fd9\u662f\u56e0\u4e3a\uff0c\u65e0\u8bba\u662f\u8fd0\u884c\u5e73\u53f0\u8fd8\u662f\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\uff0c\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u7edf\u4e00\u770b\u4f5c\u662f\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u8fd9\u6837\u7684\u7b80\u5316\u505a\u6cd5\u5927\u5927\u964d\u4f4e\u4e86\u4f30\u7b97\u96be\u5ea6\u3002
     
    \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u6bd4\u5982\uff0c\u867d\u7136\u7b97\u6cd5 A \u548c C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u662f\u5b9e\u9645\u7684\u8fd0\u884c\u65f6\u95f4\u6709\u975e\u5e38\u5927\u7684\u5dee\u522b\u3002\u518d\u6bd4\u5982\uff0c\u867d\u7136\u7b97\u6cd5 B \u6bd4 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8981\u66f4\u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6bd4\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 B \u662f\u8981\u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 C \u7684\u3002\u5bf9\u4e8e\u4ee5\u4e0a\u60c5\u51b5\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u5224\u5b9a\u7b97\u6cd5\u6548\u7387\u9ad8\u4f4e\u3002\u7136\u800c\uff0c\u5373\u4f7f\u5b58\u5728\u8fd9\u4e9b\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u7684\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#223","title":"2.2.3. \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"
    \u8bbe\u7b97\u6cd5\u300c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u300d\u4e3a \\(T(n)\\) \uff0c\u5176\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\u3002\u4f8b\u5982\uff0c\u4ee5\u4e0b\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a
     \\[ T(n) = 3 + 2n \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nSystem.out.println(0);    // +1\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\ncout << 0 << endl;    // +1\n}\n}\n
     
    def algorithm(n: int) -> None:\na: int = 1  # +1\na = a + 1  # +1\na = a * 2  # +1\n# \u5faa\u73af n \u6b21\nfor i in range(n):  # +1\nprint(0)        # +1\n
     
    func algorithm(n int) {\na := 1      // +1\na = a + 1   // +1\na = a * 2   // +1\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {   // +1\nfmt.Println(a)         // +1\n}\n}\n
     
    function algorithm(n) {\nvar a = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
     
    function algorithm(n: number): void{\nvar a: number = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintf(\"%d\", 0);            // +1\n}\n}  
     
    void algorithm(int n)\n{\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n{\nConsole.WriteLine(0);   // +1\n}\n}\n
     
    func algorithm(n: Int) {\nvar a = 1 // +1\na = a + 1 // +1\na = a * 2 // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // +1\nprint(0) // +1\n}\n}\n
     
    \n
     
    \\(T(n)\\) \u662f\u4e2a\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u6613\u5f97\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002
     
    \u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u88ab\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 Big-\\(O\\) Notation\u300d\uff0c\u4ee3\u8868\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002
     
    \u6211\u4eec\u8981\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u672c\u8d28\u4e0a\u662f\u5728\u8ba1\u7b97\u300c\u64cd\u4f5c\u6570\u91cf\u51fd\u6570 \\(T(n)\\) \u300d\u7684\u6e10\u8fd1\u4e0a\u754c\u3002\u4e0b\u9762\u6211\u4eec\u5148\u6765\u770b\u770b\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5b9a\u4e49\u3002
     
    \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c
     
    \u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 $$ T(n) \\leq c \\cdot f(n) $$ \u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a $$ T(n) = O(f(n)) $$
     
     
     Fig. \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c 
     
    \u672c\u8d28\u4e0a\u770b\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5728\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5728 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff08\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\uff09\u3002
     
    Tip
     
    \u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u56e0\u4e3a\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\u6211\u4eec\u53ea\u9700\u8981\u4f1a\u63a8\u7b97\u5373\u53ef\uff0c\u6570\u5b66\u610f\u4e49\u53ef\u4ee5\u6162\u6162\u9886\u609f\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#224","title":"2.2.4. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"
    \u63a8\u7b97\u51fa \\(f(n)\\) \u540e\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\uff0c\u5982\u4f55\u6765\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u300c\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\u300d\uff0c\u7136\u540e\u300c\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u300d\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1) \u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"
    \u5bf9\u7740\u4ee3\u7801\uff0c\u4ece\u4e0a\u5230\u4e0b\u4e00\u884c\u4e00\u884c\u5730\u8ba1\u6570\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u5077\u61d2\u6280\u5de7\uff1a
     
       
    1. \u8df3\u8fc7\u6570\u91cf\u4e0e \\(n\\) \u65e0\u5173\u7684\u64cd\u4f5c\u3002\u56e0\u4e3a\u4ed6\u4eec\u90fd\u662f \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\uff0c\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
      2.  
    2. \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u3001\u2026\u2026\uff0c\u90fd\u53ef\u4ee5\u5316\u7b80\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
      3.  
    3. \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u4e0a\u8ff0 1. \u548c 2. \u6280\u5de7\u3002
      4.  
     
    \u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u3001\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\u3002
     \\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\] 
    \u6700\u7ec8\uff0c\u4e24\u8005\u90fd\u80fd\u63a8\u51fa\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7ed3\u679c\uff0c\u5373 \\(O(n^2)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nSystem.out.println(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nSystem.out.println(0);\n}\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\ncout << 0 << endl;\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\ncout << 0 << endl;\n}\n}\n}\n
     
    def algorithm(n: int) -> None:\na: int = 1  # +0\uff08\u6280\u5de7 1\uff09\na = a + n   # +0\uff08\u6280\u5de7 1\uff09\n# +n\uff08\u6280\u5de7 2\uff09\nfor i in range(5 * n + 1):\nprint(0)\n# +n*n\uff08\u6280\u5de7 3\uff09\nfor i in range(2 * n):\nfor j in range(n + 1):\nprint(0)\n
     
    func algorithm(n int) {\na := 1      // +0\uff08\u6280\u5de7 1\uff09\na = a + n  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i := 0; i < 5 * n + 1; i++ {\nfmt.Println(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i := 0; i < 2 * n; i++ {\nfor j := 0; j < n + 1; j++ {\nfmt.Println(0)\n}\n}\n}\n
     
    function algorithm(n) {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
     
    function algorithm(n: number): void {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
     
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprintf(\"%d\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprintf(\"%d\", 0);\n}\n}\n}\n
     
    void algorithm(int n)\n{\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++)\n{\nConsole.WriteLine(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++)\n{\nfor (int j = 0; j < n + 1; j++)\n{\nConsole.WriteLine(0);\n}\n}\n}\n
     
    func algorithm(n: Int) {\nvar a = 1 // +0\uff08\u6280\u5de7 1\uff09\na = a + n // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor _ in 0 ..< (5 * n + 1) {\nprint(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor _ in 0 ..< (2 * n) {\nfor _ in 0 ..< (n + 1) {\nprint(0)\n}\n}\n}\n
     
    \n
    "},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2) \u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\u7531\u591a\u9879\u5f0f \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u5904\u4e8e\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u5b83\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002
     
    \u4ee5\u4e0b\u8868\u683c\u7ed9\u51fa\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u6709\u4e00\u4e9b\u5938\u5f20\u7684\u503c\uff0c\u662f\u60f3\u8981\u5411\u5927\u5bb6\u5f3a\u8c03 \u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570 \u8fd9\u4e00\u7ed3\u8bba\u3002\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u90fd\u662f\u201c\u6d6e\u4e91\u201d\u3002
     \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#225","title":"2.2.5. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09
     \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\] 
     
     Fig. \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b 
     
    Tip
     
    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u5148\u805a\u7126\u5728\u7406\u89e3\u65f6\u95f4\u590d\u6742\u5ea6\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u4e0a\u3002
    "},{"location":"chapter_computational_complexity/time_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"
    \u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002
     
    \u5bf9\u4e8e\u4ee5\u4e0b\u7b97\u6cd5\uff0c\u65e0\u8bba\u64cd\u4f5c\u6570\u91cf size \u6709\u591a\u5927\uff0c\u53ea\u8981\u4e0e\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4ecd\u4e3a \\(O(1)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.py
    def constant(n: int) -> int:\n\"\"\" \u5e38\u6570\u9636 \"\"\"\ncount: int = 0\nsize: int = 100000\nfor _ in range(size):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\ncount := 0\nsize := 100000\nfor i := 0; i < size; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{constant}\n
     time_complexity.cs
    /* \u5e38\u6570\u9636 */\nint constant(int n)\n{\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\nvar count = 0\nlet size = 100_000\nfor _ in 0 ..< size {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n_ = n;\nvar count: i32 = 0;\nconst size: i32 = 100_000;\nvar i: i32 = 0;\nwhile(i<size) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"
    \u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u8f93\u5165\u6570\u636e\u5927\u5c0f\u6210\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u5e38\u51fa\u73b0\u4e8e\u5355\u5c42\u5faa\u73af\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.py
    def linear(n: int) -> int:\n\"\"\" \u7ebf\u6027\u9636 \"\"\"\ncount: int = 0\nfor _ in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\ncount := 0\nfor i := 0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{linear}\n
     time_complexity.cs
    /* \u7ebf\u6027\u9636 */\nint linear(int n)\n{\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\nvar count = 0\nfor _ in 0 ..< n {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
     
    \u300c\u904d\u5386\u6570\u7ec4\u300d\u548c\u300c\u904d\u5386\u94fe\u8868\u300d\u7b49\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\u3002
     
    Tip
     
    \u6570\u636e\u5927\u5c0f \\(n\\) \u662f\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u786e\u5b9a\u7684\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u76f4\u63a5\u5c06 \\(n\\) \u770b\u4f5c\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u4ee5\u4e0b\u904d\u5386\u6570\u7ec4\u793a\u4f8b\u4e2d\uff0c\u6570\u636e\u5927\u5c0f \\(n\\) \u4e3a\u6570\u7ec4\u7684\u957f\u5ea6\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int>& nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def array_traversal(nums: List[int]) -> int:\n\"\"\" \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\ncount: int = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor num in nums:\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor range nums {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{arrayTraversal}\n
     time_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums)\n{\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nforeach (int num in nums)\n{\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\nvar count: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (nums) |_| {\ncount += 1;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"
    \u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u8f93\u5165\u6570\u636e\u5927\u5c0f\u6210\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u90fd\u4e3a \\(O(n)\\) \uff0c\u603b\u4f53\u4e3a \\(O(n^2)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.py
    def quadratic(n: int) -> int:\n\"\"\" \u5e73\u65b9\u9636 \"\"\"\ncount: int = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i in range(n):\nfor j in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i := 0; i < n; i++ {\nfor j := 0; j < n; j++ {\ncount++\n}\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{quadratic}\n
     time_complexity.cs
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n)\n{\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++)\n{\nfor (int j = 0; j < n; j++)\n{\ncount++;\n}\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0 ..< n {\nfor _ in 0 ..< n {\ncount += 1\n}\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ncount += 1;\n}\n}\nreturn count;\n}\n
     
     
     Fig. \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u3001\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u4ee5\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af \\(n-1, n-2, \\cdots, 2, 1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(\\frac{n}{2}\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
     \\[ O((n - 1) \\frac{n}{2}) = O(n^2) \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int>& nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.py
    def bubble_sort(nums: List[int]) -> int:\n\"\"\" \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\ncount: int = 0  # \u8ba1\u6570\u5668\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(len(nums) - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp: int = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\nreturn count\n
     time_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\ncount := 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp := nums[j]\nnums[j] = nums[j+1]\nnums[j+1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{bubbleSort}\n
     time_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums)\n{\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\nvar count = 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in 0 ..< i {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\nvar count: i32 = 0;  // \u8ba1\u6570\u5668 \n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: i32 = @intCast(i32, nums.len ) - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"
    Note
     
    \u751f\u7269\u5b66\u79d1\u4e2d\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u5373\u662f\u6307\u6570\u9636\u589e\u957f\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u4e3a \\(4\\) \u4e2a\uff0c\u2026\u2026\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002
     
    \u6307\u6570\u9636\u589e\u957f\u5f97\u975e\u5e38\u5feb\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u4e00\u822c\u662f\u4e0d\u80fd\u88ab\u63a5\u53d7\u7684\u3002\u82e5\u4e00\u4e2a\u95ee\u9898\u4f7f\u7528\u300c\u66b4\u529b\u679a\u4e3e\u300d\u6c42\u89e3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(2^n)\\) \uff0c\u90a3\u4e48\u4e00\u822c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u52a8\u6001\u89c4\u5212\u300d\u6216\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u7b49\u7b97\u6cd5\u6765\u6c42\u89e3\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.py
    def exponential(n: int) -> int:\n\"\"\" \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount: int = 0\nbase: int = 1\n# cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in range(n):\nfor _ in range(base):\ncount += 1\nbase *= 2\n# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n
     time_complexity.go
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\ncount, base := 0, 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor i := 0; i < n; i++ {\nfor j := 0; j < base; j++ {\ncount++\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
     time_complexity.js
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{exponential}\n
     time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n)\n{\nint count = 0, bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++)\n{\nfor (int j = 0; j < bas; j++)\n{\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\nvar count = 0\nvar base = 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0 ..< n {\nfor _ in 0 ..< base {\ncount += 1\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
     time_complexity.zig
    // \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\nvar count: i32 = 0;\nvar bas: i32 = 1;\nvar i: i32 = 0;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < bas) : (j += 1) {\ncount += 1;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
     
     
     Fig. \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u4e0d\u65ad\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u5206\u88c2 \\(n\\) \u6b21\u540e\u505c\u6b62\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.py
    def exp_recur(n: int) -> int:\n\"\"\" \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 1: return 1\nreturn exp_recur(n - 1) + exp_recur(n - 1) + 1\n
     time_complexity.go
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n-1) + expRecur(n-1) + 1\n}\n
     time_complexity.js
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.c
    [class]{}-[func]{expRecur}\n
     time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n)\n{\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
     time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
     time_complexity.zig
    // \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"
    \u5bf9\u6570\u9636\u4e0e\u6307\u6570\u9636\u6b63\u597d\u76f8\u53cd\uff0c\u540e\u8005\u53cd\u6620\u201c\u6bcf\u8f6e\u589e\u52a0\u5230\u4e24\u500d\u7684\u60c5\u51b5\u201d\uff0c\u800c\u524d\u8005\u53cd\u6620\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u7684\u60c5\u51b5\u201d\u3002\u5bf9\u6570\u9636\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\uff0c\u65f6\u95f4\u589e\u957f\u5f97\u5f88\u6162\uff0c\u662f\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
     
    \u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u5206\u6cbb\u7b97\u6cd5\u300d\u4e2d\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u591a\u201d\u3001\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002
     
    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def logarithmic(n: float) -> int:\n\"\"\" \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount: int = 0\nwhile n > 1:\nn = n / 2\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\ncount := 0\nfor n > 1 {\nn = n / 2\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{logarithmic}\n
     time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n)\n{\nint count = 0;\nwhile (n > 1)\n{\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\nvar count = 0\nvar n = n\nwhile n > 1 {\nn = n / 2\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\nvar count: i32 = 0;\nvar n_var = n;\nwhile (n_var > 1)\n{\nn_var = n_var / 2;\ncount +=1;\n}\nreturn count;\n}\n
     
     
     Fig. \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
     
    \u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.py
    def log_recur(n: float) -> int:\n\"\"\" \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 1: return 0\nreturn log_recur(n / 2) + 1\n
     time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n/2) + 1\n}\n
     time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.c
    [class]{}-[func]{logRecur}\n
     time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n)\n{\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
     time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n: n / 2) + 1\n}\n
     time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on-log-n","title":"\u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"
    \u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002
     
    \u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(n \\log n )\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.py
    def linear_log_recur(n: float) -> int:\n\"\"\" \u7ebf\u6027\u5bf9\u6570\u9636 \"\"\"\nif n <= 1: return 1\ncount: int = linear_log_recur(n // 2) + \\\n                 linear_log_recur(n // 2)\nfor _ in range(n):\ncount += 1\nreturn count\n
     time_complexity.go
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\nif n <= 1 {\nreturn 1\n}\ncount := linearLogRecur(n/2) +\nlinearLogRecur(n/2)\nfor i := 0.0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
     time_complexity.js
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{linearLogRecur}\n
     time_complexity.cs
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n)\n{\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++)\n{\ncount++;\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 1\n}\nvar count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\nfor _ in stride(from: 0, to: n, by: 1) {\ncount += 1\n}\nreturn count\n}\n
     time_complexity.zig
    // \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\nif (n <= 1) return 1;\nvar count: i32 = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nvar i: f32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
     
     
     Fig. \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/time_complexity/#on_1","title":"\u9636\u4e58\u9636 \\(O(n!)\\)","text":"
    \u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u300c\u5168\u6392\u5217\u300d\u3002\u5373\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u5219\u65b9\u6848\u6570\u91cf\u4e3a
     \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\cdots \\times 2 \\times 1 \\] 
    \u9636\u4e58\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u2026\u2026 \uff0c\u76f4\u81f3\u5230\u7b2c \\(n\\) \u5c42\u65f6\u7ec8\u6b62\u5206\u88c2\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig time_complexity.java
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.cpp
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.py
    def factorial_recur(n: int) -> int:\n\"\"\" \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 0: return 1\ncount: int = 0\n# \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in range(n):\ncount += factorial_recur(n - 1)\nreturn count\n
     time_complexity.go
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\nif n == 0 {\nreturn 1\n}\ncount := 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor i := 0; i < n; i++ {\ncount += factorialRecur(n - 1)\n}\nreturn count\n}\n
     time_complexity.js
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.ts
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.c
    [class]{}-[func]{factorialRecur}\n
     time_complexity.cs
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n)\n{\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++)\n{\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     time_complexity.swift
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\nif n == 0 {\nreturn 1\n}\nvar count = 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0 ..< n {\ncount += factorialRecur(n: n - 1)\n}\nreturn count\n}\n
     time_complexity.zig
    // \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\nif (n == 0) return 1;\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nwhile (i < n) : (i += 1) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
     
     
     Fig. \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 
    "},{"location":"chapter_computational_complexity/time_complexity/#226","title":"2.2.6. \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"
    \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u4e3e\u4e00\u4e2a\u4f8b\u5b50\uff0c\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u6570\u7ec4 nums \uff0c\u5176\u4e2d nums \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff1b\u7b97\u6cd5\u7684\u4efb\u52a1\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a
     
       
    • \u5f53 nums = [?, ?, ..., 1]\uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u5219\u9700\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1b
      •  
    • \u5f53 nums = [1, ?, ?, ...] \uff0c\u5373\u5f53\u9996\u4e2a\u6570\u5b57\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \uff1b
      •  
     
    \u300c\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u300d\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002\u4e0e\u4e4b\u5bf9\u5e94\uff0c\u300c\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\u300d\u7528 \\(\\Omega\\) \u8bb0\u53f7\uff08Omega Notation\uff09\u6765\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig worst_best_time_complexity.java
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nInteger[] nums = new Integer[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nCollections.shuffle(Arrays.asList(nums));\n// Integer[] -> int[]\nint[] res = new int[n];\nfor (int i = 0; i < n; i++) {\nres[i] = nums[i];\n}\nreturn res;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.cpp
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\nvector<int> nums(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\nunsigned seed = chrono::system_clock::now().time_since_epoch().count();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nshuffle(nums.begin(), nums.end(), default_random_engine(seed));\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int>& nums) {\nfor (int i = 0; i < nums.size(); i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.py
    def random_numbers(n: int) -> List[int]:\n\"\"\" \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71 \"\"\"\n# \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\nnums: List[int] = [i for i in range(1, n + 1)]\n# \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrandom.shuffle(nums)\nreturn nums\ndef find_one(nums: List[int]) -> int:\n\"\"\" \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 \"\"\"\nfor i in range(len(nums)):\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1:\nreturn i\nreturn -1\n
     worst_best_time_complexity.go
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\nnums := make([]int, n)\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor i := 0; i < n; i++ {\nnums[i] = i + 1\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrand.Shuffle(len(nums), func(i, j int) {\nnums[i], nums[j] = nums[j], nums[i]\n})\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\nfor i := 0; i < len(nums); i++ {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
     worst_best_time_complexity.js
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     worst_best_time_complexity.ts
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
     worst_best_time_complexity.c
    [class]{}-[func]{randomNumbers}\n[class]{}-[func]{findOne}\n
     worst_best_time_complexity.cs
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n)\n{\nint[] nums = new int[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++)\n{\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = 0; i < nums.Length; i++)\n{\nvar index = new Random().Next(i, nums.Length);\nvar tmp = nums[i];\nvar ran = nums[index];\nnums[i] = ran;\nnums[index] = tmp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums)\n{\nfor (int i = 0; i < nums.Length; i++)\n{\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
     worst_best_time_complexity.swift
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nvar nums = Array(1 ... n)\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle()\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\nfor i in nums.indices {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
     worst_best_time_complexity.zig
    // \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\npub fn randomNumbers(comptime n: usize) [n]i32 {\nvar nums: [n]i32 = undefined;\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (nums) |*num, i| {\nnum.* = @intCast(i32, i) + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nconst rand = std.crypto.random;\nrand.shuffle(i32, &nums);\nreturn nums;\n}\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\npub fn findOne(nums: []i32) i32 {\nfor (nums) |num, i| {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (num == 1) return @intCast(i32, i);\n}\nreturn -1;\n}\n
     
    Tip
     
    \u6211\u4eec\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5f88\u5c11\u4f7f\u7528\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u56e0\u4e3a\u5f80\u5f80\u53ea\u6709\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u53cd\u4e4b\uff0c\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u6700\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u201c\u6548\u7387\u5b89\u5168\u503c\u201d\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002
     
    \u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6216\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u5728\u201c\u7279\u6b8a\u5206\u5e03\u7684\u6570\u636e\u201d\u4e2d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u5f80\u5f80\u5f88\u5c0f\uff0c\u56e0\u6b64\u5e76\u4e0d\u80fd\u6700\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u5bf9\u5730\uff0c\u300c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\uff08Theta Notation\uff09\u6765\u8868\u793a\u3002
     
    \u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5219\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(\\frac{n}{2}\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(\\frac{n}{2}) = \\Theta(n)\\) \u3002
     
    \u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5c24\u5176\u662f\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u7b80\u4fbf\u5730\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e00\u822c\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002
     
    \u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f
     
    \u5b9e\u9645\u4e2d\u6211\u4eec\u7ecf\u5e38\u4f7f\u7528\u300c\u5927 \\(O\\) \u7b26\u53f7\u300d\u6765\u8868\u793a\u300c\u5e73\u5747\u590d\u6742\u5ea6\u300d\uff0c\u8fd9\u6837\u4e25\u683c\u610f\u4e49\u4e0a\u6765\u8bf4\u662f\u4e0d\u89c4\u8303\u7684\u3002\u8fd9\u53ef\u80fd\u662f\u56e0\u4e3a \\(O\\) \u7b26\u53f7\u5b9e\u5728\u662f\u592a\u6717\u6717\u4e0a\u53e3\u4e86\u3002\u5982\u679c\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\u770b\u5230\u7c7b\u4f3c \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u7684\u8868\u8ff0\uff0c\u8bf7\u4f60\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u5373\u53ef\u3002
    "},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"
    \u6570\u636e\u7ed3\u6784\u4e3b\u8981\u53ef\u6839\u636e\u300c\u903b\u8f91\u7ed3\u6784\u300d\u548c\u300c\u7269\u7406\u7ed3\u6784\u300d\u4e24\u79cd\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002
    "},{"location":"chapter_data_structure/classification_of_data_structure/#321","title":"3.2.1. \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"
    \u300c\u903b\u8f91\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u6570\u7ec4\u548c\u94fe\u8868\u7684\u6570\u636e\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u6392\u5217\uff0c\u53cd\u6620\u4e86\u6570\u636e\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u6811\u4ece\u9876\u81f3\u5e95\u6309\u5c42\u7ea7\u6392\u5217\uff0c\u53cd\u6620\u4e86\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u7531\u7ed3\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7f51\u7edc\u5173\u7cfb\u3002
     
    \u6211\u4eec\u4e00\u822c\u5c06\u903b\u8f91\u7ed3\u6784\u5206\u4e3a\u300c\u7ebf\u6027\u300d\u548c\u300c\u975e\u7ebf\u6027\u300d\u4e24\u79cd\u3002\u201c\u7ebf\u6027\u201d\u8fd9\u4e2a\u6982\u5ff5\u5f88\u76f4\u89c2\uff0c\u5373\u8868\u660e\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u662f\u6392\u6210\u4e00\u6761\u7ebf\u7684\uff1b\u800c\u5982\u679c\u6570\u636e\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u662f\u975e\u7ebf\u6027\u7684\uff08\u4f8b\u5982\u662f\u7f51\u72b6\u6216\u6811\u72b6\u7684\uff09\uff0c\u90a3\u4e48\u5c31\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
     
       
    • \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff1b
      •  
    • \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u56fe\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff1b
      •  
     
     
     Fig. \u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 
    "},{"location":"chapter_data_structure/classification_of_data_structure/#322","title":"3.2.2. \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u79bb\u6563","text":"
    Note
     
    \u82e5\u611f\u5230\u9605\u8bfb\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u770b\u5b8c\u4e0b\u4e2a\u7ae0\u8282\u300c\u6570\u7ec4\u4e0e\u94fe\u8868\u300d\uff0c\u518d\u56de\u8fc7\u5934\u6765\u7406\u89e3\u7269\u7406\u7ed3\u6784\u7684\u542b\u4e49\u3002
     
    \u300c\u7269\u7406\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u5206\u522b\u662f \u6570\u7ec4\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8 \u548c \u94fe\u8868\u7684\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u4e0a\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u6b64\u6d88\u5f7c\u957f\u7684\u7279\u6027\u3002
     
     
     Fig. \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8 
     
    \u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u6216\u94fe\u8868\u3001\u6216\u4e24\u8005\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\u6808\u548c\u961f\u5217\uff0c\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u3001\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u800c\u4f8b\u5982\u54c8\u5e0c\u8868\uff0c\u5176\u5b9e\u73b0\u540c\u65f6\u5305\u542b\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u3002
     
       
    • \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\uff1b
      •  
    • \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\uff1b
      •  
     
    \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u300c\u9759\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u610f\u5473\u7740\u8be5\u6570\u636e\u7ed3\u6784\u5728\u5728\u88ab\u521d\u59cb\u5316\u540e\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u53cd\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8be5\u6570\u636e\u7ed3\u6784\u5728\u88ab\u521d\u59cb\u5316\u540e\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u4fee\u6539\u5176\u957f\u5ea6\u3002
     
    Tip
     
    \u6570\u7ec4\u4e0e\u94fe\u8868\u662f\u5176\u4ed6\u6240\u6709\u6570\u636e\u7ed3\u6784\u7684\u201c\u5e95\u5c42\u79ef\u6728\u201d\uff0c\u5efa\u8bae\u8bfb\u8005\u4e00\u5b9a\u8981\u591a\u82b1\u4e9b\u65f6\u95f4\u4e86\u89e3\u3002
    "},{"location":"chapter_data_structure/data_and_memory/","title":"3.1. \u00a0 \u6570\u636e\u4e0e\u5185\u5b58","text":""},{"location":"chapter_data_structure/data_and_memory/#311","title":"3.1.1. \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"
    \u8c08\u5230\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\uff0c\u6211\u4eec\u80fd\u591f\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u7b49\uff0c\u8fd9\u4e9b\u6570\u636e\u867d\u7136\u7ec4\u7ec7\u5f62\u5f0f\u4e0d\u540c\uff0c\u4f46\u90fd\u662f\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u7684\u3002
     
    \u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\u3002
     
       
    • \u300c\u6574\u6570\u300d\u6839\u636e\u4e0d\u540c\u7684\u957f\u5ea6\u5206\u4e3a byte, short, int, long \uff0c\u6839\u636e\u7b97\u6cd5\u9700\u6c42\u9009\u7528\uff0c\u5373\u5728\u6ee1\u8db3\u53d6\u503c\u8303\u56f4\u7684\u60c5\u51b5\u4e0b\u5c3d\u91cf\u51cf\u5c0f\u5185\u5b58\u7a7a\u95f4\u5360\u7528\uff1b
      •  
    • \u300c\u6d6e\u70b9\u6570\u300d\u4ee3\u8868\u5c0f\u6570\uff0c\u6839\u636e\u957f\u5ea6\u5206\u4e3a float, double \uff0c\u540c\u6837\u6839\u636e\u7b97\u6cd5\u7684\u5b9e\u9645\u9700\u6c42\u9009\u7528\uff1b
      •  
    • \u300c\u5b57\u7b26\u300d\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u5b57\u7b26\u96c6\u7684\u5f62\u5f0f\u4fdd\u5b58\u7684\uff0cchar \u7684\u503c\u5b9e\u9645\u4e0a\u662f\u6570\u5b57\uff0c\u4ee3\u8868\u5b57\u7b26\u96c6\u4e2d\u7684\u7f16\u53f7\uff0c\u8ba1\u7b97\u673a\u901a\u8fc7\u5b57\u7b26\u96c6\u67e5\u8868\u6765\u5b8c\u6210\u7f16\u53f7\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002\u5360\u7528\u7a7a\u95f4\u901a\u5e38\u4e3a 2 bytes \u6216 1 byte \uff1b
      •  
    • \u300c\u5e03\u5c14\u300d\u4ee3\u8868\u903b\u8f91\u4e2d\u7684\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\uff0c\u5176\u5360\u7528\u7a7a\u95f4\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u786e\u5b9a\uff0c\u901a\u5e38\u4e3a 1 byte \u6216 1 bit \uff1b
      •  
     \u7c7b\u522b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u53d6\u503c\u8303\u56f4 \u9ed8\u8ba4\u503c \u6574\u6570 byte 1 byte \\(-2^7\\) ~ \\(2^7 - 1\\) ( \\(-128\\) ~ \\(127\\) ) \\(0\\) short 2 bytes \\(-2^{15}\\) ~ \\(2^{15} - 1\\) \\(0\\) int 4 bytes \\(-2^{31}\\) ~ \\(2^{31} - 1\\) \\(0\\) long 8 bytes \\(-2^{63}\\) ~ \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 float 4 bytes \\(-3.4 \\times 10^{38}\\) ~ \\(3.4 \\times 10^{38}\\) \\(0.0\\) f double 8 bytes \\(-1.7 \\times 10^{308}\\) ~ \\(1.7 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 char 2 bytes / 1 byte \\(0\\) ~ \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 bool 1 byte / 1 bit \\(\\text{true}\\) \u6216 \\(\\text{false}\\) \\(\\text{false}\\) 
    Tip
     
    \u4ee5\u4e0a\u8868\u683c\u4e2d\uff0c\u52a0\u7c97\u9879\u5728\u300c\u7b97\u6cd5\u9898\u300d\u4e2d\u6700\u4e3a\u5e38\u7528\u3002\u6b64\u8868\u683c\u65e0\u9700\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_1","title":"\u6574\u6570\u8868\u793a\u65b9\u5f0f","text":"
    \u6574\u6570\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u53d8\u91cf\u4f7f\u7528\u7684\u5185\u5b58\u957f\u5ea6\uff0c\u5373\u5b57\u8282\uff08\u6216\u6bd4\u7279\uff09\u6570\u3002\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c 1 \u5b57\u8282 (byte) = 8 \u6bd4\u7279 (bit) \uff0c 1 \u6bd4\u7279\u5373 1 \u4e2a\u4e8c\u8fdb\u5236\u4f4d\u3002\u4ee5 int \u7c7b\u578b\u4e3a\u4f8b\uff1a
     
       
    1. \u6574\u6570\u7c7b\u578b int \u5360\u7528 4 bytes = 32 bits \uff0c\u56e0\u6b64\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u4e0d\u540c\u7684\u6570\u5b57\uff1b
      2.  
    2. \u5c06\u6700\u9ad8\u4f4d\u770b\u4f5c\u7b26\u53f7\u4f4d\uff0c\\(0\\) \u4ee3\u8868\u6b63\u6570\uff0c\\(1\\) \u4ee3\u8868\u8d1f\u6570\uff0c\u4ece\u800c\u53ef\u4ee5\u8868\u793a \\(2^{31}\\) \u4e2a\u6b63\u6570\u548c \\(2^{31}\\) \u4e2a\u8d1f\u6570\uff1b
      3.  
    3. \u5f53\u6240\u6709 bits \u4e3a 0 \u65f6\u4ee3\u8868\u6570\u5b57 \\(0\\) \uff0c\u4ece\u96f6\u5f00\u59cb\u589e\u5927\uff0c\u53ef\u5f97\u6700\u5927\u6b63\u6570\u4e3a \\(2^{31} - 1\\) \uff1b
      4.  
    4. \u5269\u4f59 \\(2^{31}\\) \u4e2a\u6570\u5b57\u5168\u90e8\u7528\u6765\u8868\u793a\u8d1f\u6570\uff0c\u56e0\u6b64\u6700\u5c0f\u8d1f\u6570\u4e3a \\(-2^{31}\\) \uff1b\u5177\u4f53\u7ec6\u8282\u6d89\u53ca\u5230\u5230\u201c\u6e90\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u201d\u77e5\u8bc6\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u67e5\u9605\u5b66\u4e60\uff1b  
      5.  
     
    \u5176\u5b83\u6574\u6570\u7c7b\u578b byte, short, long \u53d6\u503c\u8303\u56f4\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e0e int \u7c7b\u4f3c\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_2","title":"\u6d6e\u70b9\u6570\u8868\u793a\u65b9\u5f0f *","text":"
    Note
     
    \u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u540e\u7684 * \u7b26\u53f7\u4ee3\u8868\u9009\u8bfb\u7ae0\u8282\uff0c\u5982\u679c\u4f60\u89c9\u5f97\u7406\u89e3\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u7eed\u518d\u5355\u72ec\u653b\u514b\u3002 
     
    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u7591\u60d1\uff1a int \u548c float \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48 float \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \uff1f\u6309\u8bf4 float \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002
     
    \u5176\u5b9e\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 float \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002IEEE 754 \u6807\u51c6\u89c4\u5b9a\uff0c32-bit \u957f\u5ea6\u7684 float \u7531\u4ee5\u4e0b\u90e8\u5206\u6784\u6210\uff1a
     
       
    • \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff1b
      •  
    • \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff1b
      •  
    • \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 24 bits \uff0c\u5176\u4e2d 23 \u4f4d\u663e\u5f0f\u5b58\u50a8\uff1b
      •  
     
    \u8bbe 32-bit \u4e8c\u8fdb\u5236\u6570\u7684\u7b2c \\(i\\) \u4f4d\u4e3a \\(b_i\\) \uff0c\u5219 float \u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u5b9a\u4e49\u4e3a
     \\[ \\text { val } = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] 
    \u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a
     \\[ \\text { val }=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] 
    \u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a
     \\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\} , \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] 
     
     Fig. IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u8868\u793a\u65b9\u5f0f 
     
    \u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u6613\u5f97
     \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] 
    \u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u5f00\u59cb\u7684\u95ee\u9898\uff1afloat \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c float \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002
     
    \u6d6e\u70b9\u6570 float \u867d\u7136\u62d3\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b int \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 float \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002
     
    \u8fdb\u4e00\u6b65\u5730\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002
     \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) 
    \u7279\u522b\u5730\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u5c0f\u6570\u7cbe\u5ea6\uff1a
     
       
    • \u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\approx 1.18 \\times 10^{-38}\\) \uff1b
      •  
    • \u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23} \\approx 1.4 \\times 10^{-45}\\) \uff1b
      •  
     
    \u53cc\u7cbe\u5ea6 double \u4e5f\u91c7\u7528\u7c7b\u4f3c float \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_data_structure/data_and_memory/#_3","title":"\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb","text":"
    \u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5b83\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\uff0c\u800c\u4e0d\u662f\u201c\u6570\u636e\u201d\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u81ea\u7136\u60f3\u5230\u4f7f\u7528\u300c\u6570\u7ec4\u300d\u6570\u636e\u7ed3\u6784\u3002\u6570\u7ec4\u7684\u5b58\u50a8\u65b9\u5f0f\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u3001\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5176\u4e2d\u5b58\u50a8\u7684\u662f\u6574\u6570 int \uff0c\u8fd8\u662f\u5c0f\u6570 float \uff0c\u6216\u662f\u5b57\u7b26 char \uff0c\u5219\u4e0e\u6240\u8c13\u7684\u6570\u636e\u7684\u7ed3\u6784\u65e0\u5173\u4e86\u3002
     
    \u6362\u8a00\u4e4b\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] booleans = new boolean[5];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool booleans[5];\n
     
    \"\"\" Python \u7684 list \u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 \"\"\"\nlist = [0, 0.0, 'a', False]\n
     
    // \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar booleans = [5]bool{}\n
     
    /* JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 */\nconst array = [0, 0.0, 'a', false];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst booleans: boolean[] = [];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool booleans[10];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] booleans = new bool[5];\n
     
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet booleans = Array(repeating: Bool(), count: 5)\n
     
    \n
    "},{"location":"chapter_data_structure/data_and_memory/#312","title":"3.1.2. \u00a0 \u8ba1\u7b97\u673a\u5185\u5b58","text":"
    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u5185\u5b58\u548c\u786c\u76d8\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u5b58\u50a8\u786c\u4ef6\u8bbe\u5907\u3002\u300c\u786c\u76d8\u300d\u4e3b\u8981\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5bb9\u91cf\u8f83\u5927\uff08\u901a\u5e38\u53ef\u8fbe\u5230 TB \u7ea7\u522b\uff09\u3001\u901f\u5ea6\u8f83\u6162\u3002\u300c\u5185\u5b58\u300d\u7528\u4e8e\u8fd0\u884c\u7a0b\u5e8f\u65f6\u6682\u5b58\u6570\u636e\uff0c\u901f\u5ea6\u8f83\u5feb\uff0c\u4f46\u5bb9\u91cf\u8f83\u5c0f\uff08\u901a\u5e38\u4e3a GB \u7ea7\u522b\uff09\u3002
     
    \u7b97\u6cd5\u8fd0\u884c\u4e2d\uff0c\u76f8\u5173\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8 1 byte \u7684\u6570\u636e\uff0c\u5728\u7b97\u6cd5\u8fd0\u884c\u65f6\uff0c\u6240\u6709\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u8fd9\u4e9b\u5355\u5143\u683c\u4e2d\u3002
     
    \u7cfb\u7edf\u901a\u8fc7\u300c\u5185\u5b58\u5730\u5740 Memory Location\u300d\u6765\u8bbf\u95ee\u76ee\u6807\u5185\u5b58\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u7ed9\u8868\u683c\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u7f16\u53f7\uff0c\u4fdd\u8bc1\u6bcf\u5757\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u72ec\u7acb\u7684\u5185\u5b58\u5730\u5740\u3002\u81ea\u6b64\uff0c\u7a0b\u5e8f\u4fbf\u901a\u8fc7\u8fd9\u4e9b\u5730\u5740\uff0c\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002
     
     
     Fig. \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 
     
    \u5185\u5b58\u8d44\u6e90\u662f\u8bbe\u8ba1\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u8003\u8651\u56e0\u7d20\u3002\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5f53\u5185\u5b58\u88ab\u67d0\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u4e0d\u80fd\u88ab\u5176\u5b83\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u3002\u6211\u4eec\u9700\u8981\u6839\u636e\u5269\u4f59\u5185\u5b58\u8d44\u6e90\u7684\u60c5\u51b5\u6765\u8bbe\u8ba1\u7b97\u6cd5\u3002\u4f8b\u5982\uff0c\u82e5\u5269\u4f59\u5185\u5b58\u7a7a\u95f4\u6709\u9650\uff0c\u5219\u8981\u6c42\u7b97\u6cd5\u5360\u7528\u7684\u5cf0\u503c\u5185\u5b58\u4e0d\u80fd\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u5185\u5b58\uff1b\u82e5\u8fd0\u884c\u7684\u7a0b\u5e8f\u5f88\u591a\u3001\u7f3a\u5c11\u5927\u5757\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u5219\u8981\u6c42\u9009\u53d6\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u79bb\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002
    "},{"location":"chapter_data_structure/summary/","title":"3.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6574\u6570 byte, short, int, long \u3001\u6d6e\u70b9\u6570 float, double \u3001\u5b57\u7b26 char \u3001\u5e03\u5c14 boolean \u662f\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u51b3\u5b9a\u4e86\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u3002
      •  
    • \u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u7684\u5185\u5b58\u4e2d\u3002\u5185\u5b58\u4e2d\u6bcf\u5757\u7a7a\u95f4\u90fd\u6709\u72ec\u7acb\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u662f\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u6570\u636e\u7684\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\u4e3b\u8981\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u4e2d\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u5f62\u5f0f\u3002
      •  
    • \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u6709\u7ebf\u6027\u3001\u6811\u72b6\u3001\u7f51\u72b6\u7b49\u3002\u6211\u4eec\u4e00\u822c\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u6839\u636e\u5b9e\u73b0\u65b9\u5f0f\u7684\u4e0d\u540c\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u662f\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u3002
      •  
    • \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u6709\u4e24\u79cd\uff0c\u5206\u522b\u662f\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\uff0c\u6240\u6709\u7684\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u6216\u94fe\u8868\u3001\u6216\u4e24\u8005\u7ec4\u5408\u5b9e\u73b0\u7684\u3002
      •  
    "},{"location":"chapter_graph/graph/","title":"9.1. \u00a0 \u56fe","text":"
    \u300c\u56fe Graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 Vertex\u300d\u548c\u300c\u8fb9 Edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4f8b\u5982\uff0c\u4ee5\u4e0b\u8868\u793a\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe
     \\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\] 
     
     Fig. \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb 
     
    \u90a3\u4e48\uff0c\u56fe\u4e0e\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb\u662f\u4ec0\u4e48\uff1f\u5982\u679c\u6211\u4eec\u628a\u300c\u9876\u70b9\u300d\u770b\u4f5c\u7ed3\u70b9\uff0c\u628a\u300c\u8fb9\u300d\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u7ed3\u70b9\u7684\u6307\u9488\uff0c\u5219\u53ef\u5c06\u300c\u56fe\u300d\u770b\u6210\u4e00\u79cd\u4ece\u300c\u94fe\u8868\u300d\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u76f8\u6bd4\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4e5f\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002
    "},{"location":"chapter_graph/graph/#911","title":"9.1.1. \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b","text":"
    \u6839\u636e\u8fb9\u662f\u5426\u6709\u65b9\u5411\uff0c\u5206\u4e3a\u300c\u65e0\u5411\u56fe Undirected Graph\u300d\u548c\u300c\u6709\u5411\u56fe Directed Graph\u300d\u3002
     
       
    • \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u201c\u53cc\u5411\u201d\u7684\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\uff1b
      •  
    • \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u662f\u6709\u65b9\u5411\u7684\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\uff1b
      •  
     
     
     Fig. \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe 
     
    \u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u5206\u4e3a\u300c\u8fde\u901a\u56fe Connected Graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe Disconnected Graph\u300d\u3002
     
       
    • \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\uff1b
      •  
    • \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\uff1b
      •  
     
     
     Fig. \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe 
     
    \u6211\u4eec\u53ef\u4ee5\u7ed9\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u5f97\u5230\u300c\u6709\u6743\u56fe Weighted Graph\u300d\u3002\u4f8b\u5982\uff0c\u5728\u738b\u8005\u8363\u8000\u7b49\u6e38\u620f\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u4f7f\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002
     
     
     Fig. \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe 
    "},{"location":"chapter_graph/graph/#912","title":"9.1.2. \u00a0 \u56fe\u5e38\u7528\u672f\u8bed","text":"
       
    • \u300c\u90bb\u63a5 Adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u6b64\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u4f8b\u5982\uff0c\u4e0a\u56fe\u4e2d\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2, 3, 5 \u3002
      •  
    • \u300c\u8def\u5f84 Path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u8d70\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\uff0c\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u4f8b\u5982\uff0c\u4e0a\u56fe\u4e2d\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u4e2a\u8def\u5f84\u3002
      •  
    • \u300c\u5ea6 Degree\u300d\u8868\u793a\u4e00\u4e2a\u9876\u70b9\u5177\u6709\u591a\u5c11\u6761\u8fb9\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 In-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 Out-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
      •  
    "},{"location":"chapter_graph/graph/#913","title":"9.1.3. \u00a0 \u56fe\u7684\u8868\u793a","text":"
    \u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u6cd5\u6709\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u3002\u4ee5\u4e0b\u4f7f\u7528\u300c\u65e0\u5411\u56fe\u300d\u6765\u4e3e\u4f8b\u3002
    "},{"location":"chapter_graph/graph/#_1","title":"\u90bb\u63a5\u77e9\u9635","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 Adjacency Matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u4f7f\u7528 \\(1\\) \u6216 \\(0\\) \u6765\u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8bb0\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\) \u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u5219\u77e9\u9635\u5143\u7d20 \\(M[i][j] = 1\\) \u4ee3\u8868\u7740\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u6709\u8fb9\uff0c\u76f8\u53cd\u5730 \\(M[i][j] = 0\\) \u4ee3\u8868\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002
     
     
     Fig. \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a 
     
    \u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a
     
       
    • \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u800c\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
      •  
    • \u300c\u65e0\u5411\u56fe\u300d\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
      •  
    • \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) , \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u80fd\u591f\u8868\u793a\u300c\u6709\u6743\u56fe\u300d\u3002
      •  
     
    \u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u6765\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u5927\u3002
    "},{"location":"chapter_graph/graph/#_2","title":"\u90bb\u63a5\u8868","text":"
    \u300c\u90bb\u63a5\u8868 Adjacency List\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u7ed3\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u5373\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002
     
     
     Fig. \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a 
     
    \u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u5f80\u5f80\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u4f46\u662f\uff0c\u56e0\u4e3a\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u6240\u4ee5\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002
     
    \u89c2\u5bdf\u4e0a\u56fe\u53d1\u73b0\uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u300c\u94fe\u5730\u5740\u6cd5\u300d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u7c7b\u4f3c\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\uff0c\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u4e2d\u5e8f\u904d\u5386\u83b7\u53d6\u6709\u5e8f\u5e8f\u5217\uff1b\u8fd8\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a\u54c8\u5e0c\u8868\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u4f4e\u81f3 \\(O(1)\\) \u3002
    "},{"location":"chapter_graph/graph/#914","title":"9.1.4. \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"
    \u73b0\u5b9e\u4e2d\u7684\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u4f7f\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u5bf9\u5e94\u7684\u5f85\u6c42\u89e3\u95ee\u9898\u4e5f\u53ef\u4ee5\u88ab\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002
     \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2. \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"
    \u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u5206\u4e3a\u5bf9\u300c\u8fb9\u300d\u7684\u64cd\u4f5c\u548c\u5bf9\u300c\u9876\u70b9\u300d\u7684\u64cd\u4f5c\uff0c\u5728\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u8fd9\u4e24\u79cd\u8868\u793a\u4e0b\u7684\u5b9e\u73b0\u65b9\u5f0f\u4e0d\u540c\u3002
    "},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6709\uff1a
     
       
    • \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u8fb9\u7684\u5bf9\u5e94\u5143\u7d20\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
      •  
    • \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
      •  
    • \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 vertices \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 adjMat \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
      •  
     \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_adjacency_matrix.java
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<Integer> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new ArrayList<>();\nthis.adjMat = new ArrayList<>();\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (int[] e : edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<Integer> newRow = new ArrayList<>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.add(0);\n}\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<Integer> row : adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfBoundsException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<Integer> row : adjMat) {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat.get(i).set(j, 1);\nadjMat.get(j).set(i, 1);\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\nadjMat.get(i).set(j, 0);\nadjMat.get(j).set(i, 0);\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nSystem.out.print(\"\u9876\u70b9\u5217\u8868 = \");\nSystem.out.println(vertices);\nSystem.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
     graph_adjacency_matrix.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvector<int> vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(const vector<int>& vertices, const vector<vector<int>>& edges) {\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const vector<int>& edge : edges) {\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() const {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.push_back(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nadjMat.emplace_back(n, 0);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (vector<int>& row : adjMat) {\nrow.push_back(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.erase(vertices.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.erase(adjMat.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (vector<int>& row : adjMat) {\nrow.erase(row.begin() + index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid print() {\ncout << \"\u9876\u70b9\u5217\u8868 = \";\nPrintUtil::printVector(vertices);\ncout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\nPrintUtil::printVectorMatrix(adjMat);\n}\n};\n
     graph_adjacency_matrix.py
    class GraphAdjMat:\n\"\"\" \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b \"\"\"\n# \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices: List[int] = []\n# \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadj_mat: List[List[int]] = []\ndef __init__(self, vertices: List[int], edges: List[List[int]]) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.vertices: List[int] = []\nself.adj_mat: List[List[int]] = []\n# \u6dfb\u52a0\u9876\u70b9\nfor val in vertices:\nself.add_vertex(val)\n# \u6dfb\u52a0\u8fb9\n# \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges:\nself.add_edge(e[0], e[1])\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u9876\u70b9\u6570\u91cf \"\"\"\nreturn len(self.vertices)\ndef add_vertex(self, val: int) -> None:\n\"\"\" \u6dfb\u52a0\u9876\u70b9 \"\"\"\nn = self.size()\n# \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.append(val)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnew_row = [0] * n\nself.adj_mat.append(new_row)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in self.adj_mat:\nrow.append(0)\ndef remove_vertex(self, index: int) -> None:\n\"\"\" \u5220\u9664\u9876\u70b9 \"\"\"\nif index >= self.size():\nraise IndexError()\n# \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in self.adj_mat:\nrow.pop(index)\ndef add_edge(self, i: int, j: int) -> None:\n\"\"\" \u6dfb\u52a0\u8fb9 \"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\n# \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1\nself.adj_mat[j][i] = 1\ndef remove_edge(self, i: int, j: int) -> None:\n\"\"\" \u5220\u9664\u8fb9 \"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\nself.adj_mat[i][j] = 0\nself.adj_mat[j][i] = 0\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u90bb\u63a5\u77e9\u9635 \"\"\"\nprint(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\nprint(\"\u90bb\u63a5\u77e9\u9635 =\")\nprint_matrix(self.adj_mat)\n
     graph_adjacency_matrix.go
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices []int\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat [][]int\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n// \u6dfb\u52a0\u9876\u70b9\nn := len(vertices)\nadjMat := make([][]int, n)\nfor i := range adjMat {\nadjMat[i] = make([]int, n)\n}\n// \u521d\u59cb\u5316\u56fe\ng := &graphAdjMat{\nvertices: vertices,\nadjMat:   adjMat,\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor i := range edges {\ng.addEdge(edges[i][0], edges[i][1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\nreturn len(g.vertices)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\nn := g.size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\ng.vertices = append(g.vertices, val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnewRow := make([]int, n)\ng.adjMat = append(g.adjMat, newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i], 0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\nif index >= g.size() {\nreturn\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\ng.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\ng.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\ng.adjMat[i][j] = 1\ng.adjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\ng.adjMat[i][j] = 0\ng.adjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\nfmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\nfmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\nfor i := range g.adjMat {\nfmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n}\n}\n
     graph_adjacency_matrix.js
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices;   // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat;     // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices, edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val) {\nconst n = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow = [];\nfor (let j = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index) {\nif (index >= this.size()) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint() {\nconsole.log(\"\u9876\u70b9\u5217\u8868 = \", this.vertices);\nconsole.log(\"\u90bb\u63a5\u77e9\u9635 =\", this.adjMat);\n}\n}\n
     graph_adjacency_matrix.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices: number[];     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat: number[][];     // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices: number[], edges: number[][]) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val: number): void {\nconst n: number = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow: number[] = [];\nfor (let j: number = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index: number): void {\nif (index >= this.size()) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError(\"Index Out Of Bounds Exception\");\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint(): void {\nconsole.log(\"\u9876\u70b9\u5217\u8868 = \", this.vertices);\nconsole.log(\"\u90bb\u63a5\u77e9\u9635 =\", this.adjMat);\n}\n}\n
     graph_adjacency_matrix.c
    [class]{graphAdjMat}-[func]{}\n
     graph_adjacency_matrix.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat\n{\nList<int> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjMat(int[] vertices, int[][] edges)\n{\nthis.vertices = new List<int>();\nthis.adjMat = new List<List<int>>();\n// \u6dfb\u52a0\u9876\u70b9\nforeach (int val in vertices)\n{\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nforeach (int[] e in edges)\n{\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size()\n{\nreturn vertices.Count;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val)\n{\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.Add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = new List<int>(n);\nfor (int j = 0; j < n; j++)\n{\nnewRow.Add(0);\n}\nadjMat.Add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nforeach (List<int> row in adjMat)\n{\nrow.Add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index)\n{\nif (index >= size())\nthrow new IndexOutOfRangeException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nforeach (List<int> row in adjMat)\n{\nrow.RemoveAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j)\n{\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j)\n{\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print()\n{\nConsole.Write(\"\u9876\u70b9\u5217\u8868 = \");\nPrintUtil.PrintList(vertices);\nConsole.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
     graph_adjacency_matrix.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nprivate var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nprivate var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(vertices: [Int], edges: [[Int]]) {\nself.vertices = []\nadjMat = []\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\naddVertex(val: val)\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges {\naddEdge(i: e[0], j: e[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc size() -> Int {\nvertices.count\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc addVertex(val: Int) {\nlet n = size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.append(val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nlet newRow = Array(repeating: 0, count: n)\nadjMat.append(newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i in adjMat.indices {\nadjMat[i].append(0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc removeVertex(index: Int) {\nif index >= size() {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i in adjMat.indices {\nadjMat[i].remove(at: index)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc addEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1\nadjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc removeEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\nadjMat[i][j] = 0\nadjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc print() {\nSwift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\nSwift.print(vertices)\nSwift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\nPrintUtil.printMatrix(matrix: adjMat)\n}\n}\n
     graph_adjacency_matrix.zig
    \n
    "},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"
    \u8bbe\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u6709\uff1a
     
       
    • \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u5c3e\u90e8\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u8be2\u4e0e\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u4e0e\u6dfb\u52a0\u8fb9\u4e00\u6837\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
      •  
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\u5373\u53ef\uff0c\u5e76\u4ee5\u65b0\u589e\u9876\u70b9\u4e3a\u94fe\u8868\u5934\u7ed3\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
      •  
    • \u5220\u9664\u9876\u70b9\uff1a\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
      •  
    • \u521d\u59cb\u5316\uff1a\u9700\u8981\u5728\u90bb\u63a5\u8868\u4e2d\u5efa\u7acb \\(n\\) \u4e2a\u7ed3\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
      •  
     \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 
     
     
     
     
     
    \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u56fe\u7684\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u6ce8\u610f\u5230\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 Vertex \u7ed3\u70b9\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a
     
       
    • \u5982\u679c\u6211\u4eec\u9009\u62e9\u901a\u8fc7\u9876\u70b9\u503c\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u503c\u91cd\u590d\u7684\u9876\u70b9\u5c06\u65e0\u6cd5\u88ab\u533a\u5206\u3002
      •  
    • \u5982\u679c\u7c7b\u4f3c\u90bb\u63a5\u77e9\u9635\u90a3\u6837\uff0c\u4f7f\u7528\u9876\u70b9\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\u3002\u90a3\u4e48\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u5176\u4e2d \\(> i\\) \u7684\u7d22\u5f15\u5168\u90e8\u6267\u884c \\(-1\\) \uff0c\u8fd9\u6837\u64cd\u4f5c\u6548\u7387\u592a\u4f4e\u3002
      •  
    • \u56e0\u6b64\u6211\u4eec\u8003\u8651\u5f15\u5165\u9876\u70b9\u7c7b Vertex \uff0c\u4f7f\u5f97\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684\u5bf9\u8c61\uff0c\u6b64\u65f6\u5220\u9664\u9876\u70b9\u65f6\u5c31\u65e0\u9700\u6539\u52a8\u5176\u4f59\u9876\u70b9\u4e86\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_adjacency_list.java
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new HashMap<>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (Vertex[] edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList.get(vet1).add(vet2);\nadjList.get(vet2).add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList.get(vet1).remove(vet2);\nadjList.get(vet2).remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.containsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.put(vet, new ArrayList<>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet))\nthrow new IllegalArgumentException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (List<Vertex> list : adjList.values()) {\nlist.remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nSystem.out.println(\"\u90bb\u63a5\u8868 =\");\nfor (Map.Entry<Vertex, List<Vertex>> entry : adjList.entrySet()) {\nList<Integer> tmp = new ArrayList<>();\nfor (Vertex vertex : entry.getValue())\ntmp.add(vertex.val);\nSystem.out.println(entry.getKey().val + \": \" + tmp + \",\");\n}\n}\n}\n
     graph_adjacency_list.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\npublic:\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nunordered_map<Vertex*, vector<Vertex*>> adjList;\n/* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u7ed3\u70b9 */\nvoid remove(vector<Vertex*> &vec, Vertex *vet) {\nfor (int i = 0; i < vec.size(); i++) {\nif (vec[i] == vet) {\nvec.erase(vec.begin() + i);\nbreak;\n}\n}\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(const vector<vector<Vertex*>>& edges) {\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const vector<Vertex*>& edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() { return adjList.size(); }\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex* vet1, Vertex* vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].push_back(vet2);\nadjList[vet2].push_back(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex* vet1, Vertex* vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5220\u9664\u8fb9 vet1 - vet2\nremove(adjList[vet1], vet2);\nremove(adjList[vet2], vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex* vet) {\nif (adjList.count(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = vector<Vertex*>();\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex* vet) {\nif (!adjList.count(vet))\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.erase(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (auto& [key, vec] : adjList) {\nremove(vec, vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid print() {\ncout << \"\u90bb\u63a5\u8868 =\" << endl;\nfor (auto& adj  : adjList) {\nconst auto& key= adj.first;\nconst auto& vec = adj.second;\ncout << key->val << \": \";\nPrintUtil::printVector(vetsToVals(vec));\n}\n}\n};\n
     graph_adjacency_list.py
    class GraphAdjList:\n\"\"\" \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b \"\"\"\ndef __init__(self, edges: List[List[Vertex]]) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nself.adj_list: Dict = {}\n# \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges:\nself.add_vertex(edge[0])\nself.add_vertex(edge[1])\nself.add_edge(edge[0], edge[1])\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u9876\u70b9\u6570\u91cf \"\"\"\nreturn len(self.adj_list)\ndef add_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\" \u6dfb\u52a0\u8fb9 \"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError\n# \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list[vet1].append(vet2)\nself.adj_list[vet2].append(vet1)\ndef remove_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\" \u5220\u9664\u8fb9 \"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError\n# \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list[vet1].remove(vet2)\nself.adj_list[vet2].remove(vet1)\ndef add_vertex(self, vet: Vertex) -> None:\n\"\"\" \u6dfb\u52a0\u9876\u70b9 \"\"\"\nif vet in self.adj_list:\nreturn\n# \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list[vet] = []\ndef remove_vertex(self, vet: Vertex) -> None:\n\"\"\" \u5220\u9664\u9876\u70b9 \"\"\"\nif vet not in self.adj_list:\nraise ValueError\n# \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.pop(vet)\n# \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor vertex in self.adj_list:\nif vet in self.adj_list[vertex]:\nself.adj_list[vertex].remove(vet)\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u90bb\u63a5\u8868 \"\"\"\nprint(\"\u90bb\u63a5\u8868 =\")\nfor vertex in self.adj_list:\ntmp = [v.val for v in self.adj_list[vertex]]\nprint(f\"{vertex.val}: {tmp},\")\n
     graph_adjacency_list.go
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList map[Vertex][]Vertex\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\ng := &graphAdjList{\nadjList: make(map[Vertex][]Vertex),\n}\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor _, edge := range edges {\ng.addVertex(edge[0])\ng.addVertex(edge[1])\ng.addEdge(edge[0], edge[1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\nreturn len(g.adjList)\n}\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\ng.adjList[vet1] = append(g.adjList[vet1], vet2)\ng.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nDeleteSliceElms(g.adjList[vet1], vet2)\nDeleteSliceElms(g.adjList[vet2], vet1)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif ok {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ng.adjList[vet] = make([]Vertex, 0)\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif !ok {\npanic(\"error\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\ndelete(g.adjList, vet)\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor _, list := range g.adjList {\nDeleteSliceElms(list, vet)\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\nvar builder strings.Builder\nfmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\nfor k, v := range g.adjList {\nbuilder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\nfor _, vet := range v {\nbuilder.WriteString(strconv.Itoa(vet.Val) + \" \")\n}\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
     graph_adjacency_list.js
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1, vet2) {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1, vet2) {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet) {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet) {\nif (!this.adjList.has(vet)) {\nthrow new Error(\"Illegal Argument Exception\");\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint() {\nconsole.log(\"\u90bb\u63a5\u8868 =\");\nfor (const [key, value] of this.adjList) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + \": \" + tmp.join());\n}\n}\n}\n
     graph_adjacency_list.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList: Map<Vertex, Vertex[]>;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges: Vertex[][]) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1: Vertex, vet2: Vertex): void {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1: Vertex, vet2: Vertex): void {\nif (!this.adjList.has(vet1) || !this.adjList.has(vet2) || vet1 === vet2) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet: Vertex): void {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet: Vertex): void {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index: number = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint(): void {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList.entries()) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
     graph_adjacency_list.c
    [class]{graphAdjList}-[func]{}\n
     graph_adjacency_list.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList\n{\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic Dictionary<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjList(Vertex[][] edges)\n{\nthis.adjList = new Dictionary<Vertex, List<Vertex>>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nforeach (Vertex[] edge in edges)\n{\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size()\n{\nreturn adjList.Count;\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2)\n{\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].Add(vet2);\nadjList[vet2].Add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2)\n{\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1].Remove(vet2);\nadjList[vet2].Remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet)\n{\nif (adjList.ContainsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.Add(vet, new List<Vertex>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet)\n{\nif (!adjList.ContainsKey(vet))\nthrow new InvalidOperationException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.Remove(vet);\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nforeach (List<Vertex> list in adjList.Values)\n{\nlist.Remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print()\n{\nConsole.WriteLine(\"\u90bb\u63a5\u8868 =\");\nforeach (KeyValuePair<Vertex, List<Vertex>> entry in adjList)\n{\nList<int> tmp = new List<int>();\nforeach (Vertex vertex in entry.Value)\ntmp.Add(vertex.Val);\nConsole.WriteLine(entry.Key.Val + \": [\" + string.Join(\", \", tmp) + \"],\");\n}\n}\n}\n
     graph_adjacency_list.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic private(set) var adjList: [Vertex: [Vertex]]\n/* \u6784\u9020\u65b9\u6cd5 */\npublic init(edges: [[Vertex]]) {\nadjList = [:]\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\naddVertex(vet: edge[0])\naddVertex(vet: edge[1])\naddEdge(vet1: edge[0], vet2: edge[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic func size() -> Int {\nadjList.count\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic func addEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]?.append(vet2)\nadjList[vet2]?.append(vet1)\n}\n/* \u5220\u9664\u8fb9 */\npublic func removeEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]?.removeAll(where: { $0 == vet2 })\nadjList[vet2]?.removeAll(where: { $0 == vet1 })\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic func addVertex(vet: Vertex) {\nif adjList[vet] != nil {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = []\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic func removeVertex(vet: Vertex) {\nif adjList[vet] == nil {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.removeValue(forKey: vet)\n// \u904d\u5386\u5176\u5b83\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor key in adjList.keys {\nadjList[key]?.removeAll(where: { $0 == vet })\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic func print() {\nSwift.print(\"\u90bb\u63a5\u8868 =\")\nfor entry in adjList {\nvar tmp: [Int] = []\nfor vertex in entry.value {\ntmp.append(vertex.val)\n}\nSwift.print(\"\\(entry.key.val): \\(tmp),\")\n}\n}\n}\n
     graph_adjacency_list.zig
    [class]{GraphAdjList}-[func]{}\n
    "},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3. \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"
    \u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u4e0b\u8868\u4e3a\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\u3002
     \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) 
    \u89c2\u5bdf\u4e0a\u8868\uff0c\u8c8c\u4f3c\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u603b\u7ed3\u4ee5\u4e0a\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
    "},{"location":"chapter_graph/graph_traversal/","title":"9.3. \u00a0 \u56fe\u7684\u904d\u5386","text":"
    \u56fe\u4e0e\u6811\u7684\u5173\u7cfb
     
    \u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u53ef\u4ee5\u4ee3\u8868\u4efb\u610f\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u672c\u8d28\u4e0a\uff0c\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u662f\u56fe\u7684\u4e00\u7c7b\u7279\u4f8b\u3002\u90a3\u4e48\u663e\u7136\uff0c\u6811\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u4e24\u8005\u7684\u65b9\u6cd5\u662f\u975e\u5e38\u7c7b\u4f3c\u7684\uff0c\u5efa\u8bae\u4f60\u5728\u5b66\u4e60\u672c\u7ae0\u8282\u7684\u8fc7\u7a0b\u4e2d\u5c06\u4e24\u8005\u878d\u4f1a\u8d2f\u901a\u3002
     
    \u300c\u56fe\u300d\u4e0e\u300c\u6811\u300d\u90fd\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u641c\u7d22\u7b97\u6cd5\u300d\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002
     
    \u7c7b\u4f3c\u5730\uff0c\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u5206\u4e3a\u4e24\u79cd\uff0c\u5373\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 Breadth-First Traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Travsersal\u300d\uff0c\u4e5f\u79f0\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 Depth-First Search\u300d\uff0c\u7b80\u79f0\u4e3a BFS \u548c DFS \u3002
    "},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1. \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u4f18\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\u5f00\u59cb\u8bbf\u95ee\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u968f\u540e\u904d\u5386\u4e0b\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026
     
     
     Fig. \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 
    "},{"location":"chapter_graph/graph_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    BFS \u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u662f\u5f02\u66f2\u540c\u5de5\u7684\u3002
     
       
    1. \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 startVet \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\uff1b
      2.  
    2. \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5f39\u51fa\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u5e76\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\uff1b
      3.  
    3. \u5faa\u73af 2. \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6210\u540e\u7ed3\u675f\uff1b
      4.  
     
    \u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u54ea\u4e9b\u7ed3\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_bfs.java
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>() {{ add(startVet); }};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new LinkedList<>() {{ offer(startVet); }};\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.isEmpty()) {\nVertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.cpp
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex*> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex*> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex*> visited = { startVet };\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nqueue<Vertex*> que;\nque.push(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.empty()) {\nVertex *vet = que.front();\nque.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (auto adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;           // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.push(adjVet);       // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.py
    def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:\n\"\"\" \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS \"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set([start_vet])\n# \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nque = collections.deque([start_vet])\n# \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile len(que) > 0:\nvet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet)      # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adj_vet in graph.adj_list[vet]:\nif adj_vet in visited:\ncontinue         # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adj_vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n
     graph_bfs.go
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\nvisited[startVet] = struct{}{}\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\nqueue := make([]Vertex, 0)\nqueue = append(queue, startVet)\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nfor len(queue) > 0 {\n// \u961f\u9996\u9876\u70b9\u51fa\u961f\nvet := queue[0]\nqueue = queue[1:]\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres = append(res, vet)\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nif !isExist {\nqueue = append(queue, adjVet)\nvisited[adjVet] = struct{}{}\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_bfs.js
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift();    // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet);              // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue;           // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet);       // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet);    // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.ts
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
     graph_bfs.c
    [class]{}-[func]{graphBFS}\n
     graph_bfs.cs
    [class]{graph_bfs}-[func]{graphBFS}\n
     graph_bfs.swift
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = [startVet]\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nvar que: [Vertex] = [startVet]\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.isEmpty {\nlet vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_bfs.zig
    [class]{}-[func]{graphBFS}\n
     
    \u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u4ee5\u4e0b\u52a8\u753b\u56fe\u793a\u6765\u52a0\u6df1\u7406\u89e3\u3002
     <1><2><3><4><5><6><7><8><9><10><11> 
     
     
     
     
     
     
     
     
     
     
     
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
     
    \u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\) , \\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u3001\u9876\u70b9 \\(2\\) , \\(4\\) , \\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3001\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026
    "},{"location":"chapter_graph/graph_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u3001\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \uff0c\u961f\u5217 que \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2. \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u4e0d\u65ad\u5730\u8bbf\u95ee\u5f53\u524d\u7ed3\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u56de\u6eaf\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5e95 + \u56de\u6eaf\uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u65f6\u7ed3\u675f\u3002
     
     
     Fig. \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 
    "},{"location":"chapter_graph/graph_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    \u8fd9\u79cd\u201c\u8d70\u5230\u5934 + \u56de\u6eaf\u201d\u7684\u7b97\u6cd5\u5f62\u5f0f\u4e00\u822c\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e BFS \u7c7b\u4f3c\uff0c\u5728 DFS \u4e2d\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig graph_dfs.java
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.cpp
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList& graph, unordered_set<Vertex*>& visited, vector<Vertex*>& res, Vertex* vet) {\nres.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex* adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;     // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex*> graphDFS(GraphAdjList& graph, Vertex* startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex*> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex*> visited;\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.py
    def dfs(graph: GraphAdjList, visited: Set[Vertex], res: List[Vertex], vet: Vertex):\n\"\"\" \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 \"\"\"\nres.append(vet)     # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adj_list[vet]:\nif adjVet in visited:\ncontinue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n# \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet)\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:\n\"\"\" \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \"\"\"\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set()\ndfs(graph, visited, res, start_vet)\nreturn res\n
     graph_dfs.go
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n// append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n*res = append(*res, vet)\nvisited[vet] = struct{}{}\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nif !isExist {\ndfs(g, visited, res, adjVet)\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\ndfs(g, visited, &res, startVet)\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
     graph_dfs.js
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\nres.push(vet);      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.ts
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(graph: GraphAdjList, visited: Set<Vertex>, res: Vertex[], vet: Vertex): void {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
     graph_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
     graph_dfs.cs
    [class]{graph_dfs}-[func]{dfs}\n[class]{graph_dfs}-[func]{graphDFS}\n
     graph_dfs.swift
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = []\ndfs(graph: graph, visited: &visited, res: &res, vet: startVet)\nreturn res\n}\n
     graph_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
     
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5176\u4e2d
     
       
    • \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u4ee3\u8868\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\uff1b
      •  
    • \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u4ee3\u8868\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u9012\u5f52\u65b9\u6cd5\u7684\u4f4d\u7f6e\uff1b
      •  
     
    \u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6a21\u62df\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002
     <1><2><3><4><5><6><7><8><9><10><11> 
     
     
     
     
     
     
     
     
     
     
     
    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
     
    \u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u884c\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
     
    \u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u4f53\u73b0\u4e09\u79cd\u4e0d\u540c\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u800c\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
    "},{"location":"chapter_graph/graph_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u88ab\u8bbf\u95ee\u4e00\u6b21\uff1b\u6240\u6709\u8fb9\u90fd\u88ab\u8bbf\u95ee\u4e86 \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_graph/summary/","title":"9.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
      •  
    • \u76f8\u6bd4\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4e5f\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002
      •  
    • \u6709\u5411\u56fe\u7684\u8fb9\u5b58\u5728\u65b9\u5411\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u90fd\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
      •  
    • \u90bb\u63a5\u77e9\u9635\u4f7f\u7528\u65b9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u4f7f\u7528 \\(1\\) \u6216 \\(0\\) \u6765\u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u7684\u589e\u5220\u67e5\u64cd\u4f5c\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u5360\u7528\u7a7a\u95f4\u5927\u3002
      •  
    • \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
      •  
    • \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
      •  
    • \u4ece\u7b97\u6cd5\u601d\u60f3\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d
      •  
    • \u56fe\u53ef\u4ee5\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u4f8b\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
      •  
    • \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
      •  
    • \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
      •  
    • \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_hashing/hash_collision/","title":"6.2. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570\u5e94\u8be5\u4e3a\u6bcf\u4e2a\u8f93\u5165\u4ea7\u751f\u552f\u4e00\u7684\u8f93\u51fa\uff0c\u4f7f\u5f97 key \u548c value \u4e00\u4e00\u5bf9\u5e94\u3002\u800c\u5b9e\u9645\u4e0a\uff0c\u5f80\u5f80\u5b58\u5728\u5411\u54c8\u5e0c\u51fd\u6570\u8f93\u5165\u4e0d\u540c\u7684 key \u800c\u4ea7\u751f\u76f8\u540c\u8f93\u51fa\u7684\u60c5\u51b5\uff0c\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4ece\u800c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002
     
    \u90a3\u4e48\uff0c\u4e3a\u4ec0\u4e48\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f\u672c\u8d28\u4e0a\u770b\uff0c\u7531\u4e8e\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u53ef\u907f\u514d\u5730\u4f1a\u51fa\u73b0\u591a\u4e2a\u8f93\u5165\u4ea7\u751f\u76f8\u540c\u8f93\u51fa\u7684\u60c5\u51b5\uff0c\u5373\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u4e00\u4e2a\u56fa\u5b9a\u5927\u5c0f\u7684\u6570\u7ec4\uff0c\u90a3\u4e48\u5fc5\u5b9a\u4f1a\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u5230\u540c\u4e00\u4e2a\u6570\u7ec4\u7d22\u5f15\u3002
     
    \u4e3a\u4e86\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff0c\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u8868\u6269\u5bb9\u6765\u51cf\u5c0f\u51b2\u7a81\u6982\u7387\u3002\u6781\u7aef\u60c5\u51b5\u4e0b\uff0c\u5f53\u8f93\u5165\u7a7a\u95f4\u548c\u8f93\u51fa\u7a7a\u95f4\u5927\u5c0f\u76f8\u7b49\u65f6\uff0c\u54c8\u5e0c\u8868\u5c31\u7b49\u4ef7\u4e8e\u6570\u7ec4\u4e86\uff0c\u6bcf\u4e2a key \u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u53ef\u8c13\u201c\u5927\u529b\u51fa\u5947\u8ff9\u201d\u3002
     
    \u53e6\u4e00\u65b9\u9762\uff0c\u8003\u8651\u901a\u8fc7\u4f18\u5316\u54c8\u5e0c\u8868\u7684\u8868\u793a\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e38\u89c1\u7684\u65b9\u6cd5\u6709\u300c\u94fe\u5f0f\u5730\u5740\u300d\u548c\u300c\u5f00\u653e\u5bfb\u5740\u300d\u3002
    "},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1. \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9","text":"
    \u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6876\u6570\u91cf \\(n\\) \u53d6\u4f59\uff0c\u4ee5\u5c06\u54c8\u5e0c\u503c\u6620\u5c04\u5230\u6876\u7684\u7d22\u5f15\u8303\u56f4\uff0c\u4ece\u800c\u5c06 key \u653e\u5165\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff08\u5373 \\(n\\) \u8d8a\u5927\uff09\u65f6\uff0c\u591a\u4e2a key \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002
     
    \u56e0\u6b64\uff0c\u5728\u54c8\u5e0c\u8868\u5185\u7684\u51b2\u7a81\u6574\u4f53\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u7f16\u7a0b\u8bed\u8a00\u4e00\u822c\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u79fb\u52a8\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u5f00\u9500\u5f88\u5927\u3002
     
    \u7f16\u7a0b\u8bed\u8a00\u4e00\u822c\u4f7f\u7528\u300c\u8d1f\u8f7d\u56e0\u5b50 Load Factor\u300d\u6765\u8bc4\u4f30\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u5e38\u7528\u4f5c\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u6bd4\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50 \\(> 0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06 HashMap \u5bb9\u91cf\u6269\u5145\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
    "},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2. \u00a0 \u94fe\u5f0f\u5730\u5740","text":"
    \u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u53ea\u80fd\u5b58\u50a8\u4e00\u4e2a\u5143\u7d20\uff08\u5373\u952e\u503c\u5bf9\uff09\u3002\u8003\u8651\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u6210\u4e00\u4e2a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u90fd\u5b58\u50a8\u5728\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002
     
     
     Fig. \u94fe\u5f0f\u5730\u5740 
     
    \u94fe\u5f0f\u5730\u5740\u4e0b\uff0c\u54c8\u5e0c\u8868\u64cd\u4f5c\u65b9\u6cd5\u4e3a\uff1a
     
       
    • \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 key \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6570\u7ec4\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u7ed3\u70b9\uff0c\u518d\u901a\u8fc7\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 key \u6765\u67e5\u627e\u952e\u503c\u5bf9\u3002
      •  
    • \u6dfb\u52a0\u5143\u7d20\uff1a\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u518d\u5c06\u7ed3\u70b9\uff08\u5373\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u5934\u90e8\u5373\u53ef\u3002
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u540c\u6837\u5148\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u518d\u904d\u5386\u94fe\u8868\u67e5\u627e\u5bf9\u5e94\u7ed3\u70b9\uff0c\u5220\u9664\u4e4b\u5373\u53ef\u3002
      •  
     
    \u94fe\u5f0f\u5730\u5740\u867d\u7136\u89e3\u51b3\u4e86\u54c8\u5e0c\u51b2\u7a81\u95ee\u9898\uff0c\u4f46\u4ecd\u5b58\u5728\u5c40\u9650\u6027\uff0c\u5305\u62ec\uff1a
     
       
    • \u5360\u7528\u7a7a\u95f4\u53d8\u5927\uff0c\u56e0\u4e3a\u94fe\u8868\u6216\u4e8c\u53c9\u6811\u5305\u542b\u7ed3\u70b9\u6307\u9488\uff0c\u76f8\u6bd4\u4e8e\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\uff1b
      •  
    • \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\uff1b
      •  
     
    \u4e3a\u4e86\u63d0\u5347\u64cd\u4f5c\u6548\u7387\uff0c\u53ef\u4ee5\u628a\u300c\u94fe\u8868\u300d\u8f6c\u5316\u4e3a\u300cAVL \u6811\u300d\u6216\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
    "},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3. \u00a0 \u5f00\u653e\u5bfb\u5740","text":"
    \u300c\u5f00\u653e\u5bfb\u5740\u300d\u4e0d\u5f15\u5165\u989d\u5916\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u6839\u636e\u63a2\u6d4b\u65b9\u6cd5\u7684\u4e0d\u540c\uff0c\u4e3b\u8981\u5206\u4e3a \u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u3002
    "},{"location":"chapter_hashing/hash_collision/#_1","title":"\u7ebf\u6027\u63a2\u6d4b","text":"
    \u300c\u7ebf\u6027\u63a2\u6d4b\u300d\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u67e5\u627e\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002
     
    \u63d2\u5165\u5143\u7d20\uff1a\u5982\u679c\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u4e00\u822c\u53d6 1 \uff09\uff0c\u76f4\u5230\u627e\u5230\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u5219\u5c06\u5143\u7d20\u63d2\u5165\u5230\u8be5\u7a7a\u4f4d\u4e2d\u3002
     
    \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u6267\u884c\u7ebf\u6027\u67e5\u627e\uff0c\u4f1a\u9047\u5230\u4e24\u79cd\u60c5\u51b5\uff1a
     
       
    1. \u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de value \u5373\u53ef\uff1b
      2.  
    2. \u82e5\u9047\u5230\u7a7a\u4f4d\uff0c\u5219\u8bf4\u660e\u67e5\u627e\u952e\u503c\u5bf9\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff1b
      3.  
     
     
     Fig. \u7ebf\u6027\u63a2\u6d4b 
     
    \u7ebf\u6027\u63a2\u6d4b\u5b58\u5728\u4ee5\u4e0b\u7f3a\u9677\uff1a
     
       
    • \u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u5220\u9664\u5143\u7d20\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5185\u51fa\u73b0\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u5728\u67e5\u627e\u5176\u4ed6\u5143\u7d20\u65f6\uff0c\u8be5\u7a7a\u4f4d\u6709\u53ef\u80fd\u5bfc\u81f4\u7a0b\u5e8f\u8ba4\u4e3a\u5143\u7d20\u4e0d\u5b58\u5728\uff08\u5373\u4e0a\u8ff0\u7b2c 2. \u79cd\u60c5\u51b5\uff09\u3002\u56e0\u6b64\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u6807\u8bb0\u5220\u9664\u5143\u7d20\u3002
      •  
    • \u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u6570\u7ec4\u5185\u88ab\u5360\u7528\u7684\u8fde\u7eed\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u8fdb\u8fd9\u4e00\u4f4d\u7f6e\u7684\u201c\u805a\u5806\u751f\u957f\u201d\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u7684\u52a3\u5316\u3002
      •  
    "},{"location":"chapter_hashing/hash_collision/#_2","title":"\u591a\u6b21\u54c8\u5e0c","text":"
    \u987e\u540d\u601d\u4e49\uff0c\u300c\u591a\u6b21\u54c8\u5e0c\u300d\u7684\u601d\u8def\u662f\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) , \\(f_2(x)\\) , \\(f_3(x)\\) , \\(\\cdots\\) \u8fdb\u884c\u63a2\u6d4b\u3002
     
    \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002
     
    \u67e5\u627e\u5143\u7d20\uff1a\u4ee5\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u67e5\u627e\uff0c\u5b58\u5728\u4e24\u79cd\u60c5\u51b5\uff1a
     
       
    1. \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de\u4e4b\uff1b
      2.  
    2. \u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64\u5143\u7d20\uff1b
      3.  
     
    \u76f8\u6bd4\u4e8e\u300c\u7ebf\u6027\u63a2\u6d4b\u300d\uff0c\u300c\u591a\u6b21\u54c8\u5e0c\u300d\u65b9\u6cd5\u66f4\u4e0d\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4ee3\u4ef7\u662f\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u989d\u5916\u8ba1\u7b97\u91cf\u3002
     
    \u5de5\u4e1a\u754c\u65b9\u6848
     
    Java \u91c7\u7528\u300c\u94fe\u5f0f\u5730\u5740\u300d\u3002\u5728 JDK 1.8 \u4e4b\u540e\uff0cHashMap \u5185\u6570\u7ec4\u957f\u5ea6\u5927\u4e8e 64 \u65f6\uff0c\u957f\u5ea6\u5927\u4e8e 8 \u7684\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002
     
    Python \u91c7\u7528\u300c\u5f00\u653e\u5bfb\u5740\u300d\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002 
     
    Golang \u91c7\u7528\u300c\u94fe\u5f0f\u5730\u5740\u300d\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u4fdd\u8bc1\u6027\u80fd\u3002
    "},{"location":"chapter_hashing/hash_map/","title":"6.1. \u00a0 \u54c8\u5e0c\u8868","text":"
    \u54c8\u5e0c\u8868\u901a\u8fc7\u5efa\u7acb\u300c\u952e key\u300d\u548c\u300c\u503c value\u300d\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u3002\u5177\u4f53\u5730\uff0c\u8f93\u5165\u4e00\u4e2a key \uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5e76\u83b7\u53d6 value \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
     
    \u4f8b\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5b66\u751f\u7684\u6570\u636e\u5e93\uff0c\u6bcf\u4e2a\u5b66\u751f\u6709\u201c\u59d3\u540d name \u201d\u548c\u201c\u5b66\u53f7 id \u201d\u4e24\u9879\u6570\u636e\uff0c\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u67e5\u8be2\u529f\u80fd\uff1a\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\uff0c\u5219\u53ef\u4ee5\u4f7f\u7528\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002
     
     
     Fig. \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a 
    "},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1. \u00a0 \u54c8\u5e0c\u8868\u6548\u7387","text":"
    \u9664\u4e86\u54c8\u5e0c\u8868\u4e4b\u5916\uff0c\u8fd8\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u6570\u636e\u7ed3\u6784\u6765\u5b9e\u73b0\u4e0a\u8ff0\u67e5\u8be2\u529f\u80fd\uff1a
     
       
    1. \u65e0\u5e8f\u6570\u7ec4\uff1a\u6bcf\u4e2a\u5143\u7d20\u4e3a  [\u5b66\u53f7, \u59d3\u540d] \uff1b
      2.  
    2. \u6709\u5e8f\u6570\u7ec4\uff1a\u5c06 1. \u4e2d\u7684\u6570\u7ec4\u6309\u7167\u5b66\u53f7\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff1b
      3.  
    3. \u94fe\u8868\uff1a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u503c\u4e3a [\u5b66\u53f7, \u59d3\u540d] \uff1b
      4.  
    4. \u4e8c\u53c9\u641c\u7d22\u6811\uff1a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u503c\u4e3a [\u5b66\u53f7, \u59d3\u540d] \uff0c\u6839\u636e\u5b66\u53f7\u5927\u5c0f\u6765\u6784\u5efa\u6811\uff1b
      5.  
     
    \u4f7f\u7528\u4e0a\u8ff0\u65b9\u6cd5\uff0c\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5982\u4e0b\u8868\u6240\u793a\uff08\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\uff0c\u8be6\u89e3\u53ef\u89c1 \u4e8c\u53c9\u641c\u7d22\u6811\u7ae0\u8282\uff09\u3002\u65e0\u8bba\u662f\u67e5\u627e\u5143\u7d20\u3001\u8fd8\u662f\u589e\u5220\u5143\u7d20\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u5168\u9762\u80dc\u51fa\uff01
     \u65e0\u5e8f\u6570\u7ec4 \u6709\u5e8f\u6570\u7ec4 \u94fe\u8868 \u4e8c\u53c9\u641c\u7d22\u6811 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\)"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2. \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"
    \u54c8\u5e0c\u8868\u7684\u57fa\u672c\u64cd\u4f5c\u5305\u62ec \u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u4e0e\u5220\u9664\u952e\u503c\u5bf9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hash_map.java
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   map.put(15937, \"\u5c0f\u5570\");   map.put(16750, \"\u5c0f\u7b97\");   map.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
     hash_map.cpp
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
     hash_map.py
    \"\"\" \u521d\u59cb\u5316\u54c8\u5e0c\u8868 \"\"\"\nmapp: Dict = {}\n\"\"\" \u6dfb\u52a0\u64cd\u4f5c \"\"\"\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n\"\"\" \u67e5\u8be2\u64cd\u4f5c \"\"\"\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = mapp[15937]\n\"\"\" \u5220\u9664\u64cd\u4f5c \"\"\"\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmapp.pop(10583)\n
     hash_map.go
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nmapp := make(map[int]string)\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := mapp[15937]\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(mapp, 10583)\n
     hash_map.js
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new ArrayHashMap();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
     hash_map.ts
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
     hash_map.c
    \n
     hash_map.cs
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, String> map = new ();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.Add(12836, \"\u5c0f\u54c8\");\nmap.Add(15937, \"\u5c0f\u5570\");\nmap.Add(16750, \"\u5c0f\u7b97\");\nmap.Add(13276, \"\u5c0f\u6cd5\");\nmap.Add(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
     hash_map.swift
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
     hash_map.zig
    \n
     
    \u904d\u5386\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u65b9\u5f0f\uff0c\u5373 \u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u3001\u904d\u5386\u503c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hash_map.java
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\nSystem.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\nSystem.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\nSystem.out.println(val);\n}\n
     hash_map.cpp
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\ncout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (auto key: map) {\ncout << key.first << endl;\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (auto val: map) {\ncout << val.second << endl;\n}\n
     hash_map.py
    \"\"\" \u904d\u5386\u54c8\u5e0c\u8868 \"\"\"\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in mapp.items():\nprint(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in mapp.keys():\nprint(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in mapp.values():\nprint(value)\n
     hash_map_test.go
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range mapp {\nfmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range mapp {\nfmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range mapp {\nfmt.Println(value)\n}\n
     hash_map.js
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
     hash_map.ts
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
     hash_map.c
    \n
     hash_map.cs
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map)\n{\nConsole.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys)\n{\nConsole.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (String val in map.Values)\n{\nConsole.WriteLine(val);\n}\n
     hash_map.swift
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\nprint(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\nprint(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\nprint(value)\n}\n
     hash_map.zig
    \n
    "},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3. \u00a0 \u54c8\u5e0c\u51fd\u6570","text":"
    \u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\uff0c\u5e76\u4e14\u53ef\u80fd\u5305\u542b\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff08\u7ea2\u9ed1\u6811\uff09\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6027\u80fd\uff08\u4e0b\u8282\u4f1a\u8ba8\u8bba\uff09\u3002
     
    \u9996\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u300c\u6570\u7ec4\u300d\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u6839\u636e\u4e60\u60ef\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 Bucket\u300d\uff0c\u7528\u4e8e\u5b58\u50a8\u952e\u503c\u5bf9\u3002
     
    \u6211\u4eec\u5c06\u952e\u503c\u5bf9 key, value \u5305\u88c5\u6210\u4e00\u4e2a\u7c7b Entry \uff0c\u5e76\u5c06\u6240\u6709 Entry \u90fd\u653e\u5165\u6570\u7ec4\u4e2d\uff0c\u90a3\u4e48\u6bcf\u4e2a Entry \u5728\u6570\u7ec4\u4e2d\u90fd\u6709\u552f\u4e00\u7684\u7d22\u5f15\u3002\u800c\u4e3a\u4e86\u5efa\u7acb key \u548c\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\uff0c\u6211\u4eec\u9700\u8981\u4f7f\u7528\u300c\u54c8\u5e0c\u51fd\u6570 Hash Function\u300d\u3002
     
    \u8bbe\u54c8\u5e0c\u8868\u7684\u6570\u7ec4\u4e3a buckets \uff0c\u54c8\u5e0c\u51fd\u6570\u4e3a f(x) \uff0c\u90a3\u4e48\u67e5\u8be2\u64cd\u4f5c\u7684\u6b65\u9aa4\u4e3a\uff1a
     
       
    1. \u8f93\u5165 key \uff0c\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u51fa\u7d22\u5f15 index \uff0c\u5373 index = f(key) \uff1b
      2.  
    2. \u901a\u8fc7\u7d22\u5f15\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5230\u952e\u503c\u5bf9 entry \uff0c\u5373 entry = buckets[index] \uff0c\u5e76\u5728 entry \u4e2d\u83b7\u53d6\u5230 value \u5373\u53ef\uff1b
      3.  
     
    \u4ee5\u4e0a\u8ff0\u5b66\u751f\u6570\u636e key \u5b66\u53f7 -> value \u59d3\u540d \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u54c8\u5e0c\u51fd\u6570\u300d\u8bbe\u8ba1\u4e3a
     \\[ f(x) = x \\% 100 \\] 
     
     Fig. \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_hash_map.java
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry {\npublic int key;\npublic String val;\npublic Entry(int key, String val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Entry> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new ArrayList<>();\nfor (int i = 0; i < 100; i++) {\nbuckets.add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\nEntry pair = buckets.get(index);\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\nEntry pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets.set(index, pair);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets.set(index, null);\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Entry> entrySet() {\nList<Entry> entrySet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nentrySet.add(pair);\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<Integer> keySet() {\nList<Integer> keySet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nkeySet.add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet() {\nList<String> valueSet = new ArrayList<>();\nfor (Entry pair : buckets) {\nif (pair != null)\nvalueSet.add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Entry kv: entrySet()) {\nSystem.out.println(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
     array_hash_map.cpp
    /* \u952e\u503c\u5bf9 int->String */\nstruct Entry {\npublic:\nint key;\nstring val;\nEntry(int key, string val) {\nthis->key = key;\nthis->val = val;\n}\n};\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate:\nvector<Entry*> buckets;\npublic:\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = vector<Entry*>(100);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\nEntry* pair = buckets[index];\nif (pair == nullptr)\nreturn nullptr;\nreturn pair->val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\nEntry* pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a nullptr \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nullptr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvector<Entry*> entrySet() {\nvector<Entry*> entrySet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr) {\nentrySet.push_back(pair);\n}\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvector<int> keySet() {\nvector<int> keySet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr) {\nkeySet.push_back(pair->key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvector<string> valueSet() {\nvector<string> valueSet;\nfor (Entry* pair: buckets) {\nif (pair != nullptr){\nvalueSet.push_back(pair->val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Entry* kv: entrySet()) {\ncout << kv->key << \" -> \" << kv->val << endl;\n}\n}\n};\n
     array_hash_map.py
    class Entry:\n\"\"\" \u952e\u503c\u5bf9 int->String \"\"\"\ndef __init__(self, key: int, val: str):\nself.key = key\nself.val = val\nclass ArrayHashMap:\n\"\"\" \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets: List[Optional[Entry]] = [None] * 100\ndef hash_func(self, key: int) -> int:\n\"\"\" \u54c8\u5e0c\u51fd\u6570 \"\"\"\nindex: int = key % 100\nreturn index\ndef get(self, key: int) -> str:\n\"\"\" \u67e5\u8be2\u64cd\u4f5c \"\"\"\nindex: int = self.hash_func(key)\npair: Entry = self.buckets[index]\nif pair is None:\nreturn None\nreturn pair.val\ndef put(self, key: int, val: str) -> None:\n\"\"\" \u6dfb\u52a0\u64cd\u4f5c \"\"\"\npair = Entry(key, val)\nindex: int = self.hash_func(key)\nself.buckets[index] = pair\ndef remove(self, key: int) -> None:\n\"\"\" \u5220\u9664\u64cd\u4f5c \"\"\"\nindex: int = self.hash_func(key)\n# \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None\ndef entry_set(self) -> List[Entry]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 \"\"\"\nresult: List[Entry] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair)\nreturn result\ndef key_set(self) -> List[int]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u952e \"\"\"\nresult: List[int] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.key)\nreturn result\ndef value_set(self) -> List[str]:\n\"\"\" \u83b7\u53d6\u6240\u6709\u503c \"\"\"\nresult: List[str] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.val)\nreturn result\ndef print(self) -> None:\n\"\"\" \u6253\u5370\u54c8\u5e0c\u8868 \"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\n
     array_hash_map.go
    /* \u952e\u503c\u5bf9 int->String */\ntype entry struct {\nkey int\nval string\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\nbuckets []*entry\n}\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets := make([]*entry, 100)\nreturn &arrayHashMap{buckets: buckets}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\nindex := key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\nindex := a.hashFunc(key)\npair := a.buckets[index]\nif pair == nil {\nreturn \"Not Found\"\n}\nreturn pair.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\npair := &entry{key: key, val: val}\nindex := a.hashFunc(key)\na.buckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\nindex := a.hashFunc(key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\na.buckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) entrySet() []*entry {\nvar pairs []*entry\nfor _, pair := range a.buckets {\nif pair != nil {\npairs = append(pairs, pair)\n}\n}\nreturn pairs\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\nvar keys []int\nfor _, pair := range a.buckets {\nif pair != nil {\nkeys = append(keys, pair.key)\n}\n}\nreturn keys\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\nvar values []string\nfor _, pair := range a.buckets {\nif pair != nil {\nvalues = append(values, pair.val)\n}\n}\nreturn values\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\nfor _, pair := range a.buckets {\nif pair != nil {\nfmt.Println(pair.key, \"->\", pair.val)\n}\n}\n}\n
     array_hash_map.js
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Entry {\nconstructor(key, val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n#buckets;\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.#buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nlet index = this.#hashFunc(key);\nlet entry = this.#buckets[index];\nif (entry === null) return null;\nreturn entry.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nset(key, val) {\nlet index = this.#hashFunc(key);\nthis.#buckets[index] = new Entry(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\ndelete(key) {\nlet index = this.#hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.#buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nentries() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nkeys() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvalues() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nlet entrySet = this.entries();\nfor (const entry of entrySet) {\nif (!entry) continue;\nconsole.info(`${entry.key} -> ${entry.val}`);\n}\n}\n}\n
     array_hash_map.ts
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Entry {\npublic key: number;\npublic val: string;\nconstructor(key: number, val: string) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate readonly buckets: (Entry | null)[];\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.buckets = (new Array(100)).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic get(key: number): string | null {\nlet index = this.hashFunc(key);\nlet entry = this.buckets[index];\nif (entry === null) return null;\nreturn entry.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic set(key: number, val: string) {\nlet index = this.hashFunc(key);\nthis.buckets[index] = new Entry(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic delete(key: number) {\nlet index = this.hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic entries(): (Entry | null)[] {\nlet arr: (Entry | null)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic keys(): (number | undefined)[] {\nlet arr: (number | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic values(): (string | undefined)[] {\nlet arr: (string | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic print() {\nlet entrySet = this.entries();\nfor (const entry of entrySet) {\nif (!entry) continue;\nconsole.info(`${entry.key} -> ${entry.val}`);\n}\n}\n}\n
     array_hash_map.c
    [class]{entry}-[func]{}\n[class]{arrayHashMap}-[func]{}\n
     array_hash_map.cs
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry\n{\npublic int key;\npublic String val;\npublic Entry(int key, String val)\n{\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap\n{\nprivate List<Entry?> buckets;\npublic ArrayHashMap()\n{\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new();\nfor (int i = 0; i < 100; i++)\n{\nbuckets.Add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key)\n{\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String? get(int key)\n{\nint index = hashFunc(key);\nEntry? pair = buckets[index];\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val)\n{\nEntry pair = new Entry(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key)\n{\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Entry> entrySet()\n{\nList<Entry> entrySet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nentrySet.Add(pair);\n}\nreturn entrySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<int> keySet()\n{\nList<int> keySet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nkeySet.Add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet()\n{\nList<String> valueSet = new();\nforeach (Entry? pair in buckets)\n{\nif (pair != null)\nvalueSet.Add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print()\n{\nforeach (Entry kv in entrySet())\n{\nConsole.WriteLine(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
     array_hash_map.swift
    /* \u952e\u503c\u5bf9 int->String */\nclass Entry {\nvar key: Int\nvar val: String\ninit(key: Int, val: String) {\nself.key = key\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate var buckets: [Entry?] = []\ninit() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nfor _ in 0 ..< 100 {\nbuckets.append(nil)\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate func hashFunc(key: Int) -> Int {\nlet index = key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet pair = buckets[index]\nreturn pair?.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\nlet pair = Entry(key: key, val: val)\nlet index = hashFunc(key: key)\nbuckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nfunc entrySet() -> [Entry] {\nvar entrySet: [Entry] = []\nfor pair in buckets {\nif let pair = pair {\nentrySet.append(pair)\n}\n}\nreturn entrySet\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc keySet() -> [Int] {\nvar keySet: [Int] = []\nfor pair in buckets {\nif let pair = pair {\nkeySet.append(pair.key)\n}\n}\nreturn keySet\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc valueSet() -> [String] {\nvar valueSet: [String] = []\nfor pair in buckets {\nif let pair = pair {\nvalueSet.append(pair.val)\n}\n}\nreturn valueSet\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor entry in entrySet() {\nSwift.print(\"\\(entry.key) -> \\(entry.val)\")\n}\n}\n}\n
     array_hash_map.zig
    // \u952e\u503c\u5bf9 int->String\nconst Entry = struct {\nkey: usize = undefined,\nval: []const u8 = undefined,\npub fn init(key: usize, val: []const u8) Entry {\nreturn Entry {\n.key = key,\n.val = val,\n};\n}\n};\n// \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\nreturn struct {\nbuckets: ?std.ArrayList(?T) = null,\nmem_allocator: std.mem.Allocator = undefined,\nconst Self = @This();\n// \u6784\u9020\u65b9\u6cd5\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nself.mem_allocator = allocator;\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets = std.ArrayList(?T).init(self.mem_allocator);\nvar i: i32 = 0;\nwhile (i < 100) : (i += 1) {\ntry self.buckets.?.append(null);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\npub fn deinit(self: *Self) void {\nif (self.buckets != null) self.buckets.?.deinit();\n}\n// \u54c8\u5e0c\u51fd\u6570\nfn hashFunc(key: usize) usize {\nvar index = key % 100;\nreturn index;\n}\n// \u67e5\u8be2\u64cd\u4f5c\npub fn get(self: *Self, key: usize) []const u8 {\nvar index = hashFunc(key);\nvar pair = self.buckets.?.items[index];\nreturn pair.?.val;\n}\n// \u6dfb\u52a0\u64cd\u4f5c\npub fn put(self: *Self, key: usize, val: []const u8) !void {\nvar pair = Entry.init(key, val);\nvar index = hashFunc(key);\nself.buckets.?.items[index] = pair;\n}\n// \u5220\u9664\u64cd\u4f5c\npub fn remove(self: *Self, key: usize) !void {\nvar index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets.?.items[index] = null;\n}       // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\npub fn entrySet(self: *Self) !*std.ArrayList(T) {\nvar entry_set = std.ArrayList(T).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry entry_set.append(item.?);\n}\nreturn &entry_set;\n}  // \u83b7\u53d6\u6240\u6709\u952e\npub fn keySet(self: *Self) !*std.ArrayList(usize) {\nvar key_set = std.ArrayList(usize).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry key_set.append(item.?.key);\n}\nreturn &key_set;\n}  // \u83b7\u53d6\u6240\u6709\u503c\npub fn valueSet(self: *Self) !*std.ArrayList([]const u8) {\nvar value_set = std.ArrayList([]const u8).init(self.mem_allocator);\nfor (self.buckets.?.items) |item| {\nif (item == null) continue;\ntry value_set.append(item.?.val);\n}\nreturn &value_set;\n}\n// \u6253\u5370\u54c8\u5e0c\u8868\npub fn print(self: *Self) !void {\nvar entry_set = try self.entrySet();\ndefer entry_set.deinit();\nfor (entry_set.items) |item| {\nstd.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n}\n}\n};\n}\n
    "},{"location":"chapter_hashing/hash_map/#614","title":"6.1.4. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u4f1a\u53d1\u73b0\uff0c\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x \\% 100\\) \u4f1a\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u5931\u6548\u3002\u5177\u4f53\u5730\uff0c\u5f53\u8f93\u5165\u7684 key \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u7ed3\u679c\u4e5f\u76f8\u540c\uff0c\u6307\u5411\u540c\u4e00\u4e2a value \u3002\u4f8b\u5982\uff0c\u5206\u522b\u67e5\u8be2\u4e24\u4e2a\u5b66\u53f7 \\(12836\\) \u548c \\(20336\\) \uff0c\u5219\u6709
     \\[ f(12836) = f(20336) = 36 \\] 
    \u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u660e\u663e\u662f\u4e0d\u5bf9\u7684\uff0c\u6211\u4eec\u5c06\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002\u5982\u4f55\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u7684\u95ee\u9898\u5c06\u88ab\u7559\u5728\u4e0b\u7ae0\u8ba8\u8bba\u3002
     
     
     Fig. \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b 
     
    \u7efc\u4e0a\u6240\u8ff0\uff0c\u4e00\u4e2a\u4f18\u79c0\u7684\u300c\u54c8\u5e0c\u51fd\u6570\u300d\u5e94\u8be5\u5177\u5907\u4ee5\u4e0b\u7279\u6027\uff1a
     
       
    • \u5c3d\u91cf\u5c11\u5730\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\uff1b
      •  
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff0c\u8ba1\u7b97\u5c3d\u53ef\u80fd\u9ad8\u6548\uff1b
      •  
    • \u7a7a\u95f4\u4f7f\u7528\u7387\u9ad8\uff0c\u5373\u201c\u952e\u503c\u5bf9\u5360\u7528\u7a7a\u95f4 / \u54c8\u5e0c\u8868\u603b\u5360\u7528\u7a7a\u95f4\u201d\u5c3d\u53ef\u80fd\u5927\uff1b
      •  
    "},{"location":"chapter_hashing/summary/","title":"6.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u4e00\u4e2a\u952e key \uff0c\u67e5\u8be2\u5230\u503c value \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
      •  
    • \u54c8\u5e0c\u8868\u7684\u5e38\u7528\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u4e0e\u5220\u9664\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u503c\u5bf9\u7b49\u3002
      •  
    • \u54c8\u5e0c\u51fd\u6570\u5c06 key \u6620\u5c04\u5230\u6876\uff08\u6570\u7ec4\uff09\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u503c value \u3002
      •  
    • \u4e24\u4e2a\u4e0d\u540c\u7684 key \u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u53ef\u80fd\u5f97\u5230\u76f8\u540c\u7684\u6876\u7d22\u5f15\uff0c\u8fdb\u800c\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\uff0c\u5bfc\u81f4\u67e5\u8be2\u9519\u8bef\u3002
      •  
    • \u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u7684\u9014\u5f84\u6709\u4e24\u79cd\uff1a\u54c8\u5e0c\u8868\u6269\u5bb9\u3001\u4f18\u5316\u54c8\u5e0c\u8868\u7684\u8868\u793a\u65b9\u5f0f\u3002
      •  
    • \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u69fd\u6570\u91cf\uff0c\u4f53\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7684\u539f\u7406\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u5f00\u9500\u4e5f\u5f88\u5927\u3002
      •  
    • \u94fe\u5f0f\u5730\u5740\u8003\u8651\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u6210\u4e00\u4e2a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u90fd\u5b58\u50a8\u5728\u4e00\u4e2a\u94fe\u8868\u4e2d\uff0c\u4ece\u800c\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u94fe\u8868\u8fc7\u957f\u4f1a\u5bfc\u81f4\u67e5\u8be2\u6548\u7387\u53d8\u4f4e\uff0c\u53ef\u4ee5\u901a\u8fc7\u628a\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\u6765\u89e3\u51b3\u3002
      •  
    • \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u5bf9\u7ebf\u6027\u63a2\u6d4b\u4e0d\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4ee3\u4ef7\u662f\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
      •  
    • \u5728\u5de5\u4e1a\u754c\u4e2d\uff0cJava \u7684 HashMap \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3001Python \u7684 Dict \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002
      •  
    "},{"location":"chapter_heap/build_heap/","title":"8.2. \u00a0 \u5efa\u5806\u64cd\u4f5c *","text":"
    \u5982\u679c\u6211\u4eec\u60f3\u8981\u6839\u636e\u8f93\u5165\u5217\u8868\u6765\u751f\u6210\u4e00\u4e2a\u5806\uff0c\u8fd9\u6837\u7684\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5efa\u5806\u300d\u3002
    "},{"location":"chapter_heap/build_heap/#821","title":"8.2.1. \u00a0 \u4e24\u79cd\u5efa\u5806\u65b9\u6cd5","text":""},{"location":"chapter_heap/build_heap/#_1","title":"\u501f\u52a9\u5165\u5806\u65b9\u6cd5\u5b9e\u73b0","text":"
    \u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u300c\u5143\u7d20\u5165\u5806\u300d\u65b9\u6cd5\uff0c\u5148\u5efa\u7acb\u4e00\u4e2a\u7a7a\u5806\uff0c\u518d\u5c06\u5217\u8868\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u5373\u53ef\u3002
     
    \u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u5165\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5728\u4f9d\u6b21\u5165\u5806\u65f6\uff0c\u5806\u7684\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8be5\u65b9\u6cd5\u7684\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
    "},{"location":"chapter_heap/build_heap/#_2","title":"\u57fa\u4e8e\u5806\u5316\u64cd\u4f5c\u5b9e\u73b0","text":"
    \u6709\u8da3\u7684\u662f\uff0c\u5b58\u5728\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u3002\u6211\u4eec\u5148\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\uff0c\u7136\u540e\u8fed\u4ee3\u5730\u5bf9\u5404\u4e2a\u7ed3\u70b9\u6267\u884c\u300c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u300d\u3002\u5f53\u7136\uff0c\u65e0\u9700\u5bf9\u53f6\u7ed3\u70b9\u6267\u884c\u5806\u5316\uff0c\u56e0\u4e3a\u5176\u6ca1\u6709\u5b50\u7ed3\u70b9\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new ArrayList<>(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
     my_heap.cpp
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
     my_heap.py
    def __init__(self, nums: List[int]):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\n# \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nself.max_heap = nums\n# \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor i in range(self.parent(self.size() - 1), -1, -1):\nself.sift_down(i)\n
     my_heap.go
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nh := &maxHeap{data: nums}\nfor i := len(h.data) - 1; i >= 0; i-- {\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nh.siftDown(i)\n}\nreturn h\n}\n
     my_heap.js
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.#maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (let i = this.#parent(this.size() - 1); i >= 0; i--) {\nthis.#siftDown(i);\n}\n}\n
     my_heap.ts
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor (let i = this.parent(this.size() - 1); i >= 0; i--) {\nthis.siftDown(i);\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{newMaxHeap}\n
     my_heap.cs
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums)\n{\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new List<int>(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nvar size = parent(this.size() - 1);\nfor (int i = size; i >= 0; i--)\n{\nsiftDown(i);\n}\n}\n
     my_heap.swift
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nfor i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\nsiftDown(i: i)\n}\n}\n
     my_heap.zig
    // \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\nif (self.max_heap != null) return;\nself.max_heap = std.ArrayList(T).init(allocator);\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\ntry self.max_heap.?.appendSlice(nums);\n// \u5806\u5316\u9664\u53f6\u7ed3\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u7ed3\u70b9\nvar i: usize = parent(self.size() - 1) + 1;\nwhile (i > 0) : (i -= 1) {\ntry self.siftDown(i - 1);\n}\n}\n
    "},{"location":"chapter_heap/build_heap/#822","title":"8.2.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n)\\) \u5462\uff1f\u6211\u4eec\u6765\u5c55\u5f00\u63a8\u7b97\u4e00\u4e0b\u3002
     
       
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u53f6\u7ed3\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u5728\u6392\u9664\u53f6\u7ed3\u70b9\u540e\uff0c\u9700\u8981\u5806\u5316\u7ed3\u70b9\u6570\u91cf\u4e3a \\((n - 1)/2\\) \uff0c\u5373\u4e3a \\(O(n)\\) \uff1b
      •  
    • \u4ece\u9876\u81f3\u5e95\u5806\u5316\u4e2d\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u6700\u591a\u5806\u5316\u81f3\u53f6\u7ed3\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(O(\\log n)\\) \uff1b
      •  
     
    \u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u4e0d\u591f\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230 \u4e8c\u53c9\u6811\u5e95\u5c42\u7ed3\u70b9\u8fdc\u591a\u4e8e\u9876\u5c42\u7ed3\u70b9 \u7684\u6027\u8d28\u3002
     
    \u4e0b\u9762\u6211\u4eec\u6765\u5c55\u5f00\u8ba1\u7b97\u3002\u4e3a\u4e86\u51cf\u5c0f\u8ba1\u7b97\u96be\u5ea6\uff0c\u6211\u4eec\u5047\u8bbe\u6811\u662f\u4e00\u4e2a\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002\u8bbe\u4e8c\u53c9\u6811\uff08\u5373\u5806\uff09\u7ed3\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \u3002\u4e0a\u6587\u63d0\u5230\uff0c\u7ed3\u70b9\u5806\u5316\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u7ed3\u70b9\u5230\u53f6\u7ed3\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8fd9\u6b63\u662f\u201c\u7ed3\u70b9\u9ad8\u5ea6\u201d\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u7ed3\u70b9\u6570\u91cf 
     
    \u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5404\u5c42\u7684\u201c\u7ed3\u70b9\u6570\u91cf \\(\\times\\) \u7ed3\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5373\u53ef\u5f97\u5230 \u6240\u6709\u7ed3\u70b9\u7684\u5806\u5316\u7684\u8fed\u4ee3\u6b21\u6570\u603b\u548c\u3002
     \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{(h-1)}\\times1 \\] 
    \u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u6613\u5f97
     \\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\cdots + 2^{h}\\times1 \\newline \\end{aligned} \\] 
    \u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u4ee4\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97
     \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\cdots + 2^{h-1} + 2^h \\] 
    \u89c2\u5bdf\u4e0a\u5f0f\uff0c\\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a
     \\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h \\newline & = O(2^h) \\end{aligned} \\] 
    \u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\)\u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
    "},{"location":"chapter_heap/heap/","title":"8.1. \u00a0 \u5806","text":"
    \u300c\u5806 Heap\u300d\u662f\u4e00\u68f5\u9650\u5b9a\u6761\u4ef6\u4e0b\u7684\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u3002\u6839\u636e\u6210\u7acb\u6761\u4ef6\uff0c\u5806\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff1a
     
       
    • \u300c\u5927\u9876\u5806 Max Heap\u300d\uff0c\u4efb\u610f\u7ed3\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u7ed3\u70b9\u7684\u503c\uff1b
      •  
    • \u300c\u5c0f\u9876\u5806 Min Heap\u300d\uff0c\u4efb\u610f\u7ed3\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u7ed3\u70b9\u7684\u503c\uff1b
      •  
     
     
     Fig. \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 
    "},{"location":"chapter_heap/heap/#811","title":"8.1.1. \u00a0 \u5806\u672f\u8bed\u4e0e\u6027\u8d28","text":"
       
    • \u7531\u4e8e\u5806\u662f\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6700\u5e95\u5c42\u7ed3\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u5b83\u5c42\u7ed3\u70b9\u7686\u88ab\u586b\u6ee1\u3002
      •  
    • \u4e8c\u53c9\u6811\u4e2d\u7684\u6839\u7ed3\u70b9\u5bf9\u5e94\u300c\u5806\u9876\u300d\uff0c\u5e95\u5c42\u6700\u9760\u53f3\u7ed3\u70b9\u5bf9\u5e94\u300c\u5806\u5e95\u300d\u3002
      •  
    • \u5bf9\u4e8e\u5927\u9876\u5806 / \u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\uff08\u5373\u6839\u7ed3\u70b9\uff09\u7684\u503c\u6700\u5927 / \u6700\u5c0f\u3002
      •  
    "},{"location":"chapter_heap/heap/#812","title":"8.1.2. \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"
    \u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 Priority Queue\u300d\uff0c\u5176\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\u3002
     
    \u800c\u6070\u597d\uff0c\u5806\u7684\u5b9a\u4e49\u4e0e\u4f18\u5148\u961f\u5217\u7684\u64cd\u4f5c\u903b\u8f91\u5b8c\u5168\u543b\u5408\uff0c\u5927\u9876\u5806\u5c31\u662f\u4e00\u4e2a\u5143\u7d20\u4ece\u5927\u5230\u5c0f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u4f18\u5148\u961f\u5217\u300d\u548c\u300c\u5806\u300d\u7406\u89e3\u4e3a\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u6587\u4e0e\u4ee3\u7801\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u4f7f\u7528\u300c\u5806\u300d\u6765\u547d\u540d\u3002
     
    \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u786e\u5b9a\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002
     
    Tip
     
    \u7c7b\u4f3c\u4e8e\u6392\u5e8f\u4e2d\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u201c\u5927\u9876\u5806\u201d\u548c\u201c\u5c0f\u9876\u5806\u201d\u53ef\u4ec5\u901a\u8fc7\u4fee\u6539 Comparator \u6765\u4e92\u76f8\u8f6c\u6362\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig heap.java
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = heap.poll();  // 5\npeek = heap.poll();  // 4\npeek = heap.poll();  // 3\npeek = heap.poll();  // 2\npeek = heap.poll();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
     heap.cpp
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
     heap.py
    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\"\"\" \u5143\u7d20\u5165\u5806 \"\"\"\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\"\"\" \u83b7\u53d6\u5806\u9876\u5143\u7d20 \"\"\"\npeek: int = flag * max_heap[0] # 5\n\"\"\" \u5806\u9876\u5143\u7d20\u51fa\u5806 \"\"\"\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\"\"\" \u83b7\u53d6\u5806\u5927\u5c0f \"\"\"\nsize: int = len(max_heap)\n\"\"\" \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = not max_heap\n\"\"\" \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 \"\"\"\nmin_heap: List[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
     heap.go
    // Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n// Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n// \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n*h = append(*h, x.(int))\n}\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n// \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\nlast := (*h)[len(*h)-1]\n*h = (*h)[:len(*h)-1]\nreturn last\n}\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\nreturn len(*h)\n}\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n// \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\nreturn (*h)[i].(int) > (*h)[j].(int)\n}\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n(*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\nreturn (*h)[0]\n}\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nmaxHeap := &intHeap{}\nheap.Init(maxHeap)\n/* \u5143\u7d20\u5165\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\nheap.Push(maxHeap, 1)\nheap.Push(maxHeap, 3)\nheap.Push(maxHeap, 2)\nheap.Push(maxHeap, 4)\nheap.Push(maxHeap, 5)\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\ntop := maxHeap.Top()\nfmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\nheap.Pop(maxHeap) // 5\nheap.Pop(maxHeap) // 4\nheap.Pop(maxHeap) // 3\nheap.Pop(maxHeap) // 2\nheap.Pop(maxHeap) // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nsize := len(*maxHeap)\nfmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(*maxHeap) == 0\nfmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
     heap.js
    // JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.ts
    // TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.c
    \n
     heap.cs
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new PriorityQueue<int, int>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((x, y) => y - x));\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), });\n
     heap.swift
    // Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
     heap.zig
    \n
    "},{"location":"chapter_heap/heap/#813","title":"8.1.3. \u00a0 \u5806\u7684\u5b9e\u73b0","text":"
    \u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u300c\u5927\u9876\u5806\u300d\uff0c\u82e5\u60f3\u8f6c\u6362\u4e3a\u300c\u5c0f\u9876\u5806\u300d\uff0c\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u5373\u53ef\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_heap/heap/#_1","title":"\u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"
    \u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u6211\u4eec\u5b66\u8fc7\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u975e\u5e38\u9002\u5408\u4f7f\u7528\u300c\u6570\u7ec4\u300d\u6765\u8868\u793a\uff0c\u800c\u5806\u6070\u597d\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u800c\u6211\u4eec\u91c7\u7528\u300c\u6570\u7ec4\u300d\u6765\u5b58\u50a8\u300c\u5806\u300d\u3002
     
    \u4e8c\u53c9\u6811\u6307\u9488\u3002\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u7ed3\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u7ed3\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\uff0c\u800c\u7ed3\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002
     
    \u5177\u4f53\u5730\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u90a3\u4e48\u5176\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \u3001\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3001\u7236\u7ed3\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\) \uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u4ee3\u8868\u7a7a\u7ed3\u70b9\u6216\u7ed3\u70b9\u4e0d\u5b58\u5728\u3002
     
     
     Fig. \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 
     
    \u6211\u4eec\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u4ee5\u4fbf\u540e\u7eed\u4f7f\u7528\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.cpp
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n} /* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n
     my_heap.py
    def left(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -> int:\n\"\"\" \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 \"\"\"\nreturn (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n
     my_heap.go
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n// \u5411\u4e0b\u6574\u9664\nreturn (i - 1) / 2\n}\n
     my_heap.js
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\n#left(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\n#right(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\n#parent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.ts
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{left}\n[class]{maxHeap}-[func]{right}\n[class]{maxHeap}-[func]{parent}\n
     my_heap.cs
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint left(int i)\n{\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nint right(int i)\n{\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nint parent(int i)\n{\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.swift
    /* \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
     my_heap.zig
    // \u83b7\u53d6\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\nreturn 2 * i + 1;\n}\n// \u83b7\u53d6\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\nreturn 2 * i + 2;\n}\n// \u83b7\u53d6\u7236\u7ed3\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n// return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\nreturn @divFloor(i - 1, 2);\n}\n
    "},{"location":"chapter_heap/heap/#_2","title":"\u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"
    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u7ed3\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap.get(0);\n}\n
     my_heap.cpp
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
     my_heap.py
    def peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u5806\u9876\u5143\u7d20 \"\"\"\nreturn self.max_heap[0]\n
     my_heap.go
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\nreturn h.data[0]\n}\n
     my_heap.js
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\nreturn this.#maxHeap[0];\n}\n
     my_heap.ts
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\nreturn this.maxHeap[0];\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{peek}\n
     my_heap.cs
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek()\n{\nreturn maxHeap[0];\n}\n
     my_heap.swift
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nmaxHeap[0]\n}\n
     my_heap.zig
    // \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\nreturn self.max_heap.?.items[0];\n}  
    "},{"location":"chapter_heap/heap/#_3","title":"\u5143\u7d20\u5165\u5806","text":"
    \u7ed9\u5b9a\u5143\u7d20 val \uff0c\u6211\u4eec\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u5b83\u5143\u7d20\uff0c\u6b64\u65f6\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u7ecf\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u7ed3\u70b9\u5230\u6839\u7ed3\u70b9\u8fd9\u6761\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u7ed3\u70b9\uff0c\u8be5\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 Heapify\u300d\u3002
     
    \u8003\u8651\u4ece\u5165\u5806\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5177\u4f53\u5730\uff0c\u6bd4\u8f83\u63d2\u5165\u7ed3\u70b9\u4e0e\u5176\u7236\u7ed3\u70b9\u7684\u503c\uff0c\u82e5\u63d2\u5165\u7ed3\u70b9\u66f4\u5927\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\uff1b\u5e76\u5faa\u73af\u4ee5\u4e0a\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u7ed3\u70b9\uff1b\u76f4\u81f3\u8d8a\u8fc7\u6839\u7ed3\u70b9\u65f6\u7ed3\u675f\uff0c\u6216\u5f53\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u7ed3\u70b9\u65f6\u63d0\u524d\u7ed3\u675f\u3002
     <1><2><3><4><5><6> 
     
     
     
     
     
     
    \u8bbe\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u6613\u5f97\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u56e0\u800c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.cpp
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.push_back(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p =  parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(maxHeap[i], maxHeap[p]);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.py
    def push(self, val: int):\n\"\"\" \u5143\u7d20\u5165\u5806 \"\"\"\n# \u6dfb\u52a0\u7ed3\u70b9\nself.max_heap.append(val)\n# \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1)\ndef sift_up(self, i: int):\n\"\"\" \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 \"\"\"\nwhile True:\n# \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\np = self.parent(i)\n# \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 or self.max_heap[i] <= self.max_heap[p]:\nbreak\n# \u4ea4\u6362\u4e24\u7ed3\u70b9\nself.swap(i, p)\n# \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n
     my_heap.go
    /* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n// \u6dfb\u52a0\u7ed3\u70b9\nh.data = append(h.data, val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nh.siftUp(len(h.data) - 1)\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\nfor true {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\np := h.parent(i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || h.data[i].(int) <= h.data[p].(int) {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nh.swap(i, p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
     my_heap.js
    /* \u5143\u7d20\u5165\u5806 */\npush(val) {\n// \u6dfb\u52a0\u7ed3\u70b9\nthis.#maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.#siftUp(this.size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nconst p = this.#parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.#swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.ts
    /* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n// \u6dfb\u52a0\u7ed3\u70b9\nthis.maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.siftUp(this.size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nconst p = this.parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{push}\n[class]{maxHeap}-[func]{siftUp}\n
     my_heap.cs
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val)\n{\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.Add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i)\n{\nwhile (true)\n{\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nint p = parent(i);\n// \u82e5\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
     my_heap.swift
    /* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n// \u6dfb\u52a0\u7ed3\u70b9\nmaxHeap.append(val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(i: size() - 1)\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\nvar i = i\nwhile true {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nlet p = parent(i: i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || maxHeap[i] <= maxHeap[p] {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i: i, j: p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
     my_heap.zig
    // \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n// \u6dfb\u52a0\u7ed3\u70b9\ntry self.max_heap.?.append(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\ntry self.siftUp(self.size() - 1);\n}  // \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u83b7\u53d6\u7ed3\u70b9 i \u7684\u7236\u7ed3\u70b9\nvar p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u7ed3\u70b9\u201d\u6216\u201c\u7ed3\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\ntry self.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    "},{"location":"chapter_heap/heap/#_4","title":"\u5806\u9876\u5143\u7d20\u51fa\u5806","text":"
    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u6839\u7ed3\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\uff0c\u5982\u679c\u6211\u4eec\u76f4\u63a5\u5c06\u9996\u5143\u7d20\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff0c\u5219\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u90fd\u4f1a\u968f\u4e4b\u53d1\u751f\u79fb\u4f4d\uff08\u7d22\u5f15\u53d1\u751f\u53d8\u5316\uff09\uff0c\u8fd9\u6837\u540e\u7eed\u4f7f\u7528\u5806\u5316\u4fee\u590d\u5c31\u5f88\u9ebb\u70e6\u4e86\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u53d8\u52a8\uff0c\u91c7\u53d6\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\uff1a
     
       
    1. \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u5373\u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff09\uff1b
      2.  
    2. \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u56e0\u4e3a\u5df2\u7ecf\u4ea4\u6362\uff0c\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\uff1b
      3.  
    3. \u4ece\u6839\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\uff1b
      4.  
     
    \u987e\u540d\u601d\u4e49\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u4ece\u5e95\u81f3\u9876\u5806\u5316\u76f8\u53cd\uff0c\u6211\u4eec\u6bd4\u8f83\u6839\u7ed3\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u7ed3\u70b9\u7684\u503c\uff0c\u5c06\u6700\u5927\u7684\u5b50\u7ed3\u70b9\u4e0e\u6839\u7ed3\u70b9\u6267\u884c\u4ea4\u6362\uff0c\u5e76\u5faa\u73af\u4ee5\u4e0a\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u65f6\u7ed3\u675f\uff0c\u6216\u5f53\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u7ed3\u70b9\u65f6\u63d0\u524d\u7ed3\u675f\u3002
     <1><2><3><4><5><6><7><8><9><10> 
     
     
     
     
     
     
     
     
     
     
    \u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7c7b\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig my_heap.java
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new EmptyStackException();\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nint val = maxHeap.remove(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap.get(l) > maxHeap.get(ma))\nma = l;\nif (r < size() && maxHeap.get(r) > maxHeap.get(ma))\nma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.cpp
    /* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n// \u5224\u7a7a\u5904\u7406\nif (empty()) {\nthrow out_of_range(\"\u5806\u4e3a\u7a7a\");\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap[0], maxHeap[size() - 1]);\n// \u5220\u9664\u7ed3\u70b9\nmaxHeap.pop_back();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (l < size() && maxHeap[l] > maxHeap[ma]) ma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\nswap(maxHeap[i], maxHeap[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.py
    def pop(self) -> int:\n\"\"\" \u5143\u7d20\u51fa\u5806 \"\"\"\n# \u5224\u7a7a\u5904\u7406\nassert not self.is_empty()\n# \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1)\n# \u5220\u9664\u7ed3\u70b9\nval = self.max_heap.pop()\n# \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0)\n# \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\ndef sift_down(self, i: int):\n\"\"\" \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 \"\"\"\nwhile True:\n# \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nl, r, ma = self.left(i), self.right(i), i\nif l < self.size() and self.max_heap[l] > self.max_heap[ma]:\nma = l\nif r < self.size() and self.max_heap[r] > self.max_heap[ma]:\nma = r\n# \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u7ed3\u70b9\nself.swap(i, ma)\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n
     my_heap.go
    /* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n// \u5224\u7a7a\u5904\u7406\nif h.isEmpty() {\nfmt.Println(\"error\")\nreturn nil\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nh.swap(0, h.size()-1)\n// \u5220\u9664\u7ed3\u70b9\nval := h.data[len(h.data)-1]\nh.data = h.data[:len(h.data)-1]\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nh.siftDown(0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\nfor true {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a max\nl, r, max := h.left(i), h.right(i), i\nif l < h.size() && h.data[l].(int) > h.data[max].(int) {\nmax = l\n}\nif r < h.size() && h.data[r].(int) > h.data[max].(int) {\nmax = r\n}\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif max == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nh.swap(i, max)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max\n}\n}\n
     my_heap.js
    /* \u5143\u7d20\u51fa\u5806 */\npop() {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new Error(\"\u5806\u4e3a\u7a7a\");\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.#swap(0, this.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nconst val = this.#maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.#siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.#left(i),\nr = this.#right(i);\nlet ma = i;\nif (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\nif (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.#swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.ts
    /* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new RangeError('Heap is empty.');\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.swap(0, this.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nconst val = this.maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.left(i),\nr = this.right(i);\nlet ma = i;\nif (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\nif (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nthis.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.c
    [class]{maxHeap}-[func]{pop}\n[class]{maxHeap}-[func]{siftDown}\n
     my_heap.cs
    /* \u5143\u7d20\u51fa\u5806 */\nint pop()\n{\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfRangeException();\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nint val = maxHeap.Last();\nmaxHeap.RemoveAt(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i)\n{\nwhile (true)\n{\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u201c\u7ed3\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
     my_heap.swift
    /* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n// \u5224\u7a7a\u5904\u7406\nif isEmpty() {\nfatalError(\"\u5806\u4e3a\u7a7a\")\n}\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(i: 0, j: size() - 1)\n// \u5220\u9664\u7ed3\u70b9\nlet val = maxHeap.remove(at: size() - 1)\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(i: 0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = left(i: i)\nlet r = right(i: i)\nvar ma = i\nif l < size(), maxHeap[l] > maxHeap[ma] {\nma = l\n}\nif r < size(), maxHeap[r] > maxHeap[ma] {\nma = r\n}\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\nswap(i: i, j: ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n
     my_heap.zig
    // \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n// \u5224\u65ad\u5904\u7406\nif (self.isEmpty()) unreachable;\n// \u4ea4\u6362\u6839\u7ed3\u70b9\u4e0e\u6700\u53f3\u53f6\u7ed3\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\ntry self.swap(0, self.size() - 1);\n// \u5220\u9664\u7ed3\u70b9\nvar val = self.max_heap.?.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\ntry self.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n} // \u4ece\u7ed3\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u5224\u65ad\u7ed3\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u7ed3\u70b9\uff0c\u8bb0\u4e3a ma\nvar l = left(i);\nvar r = right(i);\nvar ma = i;\nif (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\nif (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n// \u82e5\u7ed3\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u7ed3\u70b9\ntry self.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    "},{"location":"chapter_heap/heap/#814","title":"8.1.4. \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"
       
    • \u4f18\u5148\u961f\u5217\u3002\u5806\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u7686\u975e\u5e38\u9ad8\u6548\u3002
      •  
    • \u5806\u6392\u5e8f\u3002\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u4f7f\u7528\u5176\u5efa\u5806\uff0c\u5e76\u4f9d\u6b21\u5168\u90e8\u5f39\u51fa\uff0c\u5219\u53ef\u4ee5\u5f97\u5230\u6709\u5e8f\u7684\u5e8f\u5217\u3002\u5f53\u7136\uff0c\u5806\u6392\u5e8f\u4e00\u822c\u65e0\u9700\u5f39\u51fa\u5143\u7d20\uff0c\u4ec5\u9700\u6bcf\u8f6e\u5c06\u5806\u9876\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u5c3e\u90e8\u5e76\u51cf\u5c0f\u5806\u7684\u957f\u5ea6\u5373\u53ef\u3002
      •  
    • \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002\u8fd9\u65e2\u662f\u4e00\u9053\u7ecf\u5178\u7b97\u6cd5\u9898\u76ee\uff0c\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u53d6\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u524d 10 \u9500\u91cf\u7684\u5546\u54c1\u7b49\u3002
      •  
    "},{"location":"chapter_heap/summary/","title":"8.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5806\u662f\u4e00\u68f5\u9650\u5b9a\u6761\u4ef6\u4e0b\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u6700\u5927\uff08\u5c0f\uff09\u3002
      •  
    • \u4f18\u5148\u961f\u5217\u5b9a\u4e49\u4e3a\u4e00\u79cd\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\u3002\u5806\u662f\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u6700\u5e38\u7528\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u548c\u5bf9\u5e94\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) \u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u3001\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
      •  
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u4e00\u822c\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
      •  
    • \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u4fee\u590d\u5806\u7684\u7279\u6027\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u4f7f\u7528\u5230\u3002
      •  
    • \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
      •  
    "},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1. \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"
    \u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\uff0c\u6211\u4eec\u4e00\u822c\u4f1a\u8054\u60f3\u5230\u6570\u5b66\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5927\u591a\u6570\u7b97\u6cd5\u5e76\u4e0d\u5305\u542b\u590d\u6742\u7684\u6570\u5b66\uff0c\u800c\u66f4\u50cf\u662f\u5728\u8003\u5bdf\u57fa\u672c\u903b\u8f91\uff0c\u800c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002
     
    \u5728\u6b63\u5f0f\u4ecb\u7ecd\u7b97\u6cd5\u4e4b\u524d\uff0c\u6211\u60f3\u544a\u8bc9\u4f60\u4e00\u4ef6\u6709\u8da3\u7684\u4e8b\uff1a\u5176\u5b9e\uff0c\u4f60\u5728\u8fc7\u53bb\u5df2\u7ecf\u5b66\u4f1a\u4e86\u5f88\u591a\u7b97\u6cd5\uff0c\u5e76\u4e14\u5df2\u7ecf\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u5c06\u4ecb\u7ecd\u4e24\u4e2a\u5177\u4f53\u4f8b\u5b50\u6765\u4f50\u8bc1\u3002
     
    \u4f8b\u4e00\uff1a\u62fc\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u6709\u8bb8\u591a\u90e8\u4ef6\u4e4b\u5916\uff0c\u8fd8\u4f1a\u9644\u9001\u8be6\u7ec6\u7684\u62fc\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e0a\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5373\u53ef\u62fc\u51fa\u590d\u6742\u7684\u79ef\u6728\u6a21\u578b\u3002
     
    \u5982\u679c\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u5927\u5927\u5c0f\u5c0f\u7684\u300c\u79ef\u6728\u300d\u5c31\u662f\u6570\u636e\u7ed3\u6784\uff0c\u800c\u300c\u62fc\u88c5\u8bf4\u660e\u4e66\u300d\u4e0a\u7684\u4e00\u7cfb\u5217\u6b65\u9aa4\u5c31\u662f\u7b97\u6cd5\u3002
     
    \u4f8b\u4e8c\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u4e2d\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u6709\u4e00\u4e2a\u5bf9\u5e94\u7684\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u7684\u82f1\u6587\u5b57\u6bcd\u8868\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u9700\u8981\u5728\u5b57\u5178\u4e2d\u67e5\u8be2\u4efb\u610f\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u4e00\u822c\u6211\u4eec\u4f1a\u8fd9\u6837\u505a\uff1a
     
       
    1. \u6253\u5f00\u5b57\u5178\u5927\u81f4\u4e00\u534a\u9875\u6570\u7684\u4f4d\u7f6e\uff0c\u67e5\u770b\u6b64\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff08\u5047\u8bbe\u4e3a \\(m\\) \uff09\uff1b
      2.  
    2. \u7531\u4e8e\u5728\u82f1\u6587\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u5728 \\(m\\) \u7684\u540e\u9762\uff0c\u56e0\u6b64\u5e94\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u4ec5\u5269\u540e\u534a\u90e8\u5206\uff1b
      3.  
    3. \u5faa\u73af\u6267\u884c\u6b65\u9aa4 1-2 \uff0c\u76f4\u5230\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u65f6\u7ec8\u6b62\u3002
      4.  
     <1><2><3><4><5> 
     
     
     
     
     
    \u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u7684\u6807\u914d\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u5927\u540d\u9f0e\u9f0e\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u3002\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u770b\u4f5c\u662f\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u300c\u6570\u7ec4\u300d\uff1b\u800c\u4ece\u7b97\u6cd5\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u6307\u4ee4\u770b\u4f5c\u662f\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7b97\u6cd5\u3002
     
    \u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\u3001\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u4e5f\u53ef\u4ee5\u7f16\u5199\u4ee3\u7801\u6765\u8c03\u7528 CPU, GPU \u6267\u884c\u7b97\u6cd5\uff0c\u4ece\u800c\u5c06\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u642c\u8fd0\u5230\u8ba1\u7b97\u673a\u4e2d\uff0c\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3\u5404\u5f0f\u5404\u6837\u7684\u590d\u6742\u95ee\u9898\u3002
     
    Tip
     
    \u8bfb\u5230\u8fd9\u91cc\uff0c\u5982\u679c\u4f60\u611f\u5230\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u6b64\u7c7b\u6982\u5ff5\u4e00\u77e5\u534a\u89e3\uff0c\u90a3\u4e48\u5c31\u592a\u597d\u4e86\uff01\u56e0\u4e3a\u8fd9\u6b63\u662f\u672c\u4e66\u5b58\u5728\u7684\u4ef7\u503c\uff0c\u63a5\u4e0b\u6765\uff0c\u672c\u4e66\u5c06\u4f1a\u4e00\u6b65\u6b65\u5730\u5f15\u5bfc\u4f60\u8fdb\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002
    "},{"location":"chapter_introduction/summary/","title":"1.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u7b97\u6cd5\u5728\u751f\u6d3b\u4e2d\u968f\u5904\u53ef\u89c1\uff0c\u5e76\u4e0d\u9ad8\u6df1\u83ab\u6d4b\u3002\u6211\u4eec\u5df2\u7ecf\u4e0d\u77e5\u4e0d\u89c9\u5730\u5b66\u4e60\u5230\u8bb8\u591a\u201c\u7b97\u6cd5\u201d\uff0c\u7528\u4e8e\u89e3\u51b3\u751f\u6d3b\u4e2d\u5927\u5927\u5c0f\u5c0f\u7684\u95ee\u9898\u3002
      •  
    • \u201c\u67e5\u5b57\u5178\u201d\u7684\u539f\u7406\u548c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4e00\u81f4\u3002\u4e8c\u5206\u4f53\u73b0\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
      •  
    • \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e24\u8005\u7d27\u5bc6\u8054\u7cfb\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u5e95\u5ea7\uff0c\u7b97\u6cd5\u662f\u53d1\u6325\u6570\u636e\u7ed3\u6784\u7684\u821e\u53f0\u3002
      •  
    • \u4e50\u9ad8\u79ef\u6728\u5bf9\u5e94\u6570\u636e\uff0c\u79ef\u6728\u5f62\u72b6\u548c\u8fde\u63a5\u5f62\u5f0f\u5bf9\u5e94\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6d41\u7a0b\u6b65\u9aa4\u5bf9\u5e94\u7b97\u6cd5\u3002
      •  
    "},{"location":"chapter_introduction/what_is_dsa/","title":"1.2. \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1. \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"
    \u300c\u7b97\u6cd5 Algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\u3002\u7b97\u6cd5\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a
     
       
    • \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u9700\u8981\u62e5\u6709\u660e\u786e\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
      •  
    • \u89e3\u5177\u6709\u786e\u5b9a\u6027\uff0c\u5373\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u65f6\uff0c\u8f93\u51fa\u4e00\u5b9a\u76f8\u540c\u3002
      •  
    • \u5177\u6709\u53ef\u884c\u6027\uff0c\u53ef\u5728\u6709\u9650\u6b65\u9aa4\u3001\u6709\u9650\u65f6\u95f4\u3001\u6709\u9650\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
      •  
    • \u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\uff0c\u5373\u53ef\u7528\u591a\u79cd\u8bed\u8a00\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"
    \u300c\u6570\u636e\u7ed3\u6784 Data Structure\u300d\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u4e3a\u4e86\u63d0\u9ad8\u6570\u636e\u5b58\u50a8\u548c\u64cd\u4f5c\u6027\u80fd\uff0c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u539f\u5219\u6709\uff1a
     
       
    • \u7a7a\u95f4\u5360\u7528\u5c3d\u53ef\u80fd\u5c0f\uff0c\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
      •  
    • \u6570\u636e\u64cd\u4f5c\u5c3d\u91cf\u5feb\uff0c\u5305\u62ec\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
      •  
    • \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002
      •  
     
    \u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\uff0c\u8fd9\u610f\u5473\u7740\u5982\u679c\u83b7\u5f97\u67d0\u65b9\u9762\u7684\u4f18\u52bf\uff0c\u5219\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u505a\u51fa\u59a5\u534f\u3002\u4f8b\u5982\uff0c\u94fe\u8868\u76f8\u5bf9\u4e8e\u6570\u7ec4\uff0c\u6570\u636e\u6dfb\u52a0\u5220\u9664\u64cd\u4f5c\u66f4\u52a0\u65b9\u4fbf\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u901f\u5ea6\uff1b\u56fe\u76f8\u5bf9\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u591a\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
    "},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3. \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"
    \u300c\u6570\u636e\u7ed3\u6784\u300d\u4e0e\u300c\u7b97\u6cd5\u300d\u662f\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u5d4c\u5408\u7684\uff0c\u4f53\u73b0\u5728\uff1a
     
       
    • \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u5e95\u5ea7\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u5bf9\u5e94\u65b9\u6cd5\u3002
      •  
    • \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u53ef\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
      •  
    • \u7b97\u6cd5\u6709\u5bf9\u5e94\u6700\u4f18\u7684\u6570\u636e\u7ed3\u6784\u3002\u7ed9\u5b9a\u7b97\u6cd5\uff0c\u4e00\u822c\u53ef\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u800c\u6700\u7ec8\u6267\u884c\u6548\u7387\u5f80\u5f80\u76f8\u5dee\u5f88\u5927\u3002
      •  
     
     
     Fig. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb 
     
    \u5982\u679c\u5c06\u300cLEGO \u4e50\u9ad8\u300d\u7c7b\u6bd4\u5230\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\uff0c\u90a3\u4e48\u53ef\u4ee5\u5f97\u5230\u4e0b\u8868\u6240\u793a\u7684\u5bf9\u5e94\u5173\u7cfb\u3002
     \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 LEGO \u4e50\u9ad8 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b 
    \u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0
     
    \u5728\u5b9e\u9645\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\u76f4\u63a5\u7b80\u79f0\u4e3a\u300c\u7b97\u6cd5\u300d\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u719f\u79f0\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u5bdf\u4e86\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u90e8\u5206\u77e5\u8bc6\u3002
    "},{"location":"chapter_preface/about_the_book/","title":"0.1. \u00a0 \u5173\u4e8e\u672c\u4e66","text":"
    \u672c\u9879\u76ee\u81f4\u529b\u4e8e\u6784\u5efa\u4e00\u672c\u5f00\u6e90\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\u3002
     
       
    • \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\uff1b
      •  
    • \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00\uff1b
      •  
    • \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\uff1b
      •  
    "},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1. \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u521d\u5b66\u8005\u300d\uff0c\u5b8c\u5168\u6ca1\u6709\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u5c11\u91cf\u5237\u9898\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6726\u80e7\u7684\u7406\u89e3\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u8fd9\u672c\u4e66\u5c31\u662f\u4e3a\u4f60\u800c\u5199\uff01
     
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u8001\u624b\u300d\uff0c\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u63a5\u89e6\u8fc7\u5927\u591a\u6570\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u4ee5\u5e2e\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002
     
    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u5927\u4f6c\u300d\uff0c\u5e0c\u671b\u53ef\u4ee5\u5f97\u5230\u4f60\u7684\u5b9d\u8d35\u610f\u89c1\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002
     
    \u524d\u7f6e\u6761\u4ef6
     
    \u60a8\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002
    "},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2. \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"
    \u672c\u4e66\u4e3b\u8981\u5185\u5bb9\u6709\uff1a
     
       
    • \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3001\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u4f30\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5305\u62ec\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
      •  
    • \u6570\u636e\u7ed3\u6784\uff1a\u5e38\u7528\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u5728\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3001\u6570\u636e\u7ed3\u6784\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u6563\u5217\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f18\u52a3\u52bf\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
      •  
    • \u7b97\u6cd5\uff1a\u67e5\u627e\u7b97\u6cd5\u3001\u6392\u5e8f\u7b97\u6cd5\u3001\u641c\u7d22\u4e0e\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u5206\u6cbb\u7b97\u6cd5\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f7f\u7528\u573a\u666f\u3001\u4f18\u52a3\u52bf\u3001\u65f6\u7a7a\u6548\u7387\u3001\u5b9e\u73b0\u65b9\u6cd5\u3001\u793a\u4f8b\u9898\u76ee\u7b49\u3002
      •  
     
     
     Fig. Hello \u7b97\u6cd5\u5185\u5bb9\u7ed3\u6784 
    "},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3. \u00a0 \u81f4\u8c22","text":"
    \u672c\u4e66\u7684\u6210\u4e66\u8fc7\u7a0b\u4e2d\uff0c\u6211\u83b7\u5f97\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a
     
       
    • \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u65f6\u60a8\u544a\u8bc9\u6211\u201c\u89c9\u5f97\u5e94\u8be5\u505a\u5c31\u53bb\u505a\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\u3002
      •  
    • \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u62c5\u4efb\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89c6\u89d2\u63d0\u51fa\u4e86\u8bb8\u591a\u5efa\u8bae\uff0c\u4f7f\u8fd9\u672c\u4e66\u66f4\u52a0\u9002\u5408\u521d\u5b66\u8005\u6765\u9605\u8bfb\u3002
      •  
    • \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e2a\u597d\u542c\u53c8\u6709\u6897\u540d\u5b57\uff0c\u76f4\u63a5\u5524\u8d77\u6211\u6700\u521d\u6572\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u56de\u5fc6\u3002
      •  
    • \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u5c01\u9762\u548c LOGO \uff0c\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u524d\u540e\u591a\u6b21\u5e2e\u5fd9\u4fee\u6539\uff0c\u8c22\u8c22\u4f60\u7684\u8010\u5fc3\u3002
      •  
    • \u611f\u8c22 @squidfunk \u7ed9\u51fa\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4f18\u79c0\u5f00\u6e90\u9879\u76ee Material-for-MkDocs \u3002
      •  
     
    \u672c\u4e66\u9f13\u52b1\u201c\u624b\u8111\u5e76\u7528\u201d\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u70b9\u4e0a\u53d7\u5230\u4e86\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u5f88\u5927\u5f71\u54cd\uff0c\u4e5f\u5728\u6b64\u5411\u5404\u4f4d\u540c\u5b66\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u8457\u4f5c\uff0c\u5305\u62ec\u4e2d\u6587\u7248\u3001\u82f1\u6587\u7248\u3001\u674e\u6c90\u8001\u5e08 bilibili \u4e3b\u9875\u3002
     
    \u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u4e0e\u6587\u7ae0\uff0c\u8fd9\u4e9b\u8457\u4f5c\u4e3a\u672c\u4e66\u4f5c\u51fa\u4e86\u5f88\u597d\u7684\u699c\u6837\uff0c\u4fdd\u8bc1\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u6b63\u786e\u6027\u4e0e\u8d28\u91cf\uff0c\u611f\u8c22\u5404\u4f4d\u8001\u5e08\u4e0e\u524d\u8f88\u7684\u7cbe\u5f69\u521b\u4f5c\uff01
     
    \u611f\u8c22\u7236\u6bcd\uff0c\u4f60\u4eec\u4e00\u8d2f\u7684\u652f\u6301\u4e0e\u9f13\u52b1\u7ed9\u4e86\u6211\u81ea\u7531\u5ea6\u6765\u505a\u8fd9\u4e9b\u6709\u8da3\u7684\u4e8b\u3002
    "},{"location":"chapter_preface/suggestions/","title":"0.2. \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"
    \u5efa\u8bae\u901a\u8bfb\u672c\u8282\u5185\u5bb9\uff0c\u4ee5\u83b7\u53d6\u6700\u4f73\u9605\u8bfb\u4f53\u9a8c\u3002
    "},{"location":"chapter_preface/suggestions/#021","title":"0.2.1. \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"
    \u603b\u4f53\u4e0a\u770b\uff0c\u6211\u8ba4\u4e3a\u53ef\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002
     
       
    1. \u7b97\u6cd5\u5165\u95e8\u3002\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u3001\u7528\u6cd5\uff0c\u5b66\u4e60\u5404\u79cd\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u3001\u6548\u7387\u7b49\u3002
      2.  
    2. \u5237\u7b97\u6cd5\u9898\u3002\u53ef\u4ee5\u5148\u4ece\u70ed\u95e8\u9898\u5355\u5f00\u5237\uff0c\u63a8\u8350\u5251\u6307 Offer\u3001LeetCode Hot 100\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u91cf\uff0c\u719f\u6089\u5927\u591a\u6570\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521a\u5f00\u59cb\u5237\u9898\u65f6\uff0c\u201c\u9057\u5fd8\u201d\u662f\u6700\u5927\u7684\u56f0\u6270\u70b9\uff0c\u4f46\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\uff0c\u8bf7\u4e0d\u8981\u62c5\u5fc3\u3002\u5b66\u4e60\u4e2d\u6709\u4e00\u79cd\u6982\u5ff5\u53eb\u201c\u5468\u671f\u6027\u56de\u987e\u201d\uff0c\u540c\u4e00\u9053\u9898\u9694\u6bb5\u65f6\u95f4\u505a\u4e00\u6b21\uff0c\u5728\u91cd\u590d 3 \u8f6e\u4ee5\u4e0a\u540e\uff0c\u5f80\u5f80\u5c31\u80fd\u7262\u8bb0\u4e8e\u5fc3\u4e86\u3002
      3.  
    3. \u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u3001\u7b97\u6cd5\u6559\u6750\uff0c\u4e0d\u65ad\u5730\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5f00\u59cb\u91c7\u7528\u8fdb\u9636\u5237\u9898\u65b9\u6848\uff0c\u4f8b\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u4e2d\u627e\u5230\u3002
      4.  
     
    \u4f5c\u4e3a\u4e00\u672c\u5165\u95e8\u6559\u7a0b\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u5bf9\u5e94\u201c\u7b2c\u4e00\u9636\u6bb5\u201d\uff0c\u81f4\u529b\u4e8e\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5f00\u5c55\u7b2c\u4e8c\u3001\u4e09\u9636\u6bb5\u7684\u5b66\u4e60\u3002
     
     
     Fig. \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf 
    "},{"location":"chapter_preface/suggestions/#022","title":"0.2.2. \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"
    \u6807\u9898\u540e\u6807\u6ce8 * \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u8f83\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002
     
    \u6587\u7ae0\u4e2d\u7684\u91cd\u8981\u540d\u8bcd\u4f1a\u7528 \u300c\u62ec\u53f7\u300d \u6807\u6ce8\uff0c\u4f8b\u5982 \u300c\u6570\u7ec4 Array\u300d \u3002\u5efa\u8bae\u8bb0\u4f4f\u8fd9\u4e9b\u540d\u8bcd\uff0c\u5305\u62ec\u82f1\u6587\u7ffb\u8bd1\uff0c\u4ee5\u4fbf\u540e\u7eed\u9605\u8bfb\u6587\u732e\u65f6\u4f7f\u7528\u3002
     
    \u91cd\u70b9\u5185\u5bb9\u3001\u603b\u8d77\u53e5\u3001\u603b\u7ed3\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u6b64\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
     
    \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 \u201c\u53cc\u5f15\u53f7\u201d \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002
     
    \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    \"\"\" \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 \"\"\"\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
     
    // \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n// \u591a\u884c\n// \u6ce8\u91ca\n
    "},{"location":"chapter_preface/suggestions/#023","title":"0.2.3. \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"
    \u89c6\u9891\u548c\u56fe\u7247\u76f8\u6bd4\u4e8e\u6587\u5b57\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\u66f4\u9ad8\uff0c\u66f4\u5bb9\u6613\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u77e5\u8bc6\u91cd\u96be\u70b9\u4f1a\u4e3b\u8981\u4ee5\u52a8\u753b\u3001\u56fe\u89e3\u7684\u5f62\u5f0f\u5448\u73b0\uff0c\u800c\u6587\u5b57\u7684\u4f5c\u7528\u5219\u662f\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002
     
    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u82e5\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u5efa\u8bae\u4f60\u4ee5\u56fe\u4e3a\u4e3b\u7ebf\uff0c\u5c06\u6587\u5b57\u5185\u5bb9\uff08\u4e00\u822c\u5728\u56fe\u7684\u4e0a\u65b9\uff09\u5bf9\u9f50\u5230\u56fe\u4e2d\u5185\u5bb9\uff0c\u7efc\u5408\u6765\u7406\u89e3\u3002
     
     
     Fig. \u52a8\u753b\u56fe\u89e3\u793a\u4f8b 
    "},{"location":"chapter_preface/suggestions/#024","title":"0.2.4. \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"
    \u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728GitHub \u4ed3\u5e93\uff0c\u6e90\u4ee3\u7801\u5305\u542b\u8be6\u7ec6\u6ce8\u91ca\uff0c\u914d\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4ee5\u76f4\u63a5\u8fd0\u884c\u3002
     
       
    • \u82e5\u5b66\u4e60\u65f6\u95f4\u7d27\u5f20\uff0c\u5efa\u8bae\u81f3\u5c11\u5c06\u6240\u6709\u4ee3\u7801\u901a\u8bfb\u5e76\u8fd0\u884c\u4e00\u904d\u3002
      •  
    • \u82e5\u65f6\u95f4\u5141\u8bb8\uff0c\u5f3a\u70c8\u5efa\u8bae\u5bf9\u7167\u7740\u4ee3\u7801\u81ea\u5df1\u6572\u4e00\u904d\u3002\u76f8\u6bd4\u4e8e\u8bfb\u4ee3\u7801\uff0c\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u65b0\u7684\u6536\u83b7\u3002
      •  
     
     
     Fig. \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b 
     
    \u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u53c2\u7167\u9644\u5f55\u6559\u7a0b\uff0c\u5982\u679c\u5df2\u6709\u53ef\u76f4\u63a5\u8df3\u8fc7\u3002
     
    \u7b2c\u4e8c\u6b65\uff1a\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u547d\u4ee4\u884c\u6765\u514b\u9686\u4ee3\u7801\u4ed3\u3002
     
    git clone https://github.com/krahets/hello-algo.git\n
     
    \u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201cDownload ZIP\u201d\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002
     
     
     Fig. \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 
     
    \u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u82e5\u4ee3\u7801\u5757\u7684\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\uff0c\u5219\u53ef\u5728\u4ed3\u5e93 codes \u6587\u4ef6\u5939\u4e2d\u627e\u5230\u5bf9\u5e94\u7684 \u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4ee5\u5e2e\u52a9\u4f60\u7701\u53bb\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u5c06\u7cbe\u529b\u96c6\u4e2d\u5728\u5b66\u4e60\u5185\u5bb9\u4e0a\u3002
     
     
     Fig. \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 
    "},{"location":"chapter_preface/suggestions/#025","title":"0.2.5. \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"
    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u201c\u60ef\u7740\u201d\u90a3\u4e9b\u5f04\u4e0d\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u7559\u4e0b\u4f60\u7684\u95ee\u9898\uff0c\u5c0f\u4f19\u4f34\u4eec\u548c\u6211\u90fd\u4f1a\u7ed9\u4e88\u89e3\u7b54\uff0c\u60a8\u4e00\u822c 2 \u65e5\u5185\u4f1a\u5f97\u5230\u56de\u590d\u3002
     
    \u540c\u65f6\uff0c\u4e5f\u5e0c\u671b\u4f60\u53ef\u4ee5\u591a\u82b1\u65f6\u95f4\u901b\u901b\u8bc4\u8bba\u533a\u3002\u4e00\u65b9\u9762\uff0c\u53ef\u4ee5\u770b\u770b\u5927\u5bb6\u9047\u5230\u4e86\u4ec0\u4e48\u95ee\u9898\uff0c\u53cd\u8fc7\u6765\u67e5\u6f0f\u8865\u7f3a\uff0c\u8fd9\u5f80\u5f80\u53ef\u4ee5\u5f15\u8d77\u66f4\u52a0\u6df1\u5ea6\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e5f\u5e0c\u671b\u4f60\u53ef\u4ee5\u6177\u6168\u5730\u89e3\u7b54\u5c0f\u4f19\u4f34\u4eec\u7684\u95ee\u9898\u3001\u5206\u4eab\u81ea\u5df1\u7684\u89c1\u89e3\uff0c\u5927\u5bb6\u4e92\u76f8\u5b66\u4e60\u4e0e\u8fdb\u6b65\uff01
     
     
     Fig. \u8bc4\u8bba\u533a\u793a\u4f8b 
    "},{"location":"chapter_preface/summary/","title":"0.3. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u672c\u4e66\u4e3b\u8981\u9762\u5411\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5bf9\u4e8e\u5df2\u7ecf\u6709\u4e00\u5b9a\u79ef\u7d2f\u7684\u540c\u5b66\uff0c\u8fd9\u672c\u4e66\u53ef\u4ee5\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u6e90\u4ee3\u7801\u53ef\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6765\u4f7f\u7528\u3002
      •  
    • \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5206\u4e3a\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u8986\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
      •  
    • \u5bf9\u4e8e\u7b97\u6cd5\u5c0f\u767d\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u662f\u975e\u5e38\u6709\u5fc5\u8981\u7684\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
      •  
    • \u4e66\u5185\u7684\u52a8\u753b\u548c\u56fe\u89e3\u5f80\u5f80\u4ecb\u7ecd\u7684\u662f\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\uff0c\u5728\u9605\u8bfb\u65f6\u5e94\u8be5\u591a\u52a0\u5173\u6ce8\u3002
      •  
    • \u5b9e\u8df5\u662f\u5b66\u4e60\u7f16\u7a0b\u7684\u6700\u4f73\u65b9\u5f0f\uff0c\u5f3a\u70c8\u63a8\u8350\u8fd0\u884c\u6e90\u4ee3\u7801\uff0c\u52a8\u624b\u6572\u4ee3\u7801\u3002
      •  
    • \u672c\u4e66\u63d0\u4f9b\u4e86\u8ba8\u8bba\u533a\uff0c\u9047\u5230\u7591\u60d1\u53ef\u4ee5\u968f\u65f6\u63d0\u95ee\u3002
      •  
    "},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"
    [1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).
     
    [2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).
     
    [3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08 C \u8bed\u8a00\u7248\uff09.
     
    [4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08 C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.
     
    [5] \u9a6c\u514b\u00b7\u827e\u4f26\u00b7\u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.
     
    [6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.
     
    [7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.
     
    [8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition). 
     
    [9] Aston Zhang, et al. Dive into Deep Learning.
    "},{"location":"chapter_searching/binary_search/","title":"10.2. \u00a0 \u4e8c\u5206\u67e5\u627e","text":"
    \u300c\u4e8c\u5206\u67e5\u627e Binary Search\u300d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u67e5\u627e\u76ee\u6807\u5143\u7d20\u3002
     
    \u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u6709\u4e24\u4e2a\u524d\u7f6e\u6761\u4ef6\uff1a
     
       
    • \u8981\u6c42\u8f93\u5165\u6570\u636e\u662f\u6709\u5e8f\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u5224\u65ad\u5927\u5c0f\u5173\u7cfb\u6765\u6392\u9664\u4e00\u534a\u7684\u641c\u7d22\u533a\u95f4\uff1b
      •  
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u4f7f\u7528\u6548\u7387\u5f88\u4f4e\uff0c\u56e0\u4e3a\u5176\u5728\u5faa\u73af\u4e2d\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\u3002
      •  
    "},{"location":"chapter_searching/binary_search/#1021","title":"10.2.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6392\u5e8f\u6570\u7ec4 nums \uff0c\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\u3002\u6570\u7ec4\u7684\u7d22\u5f15\u53d6\u503c\u8303\u56f4\u4e3a
     \\[ 0, 1, 2, \\cdots, n-1 \\] 
    \u4f7f\u7528\u300c\u533a\u95f4\u300d\u6765\u8868\u793a\u8fd9\u4e2a\u53d6\u503c\u8303\u56f4\u7684\u65b9\u6cd5\u4e3b\u8981\u6709\u4e24\u79cd\uff1a
     
       
    1. \u53cc\u95ed\u533a\u95f4 \\([0, n-1]\\) \uff0c\u5373\u4e24\u4e2a\u8fb9\u754c\u90fd\u5305\u542b\u81ea\u8eab\uff1b\u6b64\u65b9\u6cd5\u4e0b\uff0c\u533a\u95f4 \\([0, 0]\\) \u4ecd\u5305\u542b\u4e00\u4e2a\u5143\u7d20\uff1b
      2.  
    2. \u5de6\u95ed\u53f3\u5f00 \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\u3001\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\uff1b\u6b64\u65b9\u6cd5\u4e0b\uff0c\u533a\u95f4 \\([0, 0)\\) \u4e3a\u7a7a\uff1b
      3.  
    "},{"location":"chapter_searching/binary_search/#_1","title":"\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5b9e\u73b0","text":"
    \u9996\u5148\uff0c\u6211\u4eec\u5148\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u8868\u793a\uff0c\u5728\u6570\u7ec4 nums \u4e2d\u67e5\u627e\u76ee\u6807\u5143\u7d20 target \u7684\u5bf9\u5e94\u7d22\u5f15\u3002
     <1><2><3><4><5><6><7> 
     
     
     
     
     
     
     
    \u4e8c\u5206\u67e5\u627e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e0b\u7684\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int>& nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.size() - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.py
    def binary_search(nums: List[int], target: int) -> int:\n\"\"\" \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 \"\"\"\n# \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j = 0, len(nums) - 1\nwhile i <= j:\nm = (i + j) // 2        # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:    # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\nelif nums[m] > target:  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\nelse:\nreturn m            # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1                   # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j := 0, len(nums)-1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nfor i <= j {\nm := (i + j) / 2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nconst m = parseInt((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u5728 JS \u4e2d\u9700\u4f7f\u7528 parseInt \u51fd\u6570\u53d6\u6574\nif (nums[m] < target)          // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target)     // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse\nreturn m;                  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nconst m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {        // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
     binary_search.c
    [class]{}-[func]{binarySearch}\n
     binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target)\n{\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.Length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j)\n{\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i = 0\nvar j = nums.count - 1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j {\nlet m = (i + j) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i: usize = 0;\nvar j: usize = nums.items.len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = (i + j) / 2;                    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    "},{"location":"chapter_searching/binary_search/#_2","title":"\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5b9e\u73b0","text":"
    \u5f53\u7136\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5199\u51fa\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(vector<int>& nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.size();\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.py
    def binary_search1(nums: List[int], target: int) -> int:\n\"\"\" \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 \"\"\"\n# \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j = 0, len(nums)\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j:\nm = (i + j) // 2        # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:    # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\nelif nums[m] > target:  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\nelse:                   # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\nreturn -1                   # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearch1(nums []int, target int) int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j := 0, len(nums)\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nfor i < j {\nm := (i + j) / 2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearch1(nums, target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nconst m = parseInt((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u5728 JS \u4e2d\u9700\u4f7f\u7528 parseInt \u51fd\u6570\u53d6\u6574\nif (nums[m] < target)          // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target)     // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                           // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearch1(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nconst m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {        // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
     binary_search.c
    [class]{}-[func]{binarySearch1}\n
     binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearch1(int[] nums, int target)\n{\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.Length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j)\n{\nint m = (i + j) / 2;       // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearch1(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i = 0\nvar j = nums.count\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j {\nlet m = (i + j) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\nfn binarySearch1(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i: usize = 0;\nvar j: usize = nums.items.len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = (i + j) / 2;                    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    "},{"location":"chapter_searching/binary_search/#_3","title":"\u4e24\u79cd\u8868\u793a\u5bf9\u6bd4","text":"
    \u5bf9\u6bd4\u4e0b\u6765\uff0c\u4e24\u79cd\u8868\u793a\u7684\u4ee3\u7801\u5199\u6cd5\u6709\u4ee5\u4e0b\u4e0d\u540c\u70b9\uff1a
     \u8868\u793a\u65b9\u6cd5 \u521d\u59cb\u5316\u6307\u9488 \u7f29\u5c0f\u533a\u95f4 \u5faa\u73af\u7ec8\u6b62\u6761\u4ef6 \u53cc\u95ed\u533a\u95f4 \\([0, n-1]\\) \\(i = 0\\) , \\(j = n-1\\) \\(i = m + 1\\) , \\(j = m - 1\\) \\(i > j\\) \u5de6\u95ed\u53f3\u5f00 \\([0, n)\\) \\(i = 0\\) , \\(j = n\\) \\(i = m + 1\\) , \\(j = m\\) \\(i = j\\) 
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\uff0c\u7531\u4e8e\u5bf9\u5de6\u53f3\u4e24\u8fb9\u754c\u7684\u5b9a\u4e49\u662f\u76f8\u540c\u7684\uff0c\u56e0\u6b64\u7f29\u5c0f\u533a\u95f4\u7684 \\(i\\) , \\(j\\) \u5904\u7406\u65b9\u6cd5\u4e5f\u662f\u5bf9\u79f0\u7684\uff0c\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\u3002\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002
    "},{"location":"chapter_searching/binary_search/#_4","title":"\u5927\u6570\u8d8a\u754c\u5904\u7406","text":"
    \u5f53\u6570\u7ec4\u957f\u5ea6\u5f88\u5927\u65f6\uff0c\u52a0\u6cd5 \\(i + j\\) \u7684\u7ed3\u679c\u6709\u53ef\u80fd\u4f1a\u8d85\u51fa int \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u5728\u6b64\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u6362\u4e00\u79cd\u8ba1\u7b97\u4e2d\u70b9\u7684\u5199\u6cd5\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    # Python \u4e2d\u7684\u6570\u5b57\u7406\u8bba\u4e0a\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\n# \u56e0\u6b64\u65e0\u9700\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nm := (i + j) / 2\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nm := i + (j - i) / 2\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nlet m = parseInt((i + j) / 2);\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = parseInt(i + (j - i) / 2);\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa Number \u7684\u53d6\u503c\u8303\u56f4\nlet m = Math.floor((i + j) / 2);\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = Math.floor(i + (j - i) / 2);\n
     
    \n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nint m = (i + j) / 2;\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nint m = i + (j - i) / 2;\n
     
    // (i + j) \u6709\u53ef\u80fd\u8d85\u51fa int \u7684\u53d6\u503c\u8303\u56f4\nlet m = (i + j) / 2\n// \u66f4\u6362\u4e3a\u6b64\u5199\u6cd5\u5219\u4e0d\u4f1a\u8d8a\u754c\nlet m = i + (j - 1) / 2\n
     
    \n
    "},{"location":"chapter_searching/binary_search/#1022","title":"10.2.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\uff1b\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u7684\u533a\u95f4\uff0c\u56e0\u6b64\u5faa\u73af\u8f6e\u6570\u4e3a \\(\\log_2 n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 i , j \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002
    "},{"location":"chapter_searching/binary_search/#1023","title":"10.2.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u4e8c\u5206\u67e5\u627e\u6548\u7387\u5f88\u9ad8\uff0c\u4f53\u73b0\u5728\uff1a
     
       
    • \u4e8c\u5206\u67e5\u627e\u65f6\u95f4\u590d\u6742\u5ea6\u4f4e\u3002\u5bf9\u6570\u9636\u5728\u6570\u636e\u91cf\u5f88\u5927\u65f6\u5177\u6709\u5de8\u5927\u4f18\u52bf\uff0c\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700\u8981 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u4e0d\u9700\u8981\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u5bf9\u4e8e\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u6765\u5b9e\u73b0\u67e5\u627e\u7684\u7b97\u6cd5\u6765\u8bf4\uff0c\u5176\u66f4\u52a0\u8282\u7ea6\u7a7a\u95f4\u4f7f\u7528\u3002
      •  
     
    \u4f46\u5e76\u4e0d\u610f\u5473\u7740\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u5e94\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u65e0\u5e8f\u7684\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u6267\u884c\u6570\u636e\u6392\u5e8f\uff0c\u90a3\u4e48\u662f\u5f97\u4e0d\u507f\u5931\u7684\uff0c\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e00\u822c\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u5dee\u3002\u518d\u4f8b\u5982\uff0c\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4e86\u4fdd\u6301\u6570\u7ec4\u7684\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u7531\u4e8e\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u8bbf\u95ee\u7d22\u5f15\u662f \u201c\u975e\u8fde\u7eed\u201d \u7684\uff0c\u56e0\u6b64\u94fe\u8868\u6216\u8005\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u90fd\u65e0\u6cd5\u4f7f\u7528\u3002
      •  
    • \u5728\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u7684\u6027\u80fd\u66f4\u597d\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700\u8981 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5728\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
      •  
    "},{"location":"chapter_searching/hashing_search/","title":"10.3. \u00a0 \u54c8\u5e0c\u67e5\u627e","text":"
    Question
     
    \u5728\u6570\u636e\u91cf\u5f88\u5927\u65f6\uff0c\u300c\u7ebf\u6027\u67e5\u627e\u300d\u592a\u6162\uff1b\u800c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u8981\u6c42\u6570\u636e\u5fc5\u987b\u662f\u6709\u5e8f\u7684\uff0c\u5e76\u4e14\u53ea\u80fd\u5728\u6570\u7ec4\u4e2d\u5e94\u7528\u3002\u90a3\u4e48\u662f\u5426\u6709\u65b9\u6cd5\u53ef\u4ee5\u540c\u65f6\u907f\u514d\u4e0a\u8ff0\u7f3a\u70b9\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\uff0c\u6b64\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u54c8\u5e0c\u67e5\u627e\u300d\u3002
     
    \u300c\u54c8\u5e0c\u67e5\u627e Hash Searching\u300d\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u9700\u8981\u7684\u300c\u952e\u503c\u5bf9 Key Value Pair\u300d\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u5b9e\u73b0\u201c\u952e \\(\\rightarrow\\) \u503c\u201d\u6620\u5c04\u67e5\u627e\uff0c\u4f53\u73b0\u7740\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002
    "},{"location":"chapter_searching/hashing_search/#1031","title":"10.3.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u5982\u679c\u6211\u4eec\u60f3\u8981\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 target \uff0c\u83b7\u53d6\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u90a3\u4e48\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u5b9e\u73b0\u67e5\u627e\u3002
     
     
     Fig. \u54c8\u5e0c\u67e5\u627e\u6570\u7ec4\u7d22\u5f15 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hashing_search.java
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(Map<Integer, Integer> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.getOrDefault(target, -1);\n}\n
     hashing_search.cpp
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(unordered_map<int, int> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nif (map.find(target) == map.end())\nreturn -1;\nreturn map[target];\n}\n
     hashing_search.py
    def hashing_search_array(mapp: Dict[int, int], target: int) -> int:\n\"\"\" \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 \"\"\"\n# \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n# \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn mapp.get(target, -1)\n
     hashing_search.go
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc hashingSearchArray(m map[int]int, target int) int {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nif index, ok := m[target]; ok {\nreturn index\n} else {\nreturn -1\n}\n}\n
     hashing_search.js
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction hashingSearchArray(map, target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.has(target) ? map.get(target) : -1;\n}\n
     hashing_search.ts
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction hashingSearchArray(map: Map<number, number>, target: number): number {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.has(target) ? (map.get(target) as number) : -1;\n}\n
     hashing_search.c
    [class]{}-[func]{hashingSearchArray}\n
     hashing_search.cs
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint hashingSearchArray(Dictionary<int, int> map, int target)\n{\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map.GetValueOrDefault(target, -1);\n}\n
     hashing_search.swift
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc hashingSearchArray(map: [Int: Int], target: Int) -> Int {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1\nreturn map[target, default: -1]\n}\n
     hashing_search.zig
    // \u54c8\u5e0c\u67e5\u627e\uff08\u6570\u7ec4\uff09\nfn hashingSearchArray(comptime T: type, map: std.AutoHashMap(T, T), target: T) T {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7d22\u5f15\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de -1   \nif (map.getKey(target) == null) return -1;\nreturn map.get(target).?;\n}\n
     
    \u518d\u6bd4\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u7ed9\u5b9a\u4e00\u4e2a\u76ee\u6807\u7ed3\u70b9\u503c target \uff0c\u83b7\u53d6\u5bf9\u5e94\u7684\u94fe\u8868\u7ed3\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u4e5f\u53ef\u4ee5\u4f7f\u7528\u54c8\u5e0c\u67e5\u627e\u5b9e\u73b0\u3002
     
     
     Fig. \u54c8\u5e0c\u67e5\u627e\u94fe\u8868\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig hashing_search.java
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode hashingSearchLinkedList(Map<Integer, ListNode> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.getOrDefault(target, null);\n}\n
     hashing_search.cpp
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode* hashingSearchLinkedList(unordered_map<int, ListNode*> map, int target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de nullptr\nif (map.find(target) == map.end())\nreturn nullptr;\nreturn map[target];\n}\n
     hashing_search.py
    def hashing_search_linkedlist(mapp: Dict[int, ListNode], target: int) -> Optional[ListNode]:\n\"\"\" \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 \"\"\"\n# \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u5143\u7d20\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n# \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de None\nreturn mapp.get(target, None)\n
     hashing_search.go
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc hashingSearchLinkedList(m map[int]*ListNode, target int) *ListNode {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de nil\nif node, ok := m[target]; ok {\nreturn node\n} else {\nreturn nil\n}\n}\n
     hashing_search.js
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunction hashingSearchLinkedList(map, target) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.has(target) ? map.get(target) : null;\n}\n
     hashing_search.ts
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunction hashingSearchLinkedList(map: Map<number, ListNode>, target: number): ListNode | null {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.has(target) ? (map.get(target) as ListNode) : null;\n}\n
     hashing_search.c
    [class]{}-[func]{hashingSearchLinkedList}\n
     hashing_search.cs
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode? hashingSearchLinkedList(Dictionary<int, ListNode> map, int target)\n{\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map.GetValueOrDefault(target);\n}\n
     hashing_search.swift
    /* \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc hashingSearchLinkedList(map: [Int: ListNode], target: Int) -> ListNode? {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null\nreturn map[target]\n}\n
     hashing_search.zig
    // \u54c8\u5e0c\u67e5\u627e\uff08\u94fe\u8868\uff09\nfn hashingSearchLinkedList(comptime T: type, map: std.AutoHashMap(T, *inc.ListNode(T)), target: T) ?*inc.ListNode(T) {\n// \u54c8\u5e0c\u8868\u7684 key: \u76ee\u6807\u7ed3\u70b9\u503c\uff0cvalue: \u7ed3\u70b9\u5bf9\u8c61\n// \u82e5\u54c8\u5e0c\u8868\u4e2d\u65e0\u6b64 key \uff0c\u8fd4\u56de null \nif (map.getKey(target) == null) return null;\nreturn map.get(target);\n}\n
    "},{"location":"chapter_searching/hashing_search/#1032","title":"10.3.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u54c8\u5e0c\u8868\u7684\u67e5\u627e\u64cd\u4f5c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\u3002
    "},{"location":"chapter_searching/hashing_search/#1033","title":"10.3.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \uff0c\u8fd9\u610f\u5473\u7740\u65e0\u8bba\u662f\u9ad8\u9891\u589e\u5220\u8fd8\u662f\u9ad8\u9891\u67e5\u627e\u573a\u666f\uff0c\u54c8\u5e0c\u67e5\u627e\u7684\u6027\u80fd\u8868\u73b0\u90fd\u975e\u5e38\u597d\u3002\u5f53\u7136\uff0c\u4e00\u5207\u7684\u524d\u63d0\u662f\u4fdd\u8bc1\u54c8\u5e0c\u8868\u672a\u9000\u5316\u3002
     
    \u5373\u4f7f\u5982\u6b64\uff0c\u54c8\u5e0c\u67e5\u627e\u4ecd\u5b58\u5728\u4e00\u4e9b\u95ee\u9898\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u9700\u8981\u6839\u636e\u60c5\u51b5\u7075\u6d3b\u9009\u62e9\u65b9\u6cd5\u3002
     
       
    • \u8f85\u52a9\u54c8\u5e0c\u8868 \u9700\u8981\u4f7f\u7528 \\(O(n)\\) \u7684\u989d\u5916\u7a7a\u95f4\uff0c\u610f\u5473\u7740\u9700\u8981\u9884\u7559\u66f4\u591a\u7684\u8ba1\u7b97\u673a\u5185\u5b58\uff1b
      •  
    • \u5efa\u7acb\u548c\u7ef4\u62a4\u54c8\u5e0c\u8868\u9700\u8981\u65f6\u95f4\uff0c\u56e0\u6b64\u54c8\u5e0c\u67e5\u627e \u4e0d\u9002\u5408\u9ad8\u9891\u589e\u5220\u3001\u4f4e\u9891\u67e5\u627e\u7684\u4f7f\u7528\u573a\u666f\uff1b
      •  
    • \u5f53\u54c8\u5e0c\u51b2\u7a81\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \uff1b
      •  
    • \u5f53\u6570\u636e\u91cf\u5f88\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u6bd4\u54c8\u5e0c\u67e5\u627e\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u8ba1\u7b97\u54c8\u5e0c\u6620\u5c04\u51fd\u6570\u53ef\u80fd\u6bd4\u904d\u5386\u4e00\u4e2a\u5c0f\u578b\u6570\u7ec4\u66f4\u6162\uff1b
      •  
    "},{"location":"chapter_searching/linear_search/","title":"10.1. \u00a0 \u7ebf\u6027\u67e5\u627e","text":"
    \u300c\u7ebf\u6027\u67e5\u627e Linear Search\u300d\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u67e5\u627e\u65b9\u6cd5\uff0c\u5176\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u4f9d\u6b21\u8bbf\u95ee\u6bcf\u4e2a\u5143\u7d20\uff0c\u76f4\u5230\u53e6\u4e00\u7aef\u540e\u505c\u6b62\u3002
    "},{"location":"chapter_searching/linear_search/#1011","title":"10.1.1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"
    \u7ebf\u6027\u67e5\u627e\u5b9e\u8d28\u4e0a\u5c31\u662f\u904d\u5386\u6570\u636e\u7ed3\u6784 + \u5224\u65ad\u6761\u4ef6\u3002\u6bd4\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4 nums \u4e2d\u67e5\u627e\u76ee\u6807\u5143\u7d20 target \u7684\u5bf9\u5e94\u7d22\u5f15\uff0c\u90a3\u4e48\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u7ebf\u6027\u67e5\u627e\u3002
     
     
     Fig. \u5728\u6570\u7ec4\u4e2d\u7ebf\u6027\u67e5\u627e\u5143\u7d20 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linear_search.java
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(int[] nums, int target) {\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.cpp
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(vector<int>& nums, int target) {\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.size(); i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.py
    def linear_search_array(nums: List[int], target: int) -> int:\n\"\"\" \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 \"\"\"\n# \u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\nif nums[i] == target:  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn i\nreturn -1                  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
     linear_search.go
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc linearSearchArray(nums []int, target int) int {\n// \u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif nums[i] == target {\nreturn i\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     linear_search.js
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunction linearSearchArray(nums, target) {\n// \u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] === target) {\nreturn i;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1; }\n
     linear_search.ts
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09*/\nfunction linearSearchArray(nums: number[], target: number): number {\n// \u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] === target) {\nreturn i;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.c
    [class]{}-[func]{linearSearchArray}\n
     linear_search.cs
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nint linearSearchArray(int[] nums, int target)\n{\n// \u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++)\n{\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif (nums[i] == target)\nreturn i;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     linear_search.swift
    /* \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09 */\nfunc linearSearchArray(nums: [Int], target: Int) -> Int {\n// \u904d\u5386\u6570\u7ec4\nfor i in nums.indices {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nif nums[i] == target {\nreturn i\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
     linear_search.zig
    // \u7ebf\u6027\u67e5\u627e\uff08\u6570\u7ec4\uff09\nfn linearSearchArray(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u904d\u5386\u6570\u7ec4\nfor (nums.items) |num, i| {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c \u8fd4\u56de\u5176\u7d22\u5f15\nif (num == target) {\nreturn @intCast(T, i);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
     
    \u518d\u6bd4\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5728\u7ed9\u5b9a\u4e00\u4e2a\u76ee\u6807\u7ed3\u70b9\u503c target \uff0c\u8fd4\u56de\u6b64\u7ed3\u70b9\u5bf9\u8c61\uff0c\u4e5f\u53ef\u4ee5\u5728\u94fe\u8868\u4e2d\u8fdb\u884c\u7ebf\u6027\u67e5\u627e\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linear_search.java
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode linearSearchLinkedList(ListNode head, int target) {\n// \u904d\u5386\u94fe\u8868\nwhile (head != null) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val == target)\nreturn head;\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.cpp
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode* linearSearchLinkedList(ListNode* head, int target) {\n// \u904d\u5386\u94fe\u8868\nwhile (head != nullptr) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head->val == target)\nreturn head;\nhead = head->next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de nullptr\nreturn nullptr;\n}\n
     linear_search.py
    def linear_search_linkedlist(head: ListNode, target: int) -> Optional[ListNode]:\n\"\"\" \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 \"\"\"\n# \u904d\u5386\u94fe\u8868\nwhile head:\nif head.val == target: # \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nreturn head\nhead = head.next\nreturn None                # \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de None\n
     linear_search.go
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc linearSearchLinkedList(node *ListNode, target int) *ListNode {\n// \u904d\u5386\u94fe\u8868\nfor node != nil {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif node.Val == target {\nreturn node\n}\nnode = node.Next\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de nil\nreturn nil\n}\n
     linear_search.js
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09*/\nfunction linearSearchLinkedList(head, target) {\n// \u904d\u5386\u94fe\u8868\nwhile(head) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif(head.val === target) {\nreturn head;\n}\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.ts
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09*/\nfunction linearSearchLinkedList(head: ListNode | null, target: number): ListNode | null {\n// \u904d\u5386\u94fe\u8868\nwhile (head) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val === target) {\nreturn head;\n}\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.c
    [class]{}-[func]{linearSearchLinkedList}\n
     linear_search.cs
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nListNode? linearSearchLinkedList(ListNode head, int target)\n{\n// \u904d\u5386\u94fe\u8868\nwhile (head != null)\n{\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.val == target)\nreturn head;\nhead = head.next;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn null;\n}\n
     linear_search.swift
    /* \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09 */\nfunc linearSearchLinkedList(head: ListNode?, target: Int) -> ListNode? {\nvar head = head\n// \u904d\u5386\u94fe\u8868\nwhile head != nil {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif head?.val == target {\nreturn head\n}\nhead = head?.next\n}\n// \u672a\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de null\nreturn nil\n}\n
     linear_search.zig
    // \u7ebf\u6027\u67e5\u627e\uff08\u94fe\u8868\uff09\nfn linearSearchLinkedList(comptime T: type, node: ?*inc.ListNode(T), target: T) ?*inc.ListNode(T) {\nvar head = node;\n// \u904d\u5386\u94fe\u8868\nwhile (head != null) {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8fd4\u56de\u4e4b\nif (head.?.val == target) return head;\nhead = head.?.next;\n}\nreturn null;\n}\n
    "},{"location":"chapter_searching/linear_search/#1012","title":"10.1.2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u957f\u5ea6\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u65e0\u9700\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\u3002
    "},{"location":"chapter_searching/linear_search/#1013","title":"10.1.3. \u00a0 \u4f18\u70b9\u4e0e\u7f3a\u70b9","text":"
    \u7ebf\u6027\u67e5\u627e\u7684\u901a\u7528\u6027\u6781\u4f73\u3002\u7531\u4e8e\u7ebf\u6027\u67e5\u627e\u662f\u4f9d\u6b21\u8bbf\u95ee\u5143\u7d20\u7684\uff0c\u5373\u6ca1\u6709\u8df3\u8dc3\u8bbf\u95ee\u5143\u7d20\uff0c\u56e0\u6b64\u6570\u7ec4\u6216\u94fe\u8868\u7686\u9002\u7528\u3002
     
    \u7ebf\u6027\u67e5\u627e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u592a\u9ad8\u3002\u5728\u6570\u636e\u91cf \\(n\\) \u5f88\u5927\u65f6\uff0c\u67e5\u627e\u6548\u7387\u5f88\u4f4e\u3002
    "},{"location":"chapter_searching/summary/","title":"10.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u67e5\u627e\u65b9\u6cd5\uff0c\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784 + \u5224\u65ad\u6761\u4ef6\u5b9e\u73b0\u67e5\u627e\u3002
      •  
    • \u4e8c\u5206\u67e5\u627e\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u4e0d\u65ad\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u5b9e\u73b0\u67e5\u627e\uff0c\u5176\u8981\u6c42\u8f93\u5165\u6570\u636e\u662f\u6709\u5e8f\u7684\uff0c\u5e76\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u54c8\u5e0c\u67e5\u627e\u501f\u52a9\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u5e38\u6570\u9636\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u67e5\u627e\u64cd\u4f5c\uff0c\u4f53\u73b0\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u7684\u7b97\u6cd5\u601d\u60f3\u3002
      •  
    • \u4e0b\u8868\u603b\u7ed3\u5bf9\u6bd4\u4e86\u67e5\u627e\u7b97\u6cd5\u7684\u5404\u79cd\u7279\u6027\u548c\u65f6\u95f4\u590d\u6742\u5ea6\u3002
      •  
     \u7ebf\u6027\u67e5\u627e \u4e8c\u5206\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u9002\u7528\u6570\u636e\u7ed3\u6784 \u6570\u7ec4\u3001\u94fe\u8868 \u6570\u7ec4 \u6570\u7ec4\u3001\u94fe\u8868 \u8f93\u5165\u6570\u636e\u8981\u6c42 \u65e0 \u6709\u5e8f \u65e0 \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u67e5\u627e / \u63d2\u5165 / \u5220\u9664 \\(O(n)\\) / \\(O(1)\\) / \\(O(n)\\) \\(O(\\log n)\\) / \\(O(n)\\) / \\(O(n)\\) \\(O(1)\\) / \\(O(1)\\) / \\(O(1)\\) \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u67e5\u627e / \u63d2\u5165 / \u5220\u9664 \\(O(n)\\) / \\(O(1)\\) / \\(O(n)\\) \\(O(\\log n)\\) / \\(O(n)\\) / \\(O(n)\\) \\(O(n)\\) / \\(O(n)\\) / \\(O(n)\\) \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\)"},{"location":"chapter_sorting/bubble_sort/","title":"11.2. \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"
    \u300c\u5192\u6ce1\u6392\u5e8f Bubble Sort\u300d\u662f\u4e00\u79cd\u6700\u57fa\u7840\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u975e\u5e38\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u4e2a\u5b66\u4e60\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u987e\u540d\u601d\u4e49\uff0c\u300c\u5192\u6ce1\u300d\u662f\u8be5\u7b97\u6cd5\u7684\u6838\u5fc3\u64cd\u4f5c\u3002
     
    \u4e3a\u4ec0\u4e48\u53eb\u201c\u5192\u6ce1\u201d
     
    \u5728\u6c34\u4e2d\uff0c\u8d8a\u5927\u7684\u6ce1\u6ce1\u6d6e\u529b\u8d8a\u5927\uff0c\u6240\u4ee5\u6700\u5927\u7684\u6ce1\u6ce1\u4f1a\u6700\u5148\u6d6e\u5230\u6c34\u9762\u3002
     
    \u300c\u5192\u6ce1\u300d\u64cd\u4f5c\u5219\u662f\u5728\u6a21\u62df\u4e0a\u8ff0\u8fc7\u7a0b\uff0c\u5177\u4f53\u505a\u6cd5\u4e3a\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u5bf9\u6bd4\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u82e5 \u5de6\u5143\u7d20 > \u53f3\u5143\u7d20 \u5219\u5c06\u5b83\u4fe9\u4ea4\u6362\uff0c\u6700\u7ec8\u53ef\u5c06\u6700\u5927\u5143\u7d20\u79fb\u52a8\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\u3002
     
    \u5b8c\u6210\u6b64\u6b21\u5192\u6ce1\u64cd\u4f5c\u540e\uff0c\u6570\u7ec4\u6700\u5927\u5143\u7d20\u5df2\u5728\u6b63\u786e\u4f4d\u7f6e\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u6392\u5e8f\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u3002
     <1><2><3><4><5><6><7> 
     
     
     
     
     
     
    "},{"location":"chapter_sorting/bubble_sort/#1121","title":"11.2.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5b8c\u6210\u7b2c\u4e00\u8f6e\u300c\u5192\u6ce1\u300d\u540e\uff0c\u6570\u7ec4\u6700\u5927\u5143\u7d20\u5df2\u5728\u6b63\u786e\u4f4d\u7f6e\uff0c\u63a5\u4e0b\u6765\u53ea\u9700\u6392\u5e8f\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u3002
      2.  
    2. \u540c\u7406\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u300c\u5192\u6ce1\u300d\uff0c\u53ef\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c\u56e0\u800c\u5f85\u6392\u5e8f\u5143\u7d20\u53ea\u5269 \\(n - 2\\) \u4e2a\u3002
      3.  
    3. \u4ee5\u6b64\u7c7b\u63a8\u2026\u2026 \u5faa\u73af \\(n - 1\\) \u8f6e\u300c\u5192\u6ce1\u300d\uff0c\u5373\u53ef\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002
      4.  
     
     Fig. \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\n}\n}\n}\n}\n
     bubble_sort.py
    def bubble_sort(nums: List[int]) -> None:\n\"\"\" \u5192\u6ce1\u6392\u5e8f \"\"\"\nn: int = len(nums)\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(n - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\n
     bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\n}\n}\n}\n}\n
     bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.c
    [class]{}-[func]{bubbleSort}\n
     bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
     bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\n}\n}\n}\n}\n
     bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1122","title":"11.2.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u5404\u8f6e\u300c\u5192\u6ce1\u300d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\) , \\(n - 2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488\u53d8\u91cf\u4ec5\u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0d\u4ea4\u6362\u76f8\u7b49\u5143\u7d20\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5f15\u5165 flag \u4f18\u5316\u540e\uff08\u89c1\u4e0b\u6587\uff09\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002
    "},{"location":"chapter_sorting/bubble_sort/#1123","title":"11.2.3. \u00a0 \u6548\u7387\u4f18\u5316","text":"
    \u6211\u4eec\u53d1\u73b0\uff0c\u82e5\u5728\u67d0\u8f6e\u300c\u5192\u6ce1\u300d\u4e2d\u672a\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u5219\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u8003\u8651\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d flag \u6765\u76d1\u542c\u8be5\u60c5\u51b5\uff0c\u82e5\u51fa\u73b0\u5219\u76f4\u63a5\u8fd4\u56de\u3002
     
    \u4f18\u5316\u540e\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u800c\u5728\u8f93\u5165\u6570\u7ec4 \u5df2\u6392\u5e8f \u65f6\uff0c\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.length - 1; i > 0; i--) {\nboolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.size() - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.py
    def bubble_sort_with_flag(nums: List[int]) -> None:\n\"\"\" \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 \"\"\"\nn: int = len(nums)\n# \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in range(n - 1, 0, -1):\nflag: bool = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n# \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\nflag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\nif not flag:\nbreak            # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
     bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := len(nums) - 1; i > 0; i-- {\nflag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
     bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.c
    [class]{}-[func]{bubbleSortWithFlag}\n
     bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor (int i = nums.Length - 1; i > 0; i--)\n{\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nfor (int j = 0; j < i; j++)\n{\nif (nums[j] > nums[j + 1])\n{\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
     bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\nvar flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
     bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5192\u6ce1\u64cd\u4f5c\nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;\n}\n}\nif (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    "},{"location":"chapter_sorting/bucket_sort/","title":"Bucket sort","text":""},{"location":"chapter_sorting/bucket_sort/#_1","title":"\u62d3\u5c55\u5230\u6876\u6392\u5e8f","text":"
    \u5982\u679c\u6211\u4eec\u628a\u4e0a\u8ff0 bucket \u4e2d\u7684\u6bcf\u4e2a\u7d22\u5f15\u60f3\u8c61\u6210\u4e00\u4e2a\u6876\uff0c\u90a3\u4e48\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u7406\u89e3\u4e3a\u628a \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\uff0c\u518d\u6839\u636e\u6876\u4e0e\u6876\u4e4b\u95f4\u5929\u7136\u7684\u6709\u5e8f\u6027\u6765\u5b9e\u73b0\u6392\u5e8f\u3002
     
    \u4ee5\u4e0a\u89e3\u8bfb\u4fbf\u662f\u300c\u6876\u6392\u5e8f Bucket Sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u3002\u5177\u4f53\u5730\uff0c\u6876\u6392\u5e8f\u8003\u8651\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u6839\u636e\u5927\u5c0f\u8303\u56f4\u5747\u5300\u5730\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\uff0c\u7531\u4e8e\u6876\u4e4b\u95f4\u662f\u6709\u5e8f\u7684\uff0c\u56e0\u6b64\u4ec5\u9700\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u6267\u884c\u6392\u5e8f\uff0c\u6700\u7ec8\u6309\u7167\u6876\u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u5c06\u5143\u7d20\u4f9d\u6b21\u6392\u5217\uff0c\u5373\u53ef\u5f97\u5230\u6392\u5e8f\u7ed3\u679c\u3002
     
    \u5047\u8bbe\u4f7f\u7528\u300c\u5feb\u901f\u6392\u5e8f\u300d\u6765\u6392\u5e8f\u5404\u4e2a\u6876\u5185\u7684\u5143\u7d20\uff0c\u6bcf\u4e2a\u6876\u5185\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \uff0c\u5219\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u63a5\u8fd1 \\(n\\) \u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002
     
    \uff08\u56fe\uff09
     
    \u7406\u8bba\u4e0a\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \uff0c\u4f46\u524d\u63d0\u662f\u9700\u8981\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u800c\u8fd9\u662f\u4e0d\u592a\u5bb9\u6613\u505a\u5230\u7684\u3002\u5047\u8bbe\u6211\u4eec\u8981\u628a\u6dd8\u5b9d\u4e2d\u7684 \\(100\\) \u4e07\u4ef6\u5546\u54c1\u6839\u636e\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 \\(100\\) \u4e2a\u6876\u4e2d\uff0c\u7531\u4e8e\u5546\u54c1\u4ef7\u683c\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\uff0c\u6bd4\u5982 \\(1\\) ~ \\(100\\) \u5143\u7684\u5546\u54c1\u975e\u5e38\u591a\u3001\\(1\\) \u4e07\u5143\u4ee5\u4e0a\u7684\u5546\u54c1\u975e\u5e38\u5c11\u7b49\uff0c\u56e0\u6b64\u96be\u4ee5\u7b80\u5355\u5730\u8bbe\u5b9a\u5404\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u89e3\u51b3\u65b9\u6848\u6709\uff1a
     
       
    • \u5148\u521d\u6b65\u8bbe\u7f6e\u4e00\u4e2a\u5206\u754c\u7ebf\uff0c\u5c06\u5143\u7d20\u5206\u914d\u5b8c\u540e\uff0c\u628a\u5143\u7d20\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a\u591a\u4e2a\u6876\uff0c\u76f4\u81f3\u6bcf\u4e2a\u6876\u5185\u5143\u7d20\u6570\u91cf\u5408\u7406\u4e3a\u6b62\uff1b\u8be5\u505a\u6cd5\u4e00\u822c\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\uff1b
      •  
    • \u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u5df2\u77e5\u5206\u5e03\u6765\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\uff1b\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u4e0d\u4e00\u5b9a\u9700\u8981 case-by-case \u5730\u7edf\u8ba1\uff0c\u6709\u65f6\u53ef\u4ee5\u91c7\u7528\u4e00\u4e9b\u5e38\u89c1\u5206\u5e03\u6765\u8fd1\u4f3c\uff0c\u4f8b\u5982\u81ea\u7136\u754c\u7684\u6b63\u6001\u5206\u5e03\uff1b
      •  
     
    \uff08\u56fe\uff09
    "},{"location":"chapter_sorting/counting_sort/","title":"11.6. \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"
    \u524d\u9762\u4ecb\u7ecd\u7684\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e \u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5373\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u4e4b\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5b66\u4e60\u4e00\u79cd \u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5 \uff0c\u540d\u4e3a\u300c\u8ba1\u6570\u6392\u5e8f Counting Sort\u300d\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u3002
    "},{"location":"chapter_sorting/counting_sort/#1161","title":"11.6.1. \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"
    \u5148\u770b\u4e00\u4e2a\u7b80\u5355\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5143\u7d20\u7686\u4e3a \u975e\u8d1f\u6574\u6570\u3002\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u4e3a\uff1a
     
       
    1. \u904d\u5386\u8bb0\u5f55\u6570\u7ec4\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u5e76\u5efa\u7acb\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 counter \uff1b
      2.  
    2. \u501f\u52a9 counter \u7edf\u8ba1 nums \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d counter[num] \u5bf9\u5e94\u6570\u5b57 num \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 nums \uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a num\uff09\uff0c\u6bcf\u8f6e\u5c06 counter[num] \u81ea\u589e \\(1\\) \u5373\u53ef\u3002
      3.  
    3. \u7531\u4e8e counter \u7684\u5404\u4e2a\u7d22\u5f15\u662f\u5929\u7136\u6709\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u88ab\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 counter \uff0c\u6839\u636e\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5c06\u5404\u6570\u5b57\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 nums \u5373\u53ef\u3002
      4.  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u8ba1\u6570\u6392\u5e8f\u540d\u526f\u5176\u5b9e\uff0c\u662f\u901a\u8fc7\u201c\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u201d\u6765\u5b9e\u73b0\u6392\u5e8f\u7684\u3002
     
     
     Fig. counting_sort_overview 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
     counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int>& nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
     counting_sort.py
    def counting_sort_naive(nums: List[int]) -> None:\n\"\"\" \u8ba1\u6570\u6392\u5e8f \"\"\"\n# \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = 0\nfor num in nums:\nm = max(m, num)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\ni = 0\nfor num in range(m + 1):\nfor _ in range(counter[num]):\nnums[i] = num\ni += 1\n
     counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nfor i, num := 0, 0; num < m+1; num++ {\nfor j := 0; j < counter[num]; j++ {\nnums[i] = num\ni++\n}\n}\n}\n
     counting_sort.js
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.ts
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.c
    [class]{}-[func]{countingSortNaive}\n
     counting_sort.cs
    [class]{counting_sort}-[func]{countingSortNaive}\n
     counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nvar i = 0\nfor num in stride(from: 0, to: m + 1, by: 1) {\nfor _ in stride(from: 0, to: counter[num], by: 1) {\nnums[i] = num\ni += 1\n}\n}\n}\n
     counting_sort.zig
    [class]{}-[func]{countingSortNaive}\n
    "},{"location":"chapter_sorting/counting_sort/#1162","title":"11.6.2. \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u53d1\u73b0\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 3. \u5c31\u5931\u6548\u4e86\u3002\u4f8b\u5982\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u8981\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002
     
    \u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 counter \u7684\u300c\u524d\u7f00\u548c\u300d\uff0c\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 i \u5904\u7684\u524d\u7f00\u548c prefix[i] \u7b49\u4e8e\u6570\u7ec4\u524d i \u4e2a\u5143\u7d20\u4e4b\u548c\uff0c\u5373
     \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] 
    \u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u610f\u4e49\uff0cprefix[num] - 1 \u4ee3\u8868\u5143\u7d20 num \u5728\u7ed3\u679c\u6570\u7ec4 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u5f88\u5173\u952e\uff0c\u56e0\u4e3a\u5176\u7ed9\u51fa\u4e86\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 nums \u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\uff1a
     
       
    1. \u5c06 num \u586b\u5165\u6570\u7ec4 res \u7684\u7d22\u5f15 prefix[num] - 1 \u5904\uff1b
      2.  
    2. \u4ee4\u524d\u7f00\u548c prefix[num] \u81ea\u51cf \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\uff1b
      3.  
     
    \u5b8c\u6210\u904d\u5386\u540e\uff0c\u6570\u7ec4 res \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 res \u8986\u76d6\u539f\u6570\u7ec4 nums \u5373\u53ef\uff1b
     <1><2><3><4><5><6><7><8> 
     
     
     
     
     
     
     
     
    \u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
     counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int>& nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.size();\nvector<int> res(n);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums = res;\n}\n
     counting_sort.py
    def counting_sort(nums: List[int]) -> None:\n\"\"\" \u8ba1\u6570\u6392\u5e8f \"\"\"\n# \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = max(nums)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n# \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in range(m):\ncounter[i + 1] += counter[i]\n# 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n# \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn = len(nums)\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nnum = nums[i]\nres[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n# \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\n
     counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i := 0; i < m; i++ {\ncounter[i+1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn := len(nums)\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nnum := nums[i]\n// \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\nres[counter[num]-1] = num\n// \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\ncounter[num]--\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\ncopy(nums, res)\n}\n
     counting_sort.js
    [class]{}-[func]{countingSort}\n
     counting_sort.ts
    [class]{}-[func]{countingSort}\n
     counting_sort.c
    [class]{}-[func]{countingSort}\n
     counting_sort.cs
    [class]{counting_sort}-[func]{countingSort}\n
     counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in stride(from: 0, to: m, by: 1) {\ncounter[i + 1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nvar res = Array(repeating: 0, count: nums.count)\nfor i in stride(from: nums.count - 1, through: 0, by: -1) {\nlet num = nums[i]\nres[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in stride(from: 0, to: nums.count, by: 1) {\nnums[i] = res[i]\n}\n}\n
     counting_sort.zig
    [class]{}-[func]{countingSort}\n
    "},{"location":"chapter_sorting/counting_sort/#1163","title":"11.6.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 nums \u548c\u904d\u5386 counter \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u6b64\u65f6\u4f7f\u7528\u7ebf\u6027 \\(O(n)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6570\u7ec4 res \u548c counter \u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) , \\(m\\) \u3002
     
    \u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u8f85\u52a9\u6570\u7ec4 counter \u548c\u7ed3\u679c\u6570\u7ec4 res \u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u5012\u5e8f\u904d\u5386 nums \u4fdd\u6301\u4e86\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002
     
    \u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u4e0e\u5143\u7d20\u5206\u5e03\u65e0\u5173\u3002
     
    \u4e3a\u4ec0\u4e48\u662f\u7a33\u5b9a\u6392\u5e8f\uff1f
     
    \u7531\u4e8e\u5411 res \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 nums \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u201c\u7a33\u5b9a\u6392\u5e8f\u201d\uff1b\u5176\u5b9e\u6b63\u5e8f\u904d\u5386 nums \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u201c\u975e\u7a33\u5b9a\u201d\u3002
    "},{"location":"chapter_sorting/counting_sort/#1164","title":"11.6.4. \u00a0 \u5c40\u9650\u6027","text":"
    \u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u592a\u5999\u4e86\uff0c\u5494\u5494\u4e00\u901a\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4e0b\u6765\u4e86\u3002\u4f46\u5b9e\u9645\u4e0a\u4e0e\u5176\u5b83\u7b97\u6cd5\u4e00\u6837\uff0c\u8ba1\u6570\u6392\u5e8f\u4e5f\u65e0\u6cd5\u6446\u8131\u201c\u6b64\u6d88\u5f7c\u957f\u201d\u7684\u5bbf\u547d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u7684\u4ee3\u4ef7\u662f\u901a\u7528\u578b\u53d8\u5dee\u3002
     
    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u8981\u7528\u5728\u5176\u4ed6\u7c7b\u578b\u6570\u636e\u4e0a\uff0c\u5219\u8981\u6c42\u8be5\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u88ab\u8f6c\u5316\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u5373\u53ef\u3002
     
    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u6570\u636e\u8303\u56f4\u4e0d\u5927\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u5360\u7528\u7a7a\u95f4\u592a\u591a\uff1b\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u6709\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002
    "},{"location":"chapter_sorting/insertion_sort/","title":"11.3. \u00a0 \u63d2\u5165\u6392\u5e8f","text":"
    \u300c\u63d2\u5165\u6392\u5e8f Insertion Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e \u6570\u7ec4\u63d2\u5165\u64cd\u4f5c \u7684\u6392\u5e8f\u7b97\u6cd5\u3002
     
    \u300c\u63d2\u5165\u64cd\u4f5c\u300d\u539f\u7406\uff1a\u9009\u5b9a\u67d0\u4e2a\u5f85\u6392\u5e8f\u5143\u7d20\u4e3a\u57fa\u51c6\u6570 base\uff0c\u5c06 base \u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u5143\u7d20\u4f9d\u6b21\u5bf9\u6bd4\u5927\u5c0f\uff0c\u5e76\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u3002
     
    \u56de\u5fc6\u6570\u7ec4\u63d2\u5165\u64cd\u4f5c\uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 base \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u5c06 base \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002
     
     
     Fig. \u5355\u6b21\u63d2\u5165\u64cd\u4f5c 
    "},{"location":"chapter_sorting/insertion_sort/#1131","title":"11.3.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u7b2c 1 \u8f6e\u5148\u9009\u53d6\u6570\u7ec4\u7684 \u7b2c 2 \u4e2a\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      2.  
    2. \u7b2c 2 \u8f6e\u9009\u53d6 \u7b2c 3 \u4e2a\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6570\u7ec4\u524d 3 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      3.  
    3. \u4ee5\u6b64\u7c7b\u63a8\u2026\u2026\u6700\u540e\u4e00\u8f6e\u9009\u53d6 \u6570\u7ec4\u5c3e\u5143\u7d20 \u4e3a base \uff0c\u6267\u884c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u540e\uff0c\u6240\u6709\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
      4.  
     
     Fig. \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig insertion_sort.java
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.cpp
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int>& nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.size(); i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.py
    def insertion_sort(nums: List[int]) -> None:\n\"\"\" \u63d2\u5165\u6392\u5e8f \"\"\"\n# \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]   \nfor i in range(1, len(nums)):\nbase: int = nums[i]\nj: int = i - 1\n# \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0 and nums[j] > base:\nnums[j + 1] = nums[j]  # 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\nnums[j + 1] = base         # 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
     insertion_sort.go
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u5f85\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a n-1, n-2, ..., 1\nfor i := 1; i < len(nums); i++ {\nbase := nums[i]\nj := i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nfor j >= 0 && nums[j] > base {\nnums[j+1] = nums[j] // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--\n}\nnums[j+1] = base // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.js
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (let i = 1; i < nums.length; i++) {\nlet base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.ts
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (let i = 1; i < nums.length; i++) {\nconst base = nums[i];\nlet j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.c
    [class]{}-[func]{insertionSort}\n
     insertion_sort.cs
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums)\n{\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor (int i = 1; i < nums.Length; i++)\n{\nint bas = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > bas)\n{\nnums[j + 1] = nums[j]; // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = bas;         // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.swift
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nfor i in stride(from: 1, to: nums.count, by: 1) {\nlet base = nums[i]\nvar j = i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0, nums[j] > base {\nnums[j + 1] = nums[j] // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\n}\nnums[j + 1] = base // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
     insertion_sort.zig
    // \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1abase = nums[1], nums[2], ..., nums[n-1]\nvar i: usize = 1;\nwhile (i < nums.len) : (i += 1) {\nvar base = nums[i];\nvar j: usize = i;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5de6\u8fb9\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 1 and nums[j - 1] > base) : (j -= 1) {\nnums[j] = nums[j - 1];  // 1. \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n}\nnums[j] = base;             // 2. \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    "},{"location":"chapter_sorting/insertion_sort/#1132","title":"11.3.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5404\u8f6e\u63d2\u5165\u64cd\u4f5c\u5faa\u73af \\(n - 1\\) , \\(n-2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488\u53d8\u91cf\u4ec5\u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u989d\u5916\u7a7a\u95f4\u3002
     
    \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0d\u4ea4\u6362\u76f8\u7b49\u5143\u7d20\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u6700\u4f73\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)  \u3002
    "},{"location":"chapter_sorting/insertion_sort/#1133-vs","title":"11.3.3. \u00a0 \u63d2\u5165\u6392\u5e8f vs \u5192\u6ce1\u6392\u5e8f","text":"
    Question
     
    \u867d\u7136\u300c\u63d2\u5165\u6392\u5e8f\u300d\u548c\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u901f\u5ea6\u5374\u6709\u5f88\u5927\u5dee\u522b\uff0c\u8fd9\u662f\u4e3a\u4ec0\u4e48\u5462\uff1f
     
    \u56de\u987e\u590d\u6742\u5ea6\u5206\u6790\uff0c\u4e24\u4e2a\u65b9\u6cd5\u7684\u5faa\u73af\u6b21\u6570\u90fd\u662f \\(\\frac{(n - 1) n}{2}\\) \u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u300c\u5192\u6ce1\u64cd\u4f5c\u300d\u662f\u5728\u505a \u5143\u7d20\u4ea4\u6362\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\u5b9e\u73b0\uff0c\u5171 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u800c\u300c\u63d2\u5165\u64cd\u4f5c\u300d\u662f\u5728\u505a \u8d4b\u503c\uff0c\u53ea\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u53ef\u4ee5\u7c97\u7565\u4f30\u8ba1\u51fa\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u7ea6\u4e3a\u63d2\u5165\u6392\u5e8f\u7684 3 \u500d\u3002
     
    \u63d2\u5165\u6392\u5e8f\u8fd0\u884c\u901f\u5ea6\u5feb\uff0c\u5e76\u4e14\u5177\u6709\u539f\u5730\u3001\u7a33\u5b9a\u3001\u81ea\u9002\u5e94\u7684\u4f18\u70b9\uff0c\u56e0\u6b64\u5f88\u53d7\u6b22\u8fce\u3002\u5b9e\u9645\u4e0a\uff0c\u5305\u62ec Java \u5728\u5185\u7684\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u7684\u5b9e\u73b0\u90fd\u7528\u5230\u4e86\u63d2\u5165\u6392\u5e8f\u3002\u5e93\u51fd\u6570\u7684\u5927\u81f4\u601d\u8def\uff1a
     
       
    • \u5bf9\u4e8e \u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u300c\u5feb\u901f\u6392\u5e8f\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \uff1b
      •  
    • \u5bf9\u4e8e \u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u300c\u63d2\u5165\u6392\u5e8f\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1b
      •  
     
    \u5728\u6570\u7ec4\u8f83\u77ed\u65f6\uff0c\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\uff08\u5373\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\uff09\u5360\u4e3b\u5bfc\u4f5c\u7528\uff0c\u6b64\u65f6\u63d2\u5165\u6392\u5e8f\u8fd0\u884c\u5730\u66f4\u5feb\u3002\u8fd9\u4e2a\u73b0\u8c61\u4e0e\u300c\u7ebf\u6027\u67e5\u627e\u300d\u548c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7684\u60c5\u51b5\u7c7b\u4f3c\u3002
    "},{"location":"chapter_sorting/intro_to_sort/","title":"11.1. \u00a0 \u6392\u5e8f\u7b80\u4ecb","text":"
    \u300c\u6392\u5e8f\u7b97\u6cd5 Sorting Algorithm\u300d\u4f7f\u5f97\u5217\u8868\u4e2d\u7684\u6240\u6709\u5143\u7d20\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002
     
       
    • \u5f85\u6392\u5e8f\u7684\u5217\u8868\u7684 \u5143\u7d20\u7c7b\u578b \u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u3001\u6216\u5b57\u7b26\u4e32\uff1b
      •  
    • \u6392\u5e8f\u7b97\u6cd5\u53ef\u4ee5\u6839\u636e\u9700\u8981\u8bbe\u5b9a \u5224\u65ad\u89c4\u5219\uff0c\u4f8b\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u3001\u81ea\u5b9a\u4e49\u89c4\u5219\uff1b
      •  
     
     
     Fig. \u6392\u5e8f\u4e2d\u4e0d\u540c\u7684\u5143\u7d20\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219 
    "},{"location":"chapter_sorting/intro_to_sort/#1111","title":"11.1.1. \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"
    \u6392\u5e8f\u7b97\u6cd5\u4e3b\u8981\u53ef\u6839\u636e \u7a33\u5b9a\u6027 \u3001\u5c31\u5730\u6027 \u3001\u81ea\u9002\u5e94\u6027 \u3001\u6bd4\u8f83\u7c7b \u6765\u5206\u7c7b\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_1","title":"\u7a33\u5b9a\u6027","text":"
       
    • \u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u4e0d\u6539\u53d8 \u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
    • \u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u4f4d\u7f6e \u53ef\u80fd\u88ab\u6539\u53d8\u3002
      •  
     
    \u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1, 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u90a3\u4e48\u5728\u4ee5\u4e0b\u793a\u4f8b\u4e2d\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u4f1a\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e22\u5931\u3002\u56e0\u6b64\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u662f\u5f88\u597d\u7684\u7279\u6027\uff0c\u5728\u591a\u7ea7\u6392\u5e8f\u4e2d\u662f\u5fc5\u987b\u7684\u3002
     
    # \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n('A', 19)\n('B', 18)\n('C', 21)\n('D', 19)\n('E', 23)\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n('B', 18)\n('D', 19)\n('A', 19)\n('C', 21)\n('E', 23)\n
    "},{"location":"chapter_sorting/intro_to_sort/#_2","title":"\u5c31\u5730\u6027","text":"
       
    • \u300c\u539f\u5730\u6392\u5e8f\u300d\u65e0\u9700\u8f85\u52a9\u6570\u636e\uff0c\u4e0d\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff1b
      •  
    • \u300c\u975e\u539f\u5730\u6392\u5e8f\u300d\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u636e\uff0c\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff1b
      •  
     
    \u300c\u539f\u5730\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff0c\u53ef\u4ee5\u8282\u7ea6\u5185\u5b58\uff1b\u5e76\u4e14\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7531\u4e8e\u6570\u636e\u64cd\u4f5c\u51cf\u5c11\uff0c\u539f\u5730\u6392\u5e8f\u7684\u8fd0\u884c\u6548\u7387\u4e5f\u66f4\u9ad8\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_3","title":"\u81ea\u9002\u5e94\u6027","text":"
       
    • \u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d7\u8f93\u5165\u6570\u636e\u5f71\u54cd\uff0c\u5373\u6700\u4f73 / \u6700\u5dee / \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u76f8\u7b49\u3002
      •  
    • \u300c\u975e\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6052\u5b9a\uff0c\u4e0e\u8f93\u5165\u6570\u636e\u65e0\u5173\u3002
      •  
     
    \u6211\u4eec\u5e0c\u671b \u6700\u5dee = \u5e73\u5747\uff0c\u5373\u4e0d\u5e0c\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u5728\u67d0\u4e9b\u8f93\u5165\u6570\u636e\u4e0b\u53d1\u751f\u52a3\u5316\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#_4","title":"\u6bd4\u8f83\u7c7b","text":"
       
    • \u300c\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u57fa\u4e8e\u5143\u7d20\u4e4b\u95f4\u7684\u6bd4\u8f83\u7b97\u5b50\uff08\u5c0f\u4e8e\u3001\u76f8\u7b49\u3001\u5927\u4e8e\uff09\u6765\u51b3\u5b9a\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
    • \u300c\u975e\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u4e0d\u57fa\u4e8e\u5143\u7d20\u4e4b\u95f4\u7684\u6bd4\u8f83\u7b97\u5b50\u6765\u51b3\u5b9a\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\u3002
      •  
     
    \u300c\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6700\u4f18\u4e3a \\(O(n \\log n)\\) \uff1b\u800c\u300c\u975e\u6bd4\u8f83\u7c7b\u6392\u5e8f\u300d\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u4f46\u901a\u7528\u6027\u8f83\u5dee\u3002
    "},{"location":"chapter_sorting/intro_to_sort/#1112","title":"11.1.2. \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"
       
    • \u8fd0\u884c\u5feb\uff0c\u5373\u65f6\u95f4\u590d\u6742\u5ea6\u4f4e\uff1b
      •  
    • \u7a33\u5b9a\u6392\u5e8f\uff0c\u5373\u6392\u5e8f\u540e\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\u5316\uff1b
      •  
    • \u539f\u5730\u6392\u5e8f\uff0c\u5373\u8fd0\u884c\u4e2d\u4e0d\u4f7f\u7528\u989d\u5916\u7684\u8f85\u52a9\u7a7a\u95f4\uff1b
      •  
    • \u6b63\u5411\u81ea\u9002\u5e94\u6027\uff0c\u5373\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u4e0d\u4f1a\u5728\u67d0\u4e9b\u8f93\u5165\u6570\u636e\u4e0b\u53d1\u751f\u52a3\u5316\uff1b
      •  
     
    \u7136\u800c\uff0c\u6ca1\u6709\u6392\u5e8f\u7b97\u6cd5\u540c\u65f6\u5177\u5907\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u3002\u6392\u5e8f\u7b97\u6cd5\u7684\u9009\u578b\u4f7f\u7528\u53d6\u51b3\u4e8e\u5177\u4f53\u7684\u5217\u8868\u7c7b\u578b\u3001\u5217\u8868\u957f\u5ea6\u3001\u5143\u7d20\u5206\u5e03\u7b49\u56e0\u7d20\u3002
    "},{"location":"chapter_sorting/merge_sort/","title":"11.5. \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"
    \u300c\u5f52\u5e76\u6392\u5e8f Merge Sort\u300d\u662f\u7b97\u6cd5\u4e2d\u201c\u5206\u6cbb\u601d\u60f3\u201d\u7684\u5178\u578b\u4f53\u73b0\uff0c\u5176\u6709\u300c\u5212\u5206\u300d\u548c\u300c\u5408\u5e76\u300d\u4e24\u4e2a\u9636\u6bb5\uff1a
     
       
    1. \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad \u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u4f4d\u7f6e\u5212\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u5316\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\uff1b
      2.  
    2. \u5408\u5e76\u9636\u6bb5\uff1a\u5212\u5206\u5230\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u5f00\u59cb\u5411\u4e0a\u5408\u5e76\uff0c\u4e0d\u65ad\u5c06 \u5de6\u3001\u53f3\u4e24\u4e2a\u77ed\u6392\u5e8f\u6570\u7ec4 \u5408\u5e76\u4e3a \u4e00\u4e2a\u957f\u6392\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u5408\u5e76\u81f3\u539f\u6570\u7ec4\u65f6\u5b8c\u6210\u6392\u5e8f\uff1b
      3.  
     
     
     Fig. \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 
    "},{"location":"chapter_sorting/merge_sort/#1151","title":"11.5.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
    \u300c\u9012\u5f52\u5212\u5206\u300d \u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730 \u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u957f\u5ea6\u4e3a 1 \uff1b
     
       
    1. \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 mid \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [left, mid] \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [mid + 1, right] \uff09\uff1b
      2.  
    2. \u9012\u5f52\u6267\u884c 1. \u6b65\u9aa4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5212\u5206\uff1b
      3.  
     
    \u300c\u56de\u6eaf\u5408\u5e76\u300d \u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a \u6709\u5e8f\u6570\u7ec4 \uff1b
     
    \u9700\u8981\u6ce8\u610f\uff0c\u7531\u4e8e\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u6240\u4ee5 \u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002\u56e0\u6b64\uff0c\u5408\u5e76\u4efb\u52a1\u672c\u8d28\u662f\u8981 \u5c06\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002
     <1><2><3><4><5><6><7><8><9><10> 
     
     
     
     
     
     
     
     
     
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u9012\u5f52\u987a\u5e8f\u5c31\u662f\u4e8c\u53c9\u6811\u7684\u300c\u540e\u5e8f\u904d\u5386\u300d\u3002
     
       
    • \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\u3001\u518d\u9012\u5f52\u53f3\u5b50\u6811\u3001\u6700\u540e\u5904\u7406\u6839\u7ed3\u70b9\u3002
      •  
    • \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\u3001\u518d\u9012\u5f52\u53f3\u5b50\u6811\u3001\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
      •  
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig merge_sort.java
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = Arrays.copyOfRange(nums, left, right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.cpp
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(vector<int>& nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvector<int> tmp(nums.begin() + left, nums.begin() + right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int>& nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.py
    def merge(nums: List[int], left: int, mid: int, right: int) -> None:\n\"\"\" \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 \"\"\"\n# \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n# \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\n# \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy\u6a21\u5757\ntmp: List[int] = nums[left:right + 1].copy()\n# \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleft_start: int = 0\nleft_end: int = mid - left\n# \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nright_start: int = mid + 1 - left\nright_end: int = right - left\n# i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni: int = left_start\nj: int = right_start\n# \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in range(left, right + 1):\n# \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > left_end:\nnums[k] = tmp[j]\nj += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelif j > right_end or tmp[i] <= tmp[j]:\nnums[k] = tmp[i]\ni += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse:\nnums[k] = tmp[j]\nj += 1\ndef merge_sort(nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5f52\u5e76\u6392\u5e8f \"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif left >= right:\nreturn                        # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n# \u5212\u5206\u9636\u6bb5\nmid: int = (left + right) // 2    # \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid)       # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n# \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n
     merge_sort.go
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums []int, left, mid, right int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy \u6a21\u5757\ntmp := make([]int, right-left+1)\nfor i := left; i <= right; i++ {\ntmp[i-left] = nums[i]\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleftStart, leftEnd := left-left, mid-left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nrightStart, rightEnd := mid+1-left, right-left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni, j := leftStart, rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k := left; k <= right; k++ {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj++\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni++\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j]\nj++\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right {\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nmid := (left + right) / 2\nmergeSort(nums, left, mid)\nmergeSort(nums, mid+1, right)\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n}\n
     merge_sort.js
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums, left, mid, right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);   // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nlet leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nlet rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart, j = rightStart;                // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2);    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.ts
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.c
    \n
     merge_sort.cs
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right)\n{\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = nums[left..(right + 1)];\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++)\n{\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right)\n{\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
     merge_sort.swift
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = Array(nums[left ..< (right + 1)])\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left\nlet leftEnd = mid - left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left\nlet rightEnd = right - left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart\nvar j = rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in left ... right {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse {\nnums[k] = tmp[j]\nj += 1\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums: &nums, left: left, mid: mid, right: right)\n}\n
     merge_sort.zig
    // \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\ndefer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar tmp = try mem_allocator.alloc(i32, right + 1 - left);\nstd.mem.copy(i32, tmp, nums[left..right+1]);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nvar leftStart = left - left;\nvar leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nvar rightStart = mid + 1 - left;\nvar rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart;\nvar j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nvar k = left;\nwhile (k <= right) : (k += 1) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j];\nj += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if  (j > rightEnd or tmp[i] <= tmp[j]) {\nnums[k] = tmp[i];\ni += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j];\nj += 1;\n}\n}\n}\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nvar mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\ntry mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\ntry mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\ntry merge(nums, left, mid, right);\n}\n
     
    \u4e0b\u9762\u91cd\u70b9\u89e3\u91ca\u4e00\u4e0b\u5408\u5e76\u65b9\u6cd5 merge() \u7684\u6d41\u7a0b\uff1a
     
       
    1. \u521d\u59cb\u5316\u4e00\u4e2a\u8f85\u52a9\u6570\u7ec4 tmp \u6682\u5b58\u5f85\u5408\u5e76\u533a\u95f4 [left, right] \u5185\u7684\u5143\u7d20\uff0c\u540e\u7eed\u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u7684\u5143\u7d20\u6765\u5b9e\u73b0\u5408\u5e76\uff1b
      2.  
    2. \u521d\u59cb\u5316\u6307\u9488 i , j , k \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u3001\u539f\u6570\u7ec4\u7684\u9996\u5143\u7d20\uff1b
      3.  
    3. \u5faa\u73af\u5224\u65ad tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\uff0c\u5c06\u8f83\u5c0f\u7684\u5148\u8986\u76d6\u81f3 nums[k] \uff0c\u6307\u9488 i , j \u6839\u636e\u5224\u65ad\u7ed3\u679c\u4ea4\u66ff\u524d\u8fdb\uff08\u6307\u9488 k \u4e5f\u524d\u8fdb\uff09\uff0c\u76f4\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u90fd\u904d\u5386\u5b8c\uff0c\u5373\u53ef\u5b8c\u6210\u5408\u5e76\u3002
      4.  
     
    \u5408\u5e76\u65b9\u6cd5 merge() \u4ee3\u7801\u4e2d\u7684\u4e3b\u8981\u96be\u70b9\uff1a
     
       
    • nums \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a [left, right] \uff0c\u800c\u56e0\u4e3a tmp \u53ea\u590d\u5236\u4e86 nums \u8be5\u533a\u95f4\u5143\u7d20\uff0c\u6240\u4ee5 tmp \u5bf9\u5e94\u533a\u95f4\u4e3a [0, right - left] \uff0c\u9700\u8981\u7279\u522b\u6ce8\u610f\u4ee3\u7801\u4e2d\u5404\u4e2a\u53d8\u91cf\u7684\u542b\u4e49\u3002
      •  
    • \u5224\u65ad tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\u7684\u64cd\u4f5c\u4e2d\uff0c\u8fd8 \u9700\u8003\u8651\u5f53\u5b50\u6570\u7ec4\u904d\u5386\u5b8c\u6210\u540e\u7684\u7d22\u5f15\u8d8a\u754c\u95ee\u9898\uff0c\u5373 i > leftEnd \u548c j > rightEnd \u7684\u60c5\u51b5\uff0c\u7d22\u5f15\u8d8a\u754c\u7684\u4f18\u5148\u7ea7\u662f\u6700\u9ad8\u7684\uff0c\u4f8b\u5982\u5982\u679c\u5de6\u5b50\u6570\u7ec4\u5df2\u7ecf\u88ab\u5408\u5e76\u5b8c\u4e86\uff0c\u90a3\u4e48\u4e0d\u7528\u7ee7\u7eed\u5224\u65ad\uff0c\u76f4\u63a5\u5408\u5e76\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u5373\u53ef\u3002
      •  
    "},{"location":"chapter_sorting/merge_sort/#1152","title":"11.5.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
       
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \uff1a\u5212\u5206\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002
      •  
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u9700\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\u5408\u5e76\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff1b\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
      •  
    • \u975e\u539f\u5730\u6392\u5e8f\uff1a\u8f85\u52a9\u6570\u7ec4\u9700\u8981\u4f7f\u7528 \\(O(n)\\) \u989d\u5916\u7a7a\u95f4\u3002
      •  
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u65f6\u53ef\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\u3002
      •  
    • \u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5bf9\u4e8e\u4efb\u610f\u8f93\u5165\u6570\u636e\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u76f8\u540c\u3002
      •  
    "},{"location":"chapter_sorting/merge_sort/#1153","title":"11.5.3. \u00a0 \u94fe\u8868\u6392\u5e8f *","text":"
    \u5f52\u5e76\u6392\u5e8f\u6709\u4e00\u4e2a\u5f88\u7279\u522b\u7684\u4f18\u52bf\uff0c\u7528\u4e8e\u6392\u5e8f\u94fe\u8868\u65f6\u6709\u5f88\u597d\u7684\u6027\u80fd\u8868\u73b0\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u7531\u4e8e\u94fe\u8868\u53ef\u4ec5\u901a\u8fc7\u6539\u53d8\u6307\u9488\u6765\u5b9e\u73b0\u7ed3\u70b9\u589e\u5220\uff0c\u56e0\u6b64\u201c\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\u201d\u65e0\u9700\u4f7f\u7528\u989d\u5916\u7a7a\u95f4\uff0c\u5373\u56de\u6eaf\u5408\u5e76\u9636\u6bb5\u4e0d\u7528\u50cf\u6392\u5e8f\u6570\u7ec4\u4e00\u6837\u5efa\u7acb\u8f85\u52a9\u6570\u7ec4 tmp \uff1b
      •  
    • \u901a\u8fc7\u4f7f\u7528\u300c\u8fed\u4ee3\u300d\u4ee3\u66ff\u300c\u9012\u5f52\u5212\u5206\u300d\uff0c\u53ef\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\uff1b
      •  
     
    \u8be6\u60c5\u53c2\u8003\uff1a148. \u6392\u5e8f\u94fe\u8868
    "},{"location":"chapter_sorting/quick_sort/","title":"11.4. \u00a0 \u5feb\u901f\u6392\u5e8f","text":"
    \u300c\u5feb\u901f\u6392\u5e8f Quick Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u201c\u5206\u6cbb\u601d\u60f3\u201d\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u901f\u5ea6\u5f88\u5feb\u3001\u5e94\u7528\u5f88\u5e7f\u3002
     
    \u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u4e3a\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5176\u76ee\u6807\u4e3a\uff1a\u9009\u53d6\u6570\u7ec4\u67d0\u4e2a\u5143\u7d20\u4e3a \u57fa\u51c6\u6570\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u52a8\u81f3\u5176\u5de6\u8fb9\uff0c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u52a8\u81f3\u5176\u53f3\u8fb9\u3002\u300c\u54e8\u5175\u5212\u5206\u300d\u7684\u5b9e\u73b0\u6d41\u7a0b\u4e3a\uff1a
     
       
    1. \u4ee5\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 i , j \u6307\u5411\u6570\u7ec4\u4e24\u7aef\uff1b
      2.  
    2. \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4e2d\u4f7f\u7528 i / j \u5206\u522b\u5bfb\u627e\u9996\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927 / \u5c0f\u7684\u5143\u7d20\uff0c\u5e76\u4ea4\u6362\u6b64\u4e24\u5143\u7d20\uff1b
      3.  
    3. \u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 2. \uff0c\u76f4\u81f3 i , j \u76f8\u9047\u65f6\u8df3\u51fa\uff0c\u6700\u7ec8\u628a\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\uff1b
      4.  
     
    \u300c\u54e8\u5175\u5212\u5206\u300d\u6267\u884c\u5b8c\u6bd5\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e24\u4e2a\u90e8\u5206\uff0c\u5373 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3 \u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 < \u57fa\u51c6\u6570 < \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u53ea\u9700\u8981\u6392\u5e8f\u4e24\u4e2a\u5b50\u6570\u7ec4\u5373\u53ef\u3002
     <1><2><3><4><5><6><7><8><9> 
     
     
     
     
     
     
     
     
     
    \u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u601d\u60f3
     
    \u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06 \u4e00\u4e2a\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898 \u7b80\u5316\u4e3a \u4e24\u4e2a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.cpp
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int>& nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int>& nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.py
    def partition(self, nums: List[int], left: int, right: int) -> int:\n\"\"\" \u54e8\u5175\u5212\u5206 \"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
     quick_sort.go
    /* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.js
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.ts
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.c
    [class]{quickSort}-[func]{partition}\n
     quick_sort.cs
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j)\n{\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right)\n{\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j)\n{\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.swift
    /* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\nlet tmp = nums[i]\nnums[i] = nums[j]\nnums[j] = tmp\n}\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left\nvar j = right\nwhile i < j {\nwhile i < j, nums[j] >= nums[left] {\nj -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i < j, nums[i] <= nums[left] {\ni += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.zig
    // \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\nvar tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     
    \u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f
     
    \u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002
     
    \u54e8\u5175\u5212\u5206 partition() \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 nums[left] \u548c nums[i] \uff0c\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <= \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d nums[left] >= nums[i] \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 i == j \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd nums[j] == nums[i] > nums[left] \uff1b\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002
     
    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 [0, 0, 0, 0, 1] \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a [1, 0, 0, 0, 0] \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u5bf9\u7684\u3002
     
    \u518d\u6df1\u60f3\u4e00\u6b65\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 nums[right] \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002
    "},{"location":"chapter_sorting/quick_sort/#1141","title":"11.4.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
       
    1. \u9996\u5148\uff0c\u5bf9\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5f97\u5230\u5f85\u6392\u5e8f\u7684 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4\uff1b
      2.  
    2. \u63a5\u4e0b\u6765\uff0c\u5bf9 \u5de6\u5b50\u6570\u7ec4 \u548c \u53f3\u5b50\u6570\u7ec4 \u5206\u522b \u9012\u5f52\u6267\u884c\u300c\u54e8\u5175\u5212\u5206\u300d\u2026\u2026
      3.  
    3. \u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6 \u7ec8\u6b62\u9012\u5f52\uff0c\u5373\u53ef\u5b8c\u6210\u5bf9\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\uff1b
      4.  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5feb\u901f\u6392\u5e8f\u548c\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7684\u539f\u7406\u7c7b\u4f3c\uff0c\u90fd\u662f\u4ee5\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6765\u7f29\u5c0f\u5904\u7406\u533a\u95f4\u3002
     
     
     Fig. \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int>& nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.py
    def quick_sort(self, nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5feb\u901f\u6392\u5e8f \"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right:\nreturn\n# \u54e8\u5175\u5212\u5206\npivot: int = self.partition(nums, left, right)\n# \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nself.quick_sort(nums, left, pivot - 1)\nself.quick_sort(nums, pivot + 1, right)\n
     quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\npivot := q.partition(nums, left, right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nq.quickSort(nums, left, pivot-1)\nq.quickSort(nums, pivot+1, right)\n}\n
     quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
     quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
     quick_sort.c
    [class]{quickSort}-[func]{quickSort}\n
     quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right)\n{\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
     quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums: &nums, left: left, right: pivot - 1)\nquickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
     quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nvar pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1142","title":"11.4.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \uff1a\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002
     
    \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \u5c42\uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \u3002
     
    \u539f\u5730\u6392\u5e8f\uff1a\u53ea\u5728\u9012\u5f52\u4e2d\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
     
    \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u53ef\u80fd\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002
     
    \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002
    "},{"location":"chapter_sorting/quick_sort/#1143","title":"11.4.3. \u00a0 \u5feb\u6392\u4e3a\u4ec0\u4e48\u5feb\uff1f","text":"
    \u4ece\u547d\u540d\u80fd\u591f\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u4e00\u5b9a\u201c\u6709\u4e24\u628a\u5237\u5b50\u201d\u3002\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u867d\u7136\u4e0e\u300c\u5f52\u5e76\u6392\u5e8f\u300d\u548c\u300c\u5806\u6392\u5e8f\u300d\u4e00\u81f4\uff0c\u4f46\u5b9e\u9645 \u6548\u7387\u66f4\u9ad8\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a
     
       
    • \u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4e0d\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u4f46\u7edd\u5927\u90e8\u5206\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u4ee5\u8fbe\u5230 \\(O(n \\log n)\\) \u7684\u590d\u6742\u5ea6\u3002
      •  
    • \u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5165\u7f13\u5b58\u4e2d\uff0c\u8bbf\u95ee\u5143\u7d20\u6548\u7387\u5f88\u9ad8\u3002\u800c\u8bf8\u5982\u300c\u5806\u6392\u5e8f\u300d\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u5177\u6709\u6b64\u7279\u6027\u3002
      •  
    • \u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u4f4e\uff1a\u5728\u63d0\u53ca\u7684\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684 \u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362 \u4e09\u79cd\u64cd\u4f5c\u7684\u603b\u4f53\u6570\u91cf\u6700\u5c11\uff08\u7c7b\u4f3c\u4e8e\u300c\u63d2\u5165\u6392\u5e8f\u300d\u5feb\u4e8e\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u7684\u539f\u56e0\uff09\u3002
      •  
    "},{"location":"chapter_sorting/quick_sort/#1144","title":"11.4.4. \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"
    \u666e\u901a\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53d8\u5dee\u3002\u4e3e\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u53d6\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u4ece\u800c \u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u8fd9\u6837\u8fdb\u4e00\u6b65\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e86\u3002
     
    \u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u4e00\u4e0b\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u9996\u5148\uff0c\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5 \u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u4f46\u5982\u679c\u8fd0\u6c14\u5f88\u5dee\uff0c\u6bcf\u6b21\u90fd\u9009\u62e9\u5230\u6bd4\u8f83\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6548\u7387\u4f9d\u7136\u4e0d\u597d\u3002
     
    \u8fdb\u4e00\u6b65\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6 3 \u4e2a\u5019\u9009\u5143\u7d20\uff08\u4e00\u822c\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u8fd9\u6837\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u5927\u4e5f\u4e0d\u5c0f\u201d\u7684\u6982\u7387\u5c31\u5927\u5927\u63d0\u5347\u4e86\u3002\u5f53\u7136\uff0c\u5982\u679c\u6570\u7ec4\u5f88\u957f\u7684\u8bdd\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u6765\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u53d6\u8be5\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u6781\u4f4e\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.cpp
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int>& nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int>& nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.py
    def median_three(self, nums: List[int], left: int, mid: int, right: int) -> int:\n\"\"\" \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 \"\"\"\n# \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n# \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\nreturn left\nelif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\nreturn mid\nreturn right\ndef partition(self, nums: List[int], left: int, right: int) -> int:\n\"\"\" \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 \"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed: int = self.median_three(nums, left, (left + right) // 2, right)\n# \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
     quick_sort.go
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n}\nreturn right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed := q.medianThree(nums, left, (left+right)/2, right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n//\u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n//\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.js
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right])) return mid;\nelse return right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(nums, left, Math.floor((left + right) / 2), right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.ts
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums: number[], left: number, mid: number, right: number): number {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\nreturn left;\n} else if (Number(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(nums, left, Math.floor((left + right) / 2), right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.c
    [class]{quickSortMedian}-[func]{medianThree}\n[class]{quickSortMedian}-[func]{partition}\n
     quick_sort.cs
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right)\n{\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right)\n{\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j)\n{\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
     quick_sort.swift
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n} else {\nreturn right\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums: &nums, i: left, j: med)\nreturn partition(nums: &nums, left: left, right: right)\n}\n
     quick_sort.zig
    // \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\nreturn left;\n} else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nvar med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1145","title":"11.4.5. \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"
    \u666e\u901a\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u7a7a\u95f4\u6548\u7387\u53d8\u5dee\u3002\u4ecd\u7136\u4ee5\u5b8c\u5168\u5012\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u7531\u4e8e\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 0 \uff0c\u90a3\u4e48\u5c06\u5f62\u6210\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(n - 1\\) \u7684\u9012\u5f52\u6811\uff0c\u6b64\u65f6\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u5927\u5c0f\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
     
    \u4e3a\u4e86\u907f\u514d\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u5224\u65ad\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u5927\u5c0f\uff0c\u4ec5\u9012\u5f52\u6392\u5e8f\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u3002\u7531\u4e8e\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8fd9\u6837\u505a\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u5373\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u88ab\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int>& nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.py
    def quick_sort(self, nums: List[int], left: int, right: int) -> None:\n\"\"\" \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 \"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right:\n# \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot: int = self.partition(nums, left, right)\n# \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot - left < right - pivot:\nself.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1     # \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\nelse:\nself.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1    # \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
     quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nfor left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot := q.partition(nums, left, right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot-left < right-pivot {\nq.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1                 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nq.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1                 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.c
    [class]{quickSortTailCall}-[func]{quickSort}\n
     quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right)\n{\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right)\n{\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot)\n{\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n}\nelse\n{\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\nvar left = left\nvar right = right\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left) < (right - pivot) {\nquickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1 // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
     quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\nvar left = left_;\nvar right = right_;\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nvar pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                   // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                  // \u5269\u4f59\u5f85\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    "},{"location":"chapter_sorting/summary/","title":"11.7. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u589e\u52a0\u6807\u5fd7\u4f4d\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(N)\\) \u3002
      •  
    • \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u5f85\u6392\u5e8f\u533a\u95f4\u5185\u5143\u7d20\u63d2\u5165\u81f3\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u6392\u5e8f\u3002\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u867d\u4e3a \\(O(N^2)\\) \uff0c\u4f46\u56e0\u4e3a\u603b\u4f53\u64cd\u4f5c\u5c11\u800c\u5f88\u53d7\u6b22\u8fce\uff0c\u4e00\u822c\u7528\u4e8e\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u5de5\u4f5c\u3002
      •  
    • \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u4ece\u800c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(N^2)\\) \uff0c\u901a\u8fc7\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u5927\u5927\u964d\u4f4e\u52a3\u5316\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u51cf\u5c0f\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log N)\\) \u3002
      •  
    • \u5f52\u5e76\u6392\u5e8f\u5305\u542b\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u662f\u5206\u800c\u6cbb\u4e4b\u7684\u6807\u51c6\u4f53\u73b0\u3002\u5bf9\u4e8e\u5f52\u5e76\u6392\u5e8f\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \uff1b\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
      •  
    • \u4e0b\u56fe\u603b\u7ed3\u5bf9\u6bd4\u4e86\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u4e0e\u7279\u6027\u3002\u5176\u4e2d\uff0c\u6876\u6392\u5e8f\u4e2d \\(k\\) \u4e3a\u6876\u7684\u6570\u91cf\uff1b\u57fa\u6570\u6392\u5e8f\u4ec5\u9002\u7528\u4e8e\u6b63\u6574\u6570\u3001\u5b57\u7b26\u4e32\u3001\u7279\u5b9a\u683c\u5f0f\u7684\u6d6e\u70b9\u6570\uff0c\\(k\\) \u4e3a\u6700\u5927\u6570\u5b57\u7684\u4f4d\u6570\u3002
      •  
     
     Fig. \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 
     
       
    • \u603b\u4f53\u6765\u770b\uff0c\u6211\u4eec\u8ffd\u6c42\u8fd0\u884c\u5feb\u3001\u7a33\u5b9a\u3001\u539f\u5730\u3001\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7684\u6392\u5e8f\u3002\u663e\u7136\uff0c\u5982\u540c\u5176\u5b83\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e00\u6837\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e9b\u6761\u4ef6\u7684\u6392\u5e8f\u7b97\u6cd5\u5e76\u4e0d\u5b58\u5728\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u95ee\u9898\u7279\u70b9\u6765\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u3002
      •  
    "},{"location":"chapter_stack_and_queue/deque/","title":"5.3. \u00a0 \u53cc\u5411\u961f\u5217","text":"
    \u5bf9\u4e8e\u961f\u5217\uff0c\u6211\u4eec\u53ea\u80fd\u5728\u5934\u90e8\u5220\u9664\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u300c\u53cc\u5411\u961f\u5217 Deque\u300d\u66f4\u52a0\u7075\u6d3b\uff0c\u5728\u5176\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u80fd\u6267\u884c\u5143\u7d20\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002
     
     
     Fig. \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c 
    "},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1. \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"
    \u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u8bed\u8a00\u6765\u786e\u5b9a\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) 
    \u76f8\u540c\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u53cc\u5411\u961f\u5217\u7c7b\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig deque.java
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
     deque.cpp
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
     deque.py
    \"\"\" \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 \"\"\"\ndeque: Deque[int] = collections.deque()\n\"\"\" \u5143\u7d20\u5165\u961f \"\"\"\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\"\"\" \u8bbf\u95ee\u5143\u7d20 \"\"\"\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n\"\"\" \u5143\u7d20\u51fa\u961f \"\"\"\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(deque)\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(deque) == 0\n
     deque_test.go
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
     deque.js
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
     deque.ts
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
     deque.c
    \n
     deque.cs
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new LinkedList<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
     deque.swift
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
     deque.zig
    \n
    "},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2. \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"
    \u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53cc\u5411\u961f\u5217\u540c\u6837\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u6216\u6570\u7ec4\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/deque/#_1","title":"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u56de\u5fc6\u4e0a\u8282\u5185\u5bb9\uff0c\u7531\u4e8e\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u94fe\u8868\u5934\u7ed3\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\uff0c\u4ee5\u53ca\u5728\u94fe\u8868\u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0\u65b0\u7ed3\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\u3002
     
    \u800c\u53cc\u5411\u961f\u5217\u7684\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u4e0e\u51fa\u961f\u64cd\u4f5c\uff0c\u6362\u8a00\u4e4b\uff0c\u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c\u662f\u201c\u9996\u5c3e\u5bf9\u79f0\u201d\u7684\uff0c\u4e5f\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u4f7f\u7528\u300c\u53cc\u5411\u94fe\u8868\u300d\u6765\u5b9e\u73b0\u3002
     
    \u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u7ed3\u70b9\u548c\u5c3e\u7ed3\u70b9\u5206\u522b\u770b\u4f5c\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u5e76\u4e14\u5b9e\u73b0\u5728\u4e24\u7aef\u90fd\u80fd\u6dfb\u52a0\u4e0e\u5220\u9664\u7ed3\u70b9\u3002
     LinkedListDequepushLast()pushFirst()popLast()popFirst() 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_deque.java
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nint val;       // \u7ed3\u70b9\u503c\nListNode next; // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode prev; // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(int val) {\nthis.val = val;\nprev = next = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode front, rear; // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear\nprivate int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, boolean isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate Integer pop(boolean isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty())\nreturn null;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear.val;  // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic Integer popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic Integer popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic Integer peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic Integer peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_deque.cpp
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nstruct DoublyListNode {\nint val;               // \u7ed3\u70b9\u503c\nDoublyListNode *next;  // \u540e\u7ee7\u7ed3\u70b9\u6307\u9488\nDoublyListNode *prev;  // \u524d\u9a71\u7ed3\u70b9\u6307\u9488\nDoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {}\n};\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate:\nDoublyListNode *front, *rear; // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear\nint queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nLinkedListDeque() : front(nullptr), rear(nullptr) {}\n/* \u6790\u6784\u65b9\u6cd5 */\n~LinkedListDeque() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nDoublyListNode *pre, *cur = front;\nwhile (cur != nullptr) {\npre = cur;\ncur = cur->next;\ndelete pre;\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nDoublyListNode *node = new DoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront->prev = node;\nnode->next = front;\nfront = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear->next = node;\nnode->prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de -1\nif (isEmpty())\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front->val; // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nDoublyListNode *fNext = front->next;\nif (fNext != nullptr) {\nfNext->prev = nullptr;\nfront->next = nullptr;\ndelete front;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear->val; // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nDoublyListNode *rPrev = rear->prev;\nif (rPrev != nullptr) {\nrPrev->next = nullptr;\nrear->prev = nullptr;\ndelete rear;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nreturn isEmpty() ? -1 : front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nreturn isEmpty() ? -1 : rear->val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\nDoublyListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_deque.py
    class ListNode:\n\"\"\" \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 \"\"\"\ndef __init__(self, val: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.val: int = val\nself.next: Optional[ListNode] = None  # \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nself.prev: Optional[ListNode] = None  # \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nclass LinkedListDeque:\n\"\"\" \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 \"\"\"\ndef __init__(self) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.front: Optional[ListNode] = None # \u5934\u7ed3\u70b9 front\nself.rear: Optional[ListNode] = None  # \u5c3e\u7ed3\u70b9 rear\nself.__size: int = 0        # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.size() == 0\ndef push(self, num: int, is_front: bool) -> None:\n\"\"\" \u5165\u961f\u64cd\u4f5c \"\"\"\nnode = ListNode(num)\n# \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif self.is_empty():\nself.front = self.rear = node\n# \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelif is_front:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.prev = node\nnode.next = self.front\nself.front = node  # \u66f4\u65b0\u5934\u7ed3\u70b9\n# \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.next = node\nnode.prev = self.rear\nself.rear = node  # \u66f4\u65b0\u5c3e\u7ed3\u70b9\nself.__size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\ndef push_first(self, num: int) -> None:\n\"\"\" \u961f\u9996\u5165\u961f \"\"\"\nself.push(num, True)\ndef push_last(self, num: int) -> None:\n\"\"\" \u961f\u5c3e\u5165\u961f \"\"\"\nself.push(num, False)\ndef pop(self, is_front: bool) -> int:\n\"\"\" \u51fa\u961f\u64cd\u4f5c \"\"\"\n# \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\nif self.is_empty():\nreturn None\n# \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front:\nval: int = self.front.val  # \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n# \u5220\u9664\u5934\u7ed3\u70b9\nfnext: Optional[ListNode] = self.front.next\nif fnext != None:\nfnext.prev = None\nself.front.next = None\nself.front = fnext  # \u66f4\u65b0\u5934\u7ed3\u70b9\n# \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse:\nval: int = self.rear.val  # \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n# \u5220\u9664\u5c3e\u7ed3\u70b9\nrprev: Optional[ListNode] = self.rear.prev\nif rprev != None:\nrprev.next = None\nself.rear.prev = None\nself.rear = rprev  # \u66f4\u65b0\u5c3e\u7ed3\u70b9\nself.__size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\ndef pop_first(self) -> int:\n\"\"\" \u961f\u9996\u51fa\u961f \"\"\"\nreturn self.pop(True)\ndef pop_last(self) -> int:\n\"\"\" \u961f\u5c3e\u51fa\u961f \"\"\"\nreturn self.pop(False)\ndef peek_first(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nreturn None if self.is_empty() else self.front.val\ndef peek_last(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \"\"\"\nreturn None if self.is_empty() else self.rear.val\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 \"\"\"\nnode: Optional[ListNode] = self.front\nres: List[int] = [0] * self.size()\nfor i in range(self.size()):\nres[i] = node.val\nnode = node.next\nreturn res\n
     linkedlist_deque.go
    /* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\nreturn &linkedListDeque{\ndata: list.New(),\n}\n}\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\ns.data.PushFront(value)\n}\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\ns.data.PushBack(value)\n}\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_deque.js
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nprev;   // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nnext;   // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nval;    // \u7ed3\u70b9\u503c\nconstructor(val) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n#front;  // \u5934\u7ed3\u70b9 front\n#rear;   // \u5c3e\u7ed3\u70b9 rear\n#queSize;    // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\nthis.#queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.#rear.next = node;\nnode.prev = this.#rear;\nthis.#rear = node; // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.#front.prev = node;\nnode.next = this.#front;\nthis.#front = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#rear.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet temp = this.#rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.#rear.prev = null;\n}\nthis.#rear = temp;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#front.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet temp = this.#front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.#front.next = null;\n}\nthis.#front = temp;   // \u66f4\u65b0\u5934\u7ed3\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nreturn this.#queSize === 0 ? null : this.#rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nreturn this.#queSize === 0 ? null : this.#front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint() {\nconst arr = [];\nlet temp = this.#front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log(\"[\" + arr.join(\", \") + \"]\");\n}\n}\n
     linkedlist_deque.ts
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nprev: ListNode;     // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nnext: ListNode;     // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528 (\u6307\u9488)\nval: number;        // \u7ed3\u70b9\u503c\nconstructor(val: number) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate front: ListNode;    // \u5934\u7ed3\u70b9 front\nprivate rear: ListNode;     // \u5c3e\u7ed3\u70b9 rear\nprivate queSize: number;        // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.front = null;\nthis.rear = null;\nthis.queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.rear.next = node;\nnode.prev = this.rear;\nthis.rear = node; // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.front.prev = node;\nnode.next = this.front;\nthis.front = node; // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet temp: ListNode = this.rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.rear.prev = null;\n}\nthis.rear = temp;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.front.val; // \u5b58\u50a8\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet temp: ListNode = this.front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.front.next = null;\n}\nthis.front = temp;   // \u66f4\u65b0\u5934\u7ed3\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nreturn this.queSize === 0 ? null : this.rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nreturn this.queSize === 0 ? null : this.front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint(): void {\nconst arr: number[] = [];\nlet temp: ListNode = this.front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log(\"[\" + arr.join(\", \") + \"]\");\n}\n}\n
     linkedlist_deque.c
    [class]{ListNode}-[func]{}\n[class]{LinkedListDeque}-[func]{}\n
     linkedlist_deque.cs
    [class]{ListNode}-[func]{}\n[class]{LinkedListDeque}-[func]{}\n
     linkedlist_deque.swift
    /* \u53cc\u5411\u94fe\u8868\u7ed3\u70b9 */\nclass ListNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar next: ListNode? // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nvar prev: ListNode? // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\ninit(val: Int) {\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate var front: ListNode? // \u5934\u7ed3\u70b9 front\nprivate var rear: ListNode? // \u5c3e\u7ed3\u70b9 rear\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ninit() {\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate func push(num: Int, isFront: Bool) {\nlet node = ListNode(val: num)\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif isEmpty() {\nfront = node\nrear = node\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if isFront {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront?.prev = node\nnode.next = front\nfront = node // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear?.next = node\nnode.prev = rear\nrear = node // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\npush(num: num, isFront: true)\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\npush(num: num, isFront: false)\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate func pop(isFront: Bool) -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nlet val: Int\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif isFront {\nval = front!.val // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nlet fNext = front?.next\nif fNext != nil {\nfNext?.prev = nil\nfront?.next = nil\n}\nfront = fNext // \u66f4\u65b0\u5934\u7ed3\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear!.val // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nlet rPrev = rear?.prev\nif rPrev != nil {\nrPrev?.next = nil\nrear?.prev = nil\n}\nrear = rPrev // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nqueSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\npop(isFront: true)\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\npop(isFront: false)\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int? {\nisEmpty() ? nil : front?.val\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int? {\nisEmpty() ? nil : rear?.val\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_deque.zig
    // \u53cc\u5411\u94fe\u8868\u7ed3\u70b9\nfn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = undefined,     // \u7ed3\u70b9\u503c\nnext: ?*Self = null,    // \u540e\u7ee7\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nprev: ?*Self = null,    // \u524d\u9a71\u7ed3\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\n// Initialize a list node with specific value\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*ListNode(T) = null,                    // \u5934\u7ed3\u70b9 front\nrear: ?*ListNode(T) = null,                     // \u5c3e\u7ed3\u70b9 rear\nque_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u5165\u961f\u64cd\u4f5c\npub fn push(self: *Self, num: T, is_front: bool) !void {\nvar node = try self.mem_allocator.create(ListNode(T));\nnode.init(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (self.isEmpty()) {\nself.front = node;\nself.rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n} else if (is_front) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.?.prev = node;\nnode.next = self.front;\nself.front = node;  // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.?.next = node;\nnode.prev = self.rear;\nself.rear = node;   // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nself.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n} // \u961f\u9996\u5165\u961f\npub fn pushFirst(self: *Self, num: T) !void {\ntry self.push(num, true);\n} // \u961f\u5c3e\u5165\u961f\npub fn pushLast(self: *Self, num: T) !void {\ntry self.push(num, false);\n} // \u51fa\u961f\u64cd\u4f5c\npub fn pop(self: *Self, is_front: bool) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nvar val: T = undefined;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (is_front) {\nval = self.front.?.val;     // \u6682\u5b58\u5934\u7ed3\u70b9\u503c\n// \u5220\u9664\u5934\u7ed3\u70b9\nvar fNext = self.front.?.next;\nif (fNext != null) {\nfNext.?.prev = null;\nself.front.?.next = null;\n}\nself.front = fNext;         // \u66f4\u65b0\u5934\u7ed3\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = self.rear.?.val;      // \u6682\u5b58\u5c3e\u7ed3\u70b9\u503c\n// \u5220\u9664\u5c3e\u7ed3\u70b9\nvar rPrev = self.rear.?.prev;\nif (rPrev != null) {\nrPrev.?.next = null;\nself.rear.?.prev = null;\n}\nself.rear = rPrev;          // \u66f4\u65b0\u5c3e\u7ed3\u70b9\n}\nself.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n} // \u961f\u9996\u51fa\u961f\npub fn popFirst(self: *Self) T {\nreturn self.pop(true);\n} // \u961f\u5c3e\u51fa\u961f\npub fn popLast(self: *Self) T {\nreturn self.pop(false);\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peekFirst(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\npub fn peekLast(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.rear.?.val;\n}\n// \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002\u5728\u5b9e\u73b0\u961f\u5217\u7684\u57fa\u7840\u4e0a\uff0c\u589e\u52a0\u5b9e\u73b0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u65b9\u6cd5\u5373\u53ef\u3002
     ArrayDequepushLast()pushFirst()popLast()popFirst() 
     
     
     
     
     
    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_deque.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new EmptyStackException();\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate:\nvector<int> nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;         // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;       // \u53cc\u5411\u961f\u5217\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nnums.resize(capacity);\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn nums.size();\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> res(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n};\n
     array_deque.py
    class ArrayDeque:\n\"\"\" \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 \"\"\"\ndef __init__(self, capacity: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__nums: List[int] = [0] * capacity\nself.__front: int = 0\nself.__size: int = 0\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf \"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__size == 0\ndef index(self, i: int) -> int:\n\"\"\" \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 \"\"\"\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + self.capacity()) % self.capacity()\ndef push_first(self, num: int) -> None:\n\"\"\" \u961f\u9996\u5165\u961f \"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself.__front = self.index(self.__front - 1)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself.__nums[self.__front] = num\nself.__size += 1\ndef push_last(self, num: int) -> None:\n\"\"\" \u961f\u5c3e\u5165\u961f \"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear = self.index(self.__front + self.__size)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop_first(self) -> int:\n\"\"\" \u961f\u9996\u51fa\u961f \"\"\"\nnum = self.peek_first()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself.__front = self.index(self.__front + 1)\nself.__size -= 1\nreturn num\ndef pop_last(self) -> int:\n\"\"\" \u961f\u5c3e\u51fa\u961f \"\"\"\nnum = self.peek_last()\nself.__size -= 1\nreturn num\ndef peek_first(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\"\nreturn self.__nums[self.__front]\ndef peek_last(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\"\n# \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast = self.index(self.__front + self.__size - 1)\nreturn self.__nums[last]\ndef to_array(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 \"\"\"\n# \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres = []\nfor i in range(self.__size):\nres.append(self.__nums[self.index(self.__front + i)])\nreturn res\n
     array_deque.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\nreturn &arrayDeque{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + q.queCapacity) % q.queCapacity\n}\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nq.front = q.index(q.front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[q.front] = num\nq.queSize++\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear := q.index(q.front + q.queSize)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[rear] = num\nq.queSize++\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\nnum := q.peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nq.front = q.index(q.front + 1)\nq.queSize--\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\nnum := q.peekLast()\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\nif q.isEmpty() {\nreturn nil\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast := q.index(q.front + q.queSize - 1)\nreturn q.nums[last]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres := make([]int, q.queSize)\nfor i, j := 0, q.front; i < q.queSize; i++ {\nres[i] = q.nums[q.index(j)]\nj++\n}\nreturn res\n}\n
     array_deque.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n#nums;       // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front;      // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize;    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\nthis.#front = 0;\nthis.#queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.#front = this.index(this.#front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.#nums[this.#front] = num;\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear = this.index(this.#front + this.#queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst() {\nconst num = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.#front = this.index(this.#front + 1);\nthis.#queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast() {\nconst num = this.peekLast();\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\nreturn this.#nums[this.#front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.#front + this.#queSize - 1);\nreturn this.#nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res = [];\nfor (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\nres[i] = this.#nums[this.index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate nums: number[];     // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number;      // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number;    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = 0;\nthis.queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i: number): number {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.front = this.index(this.front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.nums[this.front] = num;\nthis.queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear: number = this.index(this.front + this.queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst(): number {\nconst num: number = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.front = this.index(this.front + 1);\nthis.queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast(): number {\nconst num: number = this.peekLast();\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\nreturn this.nums[this.front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nif (this.isEmpty())\nthrow new Error(\"The Deque Is Empty.\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.front + this.queSize - 1);\nreturn this.nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res: number[] = [];\nfor (let i = 0, j = this.front; i < this.queSize; i++, j++) {\nres[i] = this.nums[this.index(j)];\n}\nreturn res;\n}\n}\n
     array_deque.c
    [class]{ArrayDeque}-[func]{}\n
     array_deque.cs
    [class]{ArrayDeque}-[func]{}\n
     array_deque.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(capacity: Int) {\nnums = Array(repeating: 0, count: capacity)\nfront = 0\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate func index(i: Int) -> Int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n(i + capacity()) % capacity()\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(i: front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num\nqueSize += 1\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = index(i: front + size())\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\nlet num = peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(i: front + 1)\nqueSize -= 1\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\nlet num = peekLast()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = index(i: front + size() - 1)\nreturn nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: size())\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[index(i: j)]\n}\nreturn res\n}\n}\n
     array_deque.zig
    [class]{ArrayDeque}-[func]{}\n
    "},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3. \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"
    \u53cc\u5411\u961f\u5217\u540c\u65f6\u8868\u73b0\u51fa\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002
     
    \u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u9700\u8981\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1b\u7cfb\u7edf\u628a\u6bcf\u4e00\u6b21\u66f4\u6539\u64cd\u4f5c push \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 pop \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u6709\u9650\uff0c\u8f6f\u4ef6\u4e00\u822c\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\uff0c\u90a3\u4e48\u5f53\u6808\u7684\u957f\u5ea6 \\(> 50\\) \u65f6\uff0c\u8f6f\u4ef6\u5c31\u9700\u8981\u5728\u6808\u5e95\uff08\u5373\u961f\u9996\uff09\u6267\u884c\u5220\u9664\uff0c\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u662f\u6808\u7684\u5148\u5165\u540e\u51fa\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/","title":"5.2. \u00a0 \u961f\u5217","text":"
    \u300c\u961f\u5217 Queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u300c\u5148\u5165\u5148\u51fa first in, first out\u300d\u6570\u636e\u64cd\u4f5c\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u7684\u662f\u6392\u961f\u73b0\u8c61\uff0c\u5373\u5916\u9762\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u5904\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u4e0d\u65ad\u5730\u79bb\u5f00\u3002
     
    \u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u300c\u961f\u9996\u300d\uff0c\u961f\u5217\u5c3e\u90e8\u79f0\u4e3a\u300c\u961f\u5c3e\u300d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u961f\u300d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u961f\u300d\u3002
     
     
     Fig. \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 
    "},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1. \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"
    \u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\u3002\u9700\u8981\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u662f\u4e0d\u540c\u7684\uff0c\u5728\u8fd9\u91cc\u6211\u4eec\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002
     \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u961f\u5217\u7c7b\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig queue.java
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
     queue.cpp
    /* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
     queue.py
    \"\"\" \u521d\u59cb\u5316\u961f\u5217 \"\"\"\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u770b\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u5efa\u8bae\nque: Deque[int] = collections.deque()\n\"\"\" \u5143\u7d20\u5165\u961f \"\"\"\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nfront: int = que[0];\n\"\"\" \u5143\u7d20\u51fa\u961f \"\"\"\npop: int = que.popleft()\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(que)\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(que) == 0\n
     queue_test.go
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
     queue.js
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
     queue.ts
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
     queue.c
    \n
     queue.cs
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count() == 0;\n
     queue.swift
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
     queue.zig
    \n
    "},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2. \u00a0 \u961f\u5217\u5b9e\u73b0","text":"
    \u961f\u5217\u9700\u8981\u4e00\u79cd\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u7684\u6570\u636e\u7ed3\u6784\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u6216\u6570\u7ec4\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u6211\u4eec\u5c06\u94fe\u8868\u7684\u300c\u5934\u7ed3\u70b9\u300d\u548c\u300c\u5c3e\u7ed3\u70b9\u300d\u5206\u522b\u770b\u4f5c\u662f\u961f\u9996\u548c\u961f\u5c3e\uff0c\u5e76\u89c4\u5b9a\u961f\u5c3e\u53ea\u53ef\u6dfb\u52a0\u7ed3\u70b9\uff0c\u961f\u9996\u53ea\u53ef\u5220\u9664\u7ed3\u70b9\u3002
     LinkedListQueuepush()pop() 
     
     
     
    \u4ee5\u4e0b\u662f\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_queue.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode front, rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new EmptyStackException();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate:\nListNode *front, *rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nint queSize;\npublic:\nLinkedListQueue() {\nfront = nullptr;\nrear = nullptr;\nqueSize = 0;\n}\n~LinkedListQueue() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(front);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode* node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == nullptr) {\nfront = node;\nrear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse {\nrear->next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nListNode *tmp = front;\nfront = front->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp; queSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (size() == 0)\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn front->val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode* node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_queue.py
    class LinkedListQueue:\n\"\"\" \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__front: Optional[ListNode] = None  # \u5934\u7ed3\u70b9 front\nself.__rear: Optional[ListNode] = None   # \u5c3e\u7ed3\u70b9 rear\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn not self.__front\ndef push(self, num: int) -> None:\n\"\"\" \u5165\u961f \"\"\"\n# \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nnode = ListNode(num)\n# \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif self.__front is None:\nself.__front = node\nself.__rear = node\n# \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse:\nself.__rear.next = node\nself.__rear = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u961f \"\"\"\nnum = self.peek()\n# \u5220\u9664\u5934\u7ed3\u70b9\nself.__front = self.__front.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nif self.size() == 0:\nprint(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn False\nreturn self.__front.val\ndef to_list(self) -> List[int]:\n\"\"\" \u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nqueue = []\ntemp = self.__front\nwhile temp:\nqueue.append(temp.val)\ntemp = temp.next\nreturn queue\n
     linkedlist_queue.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\nreturn &linkedListQueue{\ndata: list.New(),\n}\n}\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\ns.data.PushBack(value)\n}\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_queue.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n#front;  // \u5934\u7ed3\u70b9 #front\n#rear;   // \u5c3e\u7ed3\u70b9 #rear\n#queSize = 0;\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (!this.#front) {\nthis.#front = node;\nthis.#rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nthis.#rear.next = node;\nthis.#rear = node;\n}\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nthis.#front = this.#front.next;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.size === 0)\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.#front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#front;\nconst res = new Array(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate front: ListNode | null; // \u5934\u7ed3\u70b9 front\nprivate rear: ListNode | null; // \u5c3e\u7ed3\u70b9 rear\nprivate queSize: number = 0;\nconstructor() {\nthis.front = null;\nthis.rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (!this.front) {\nthis.front = node;\nthis.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nthis.rear!.next = node;\nthis.rear = node;\n}\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\nif (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n// \u5220\u9664\u5934\u7ed3\u70b9\nthis.front = this.front.next;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.front;\nconst res = new Array<number>(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.c
    [class]{linkedListQueue}-[func]{}\n
     linkedlist_queue.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue\n{\nprivate ListNode? front, rear;  // \u5934\u7ed3\u70b9 front \uff0c\u5c3e\u7ed3\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue()\n{\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num)\n{\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (front == null)\n{\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n}\nelse if (rear != null)\n{\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop()\n{\nint num = peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front?.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek()\n{\nif (size() == 0 || front == null)\nthrow new Exception();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nif (front == null)\nreturn Array.Empty<int>();\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++)\n{\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_queue.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate var front: ListNode? // \u5934\u7ed3\u70b9\nprivate var rear: ListNode? // \u5c3e\u7ed3\u70b9\nprivate var _size = 0\ninit() {}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nlet node = ListNode(x: num)\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif front == nil {\nfront = node\nrear = node\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\nelse {\nrear?.next = node\nrear = node\n}\n_size += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u5220\u9664\u5934\u7ed3\u70b9\nfront = front?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn front!.val\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_queue.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*inc.ListNode(T) = null,                // \u5934\u7ed3\u70b9 front\nrear: ?*inc.ListNode(T) = null,                 // \u5c3e\u7ed3\u70b9 rear\nque_size: usize = 0,                             // \u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u5165\u961f\npub fn push(self: *Self, num: T) !void {\n// \u5c3e\u7ed3\u70b9\u540e\u6dfb\u52a0 num\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u7ed3\u70b9\u90fd\u6307\u5411\u8be5\u7ed3\u70b9\nif (self.front == null) {\nself.front = node;\nself.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u7ed3\u70b9\u6dfb\u52a0\u5230\u5c3e\u7ed3\u70b9\u540e\n} else {\nself.rear.?.next = node;\nself.rear = node;\n}\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u5220\u9664\u5934\u7ed3\u70b9\nself.front = self.front.?.next;\nself.que_size -= 1;\nreturn num;\n} // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/queue/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u6570\u7ec4\u7684\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u4f4e\u4e0b\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u4e0b\u8ff0\u7684\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002
     
    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u53d8\u91cf front \u6765\u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u53d8\u91cf queSize \u6765\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u6211\u4eec\u5b9a\u4e49 rear = front + queSize \uff0c\u8be5\u516c\u5f0f\u8ba1\u7b97\u51fa\u6765\u7684 rear \u6307\u5411\u201c\u961f\u5c3e\u5143\u7d20\u7d22\u5f15 \\(+1\\) \u201d\u7684\u4f4d\u7f6e\u3002
     
    \u5728\u8be5\u8bbe\u8ba1\u4e0b\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a [front, rear - 1] \uff0c\u8fdb\u800c
     
       
    • \u5bf9\u4e8e\u5165\u961f\u64cd\u4f5c\uff0c\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 rear \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 queSize \u81ea\u589e \\(1\\) \u5373\u53ef\uff1b
      •  
    • \u5bf9\u4e8e\u51fa\u961f\u64cd\u4f5c\uff0c\u4ec5\u9700\u5c06 front \u81ea\u589e \\(1\\) \uff0c\u5e76\u5c06 queSize \u81ea\u51cf \\(1\\) \u5373\u53ef\uff1b
      •  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5165\u961f\u4e0e\u51fa\u961f\u64cd\u4f5c\u90fd\u4ec5\u9700\u5355\u6b21\u64cd\u4f5c\u5373\u53ef\u5b8c\u6210\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(1)\\) \u3002
     ArrayQueuepush()pop() 
     
     
     
    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u5165\u961f\u4e0e\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0cfront \u548c rear \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5728\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u540e\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u8003\u8651\u5c06\u6570\u7ec4\u770b\u4f5c\u662f\u9996\u5c3e\u76f8\u63a5\u7684\uff0c\u8fd9\u6837\u7684\u6570\u7ec4\u88ab\u79f0\u4e3a\u300c\u73af\u5f62\u6570\u7ec4\u300d\u3002
     
    \u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u4ee4 front \u6216 rear \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u76f4\u63a5\u7ed5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u63a5\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u300c\u53d6\u4f59\u64cd\u4f5c\u300d\u6765\u5b9e\u73b0\uff0c\u8be6\u60c5\u8bf7\u89c1\u4ee5\u4e0b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_queue.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % capacity()];\n}\nreturn res;\n}\n}\n
     array_queue.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate:\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u961f\u5217\u957f\u5ea6\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\npublic:\nArrayQueue(int capacity) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = new int[capacity];\nqueCapacity = capacity;\nfront = queSize = 0;\n}\n~ArrayQueue() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (queSize == queCapacity) {\ncout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % queCapacity;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (empty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> arr(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\narr[i] = nums[j % queCapacity];\n}\nreturn arr;\n}\n};\n
     array_queue.py
    class ArrayQueue:\n\"\"\" \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 \"\"\"\ndef __init__(self, size: int) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__nums: List[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nself.__front: int = 0          # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nself.__size: int = 0           # \u961f\u5217\u957f\u5ea6\ndef capacity(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf \"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__size == 0\ndef push(self, num: int) -> None:\n\"\"\" \u5165\u961f \"\"\"\nassert self.__size < self.capacity(), \"\u961f\u5217\u5df2\u6ee1\"\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear: int = (self.__front + self.__size) % self.capacity()\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u961f \"\"\"\nnum: int = self.peek()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.__front = (self.__front + 1) % self.capacity()\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u961f\u5217\u4e3a\u7a7a\"\nreturn self.__nums[self.__front]\ndef to_list(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nres: List[int] = [0] * self.size()\nj: int = self.__front\nfor i in range(self.size()):\nres[i] = self.__nums[(j % self.capacity())]\nj += 1\nreturn res\n
     array_queue.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\nreturn &arrayQueue{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n// \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\nif q.queSize == q.queCapacity {\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear := (q.front + q.queSize) % q.queCapacity\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nq.nums[rear] = num\nq.queSize++\n}\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\nnum := q.peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nq.front = (q.front + 1) % q.queCapacity\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\nrear := (q.front + q.queSize)\nif rear >= q.queCapacity {\nrear %= q.queCapacity\nreturn append(q.nums[q.front:], q.nums[:rear]...)\n}\nreturn q.nums[q.front:rear]\n}\n
     array_queue.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n#nums;         // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front = 0;    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize = 0;  // \u961f\u5217\u957f\u5ea6\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#queSize == 0;\n}\n/* \u5165\u961f */\npush(num) {\nif (this.size == this.capacity) {\nconsole.log(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.#front + this.size) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.#front = (this.#front + 1) % this.capacity;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.empty())\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.#nums[this.#front];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.#front; i < this.size; i++, j++) {\narr[i] = this.#nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
     array_queue.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate nums: number[];  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = this.queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.queSize == 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\nif (this.size == this.capacity) {\nconsole.log(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.front + this.queSize) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.front = (this.front + 1) % this.capacity;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.empty())\nthrow new Error(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn this.nums[this.front];\n}\n/* \u8fd4\u56de Array */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.front; i < this.size; i++, j++) {\narr[i] = this.nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
     array_queue.c
    [class]{arrayQueue}-[func]{}\n
     array_queue.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue\n{\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity)\n{\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity()\n{\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num)\n{\nif (queSize == capacity())\n{\nConsole.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop()\n{\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek()\n{\nif (isEmpty())\nthrow new Exception();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray()\n{\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++)\n{\nres[i] = nums[j % this.capacity()];\n}\nreturn res;\n}\n}\n
     array_queue.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize = 0 // \u961f\u5217\u957f\u5ea6\ninit(capacity: Int) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = Array(repeating: 0, count: capacity)\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nqueSize == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\nif size() == capacity() {\nprint(\"\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (front + queSize) % capacity()\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: queSize)\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[j % capacity()]\n}\nreturn res\n}\n}\n
     array_queue.zig
    // \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \ncap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\nfront: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nque_size: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.cap = cap;\nself.nums = try self.mem_allocator.alloc(T, self.cap);\nstd.mem.set(T, self.nums, @as(T, 0));\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.cap;\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.que_size == 0;\n}\n// \u5165\u961f\npub fn push(self: *Self, num: T) !void {\nif (self.size() == self.capacity()) {\nstd.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nvar rear = (self.front + self.que_size) % self.capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.nums[rear] = num;\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.capacity();\nself.que_size -= 1;\nreturn num;\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.nums[self.front];\n} // \u8fd4\u56de\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nvar j: usize = self.front;\nwhile (i < self.size()) : ({ i += 1; j += 1; }) {\nres[i] = self.nums[j % self.capacity()];\n}\nreturn res;\n}\n};\n}\n
     
    \u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u5b58\u5728\u5c40\u9650\u6027\uff0c\u5373\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u4e0d\u8fc7\u8fd9\u4e2a\u95ee\u9898\u5f88\u5bb9\u6613\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u5217\u8868\uff08\u5373\u52a8\u6001\u6570\u7ec4\uff09\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"
    \u4e0e\u6808\u7684\u7ed3\u8bba\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_stack_and_queue/queue/#524","title":"5.2.4. \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"
       
    • \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c31\u88ab\u52a0\u5165\u5230\u961f\u5217\u4e4b\u4e2d\uff0c\u968f\u540e\u7cfb\u7edf\u518d\u6839\u636e\u987a\u5e8f\u4f9d\u6b21\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u65f6\uff0c\u5728\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u7684\u8ba2\u5355\uff0c\u5982\u4f55\u5904\u7406\u300c\u9ad8\u5e76\u53d1\u300d\u5219\u662f\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u601d\u8003\u7684\u95ee\u9898\u3002
      •  
    • \u5404\u79cd\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u7684\u529f\u80fd\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\u7b49\u3002
      •  
    "},{"location":"chapter_stack_and_queue/stack/","title":"5.1. \u00a0 \u6808","text":"
    \u300c\u6808 Stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u300c\u5148\u5165\u540e\u51fa first in, last out\u300d\u6570\u636e\u64cd\u4f5c\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u653e\u5728\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u9700\u8981\u62ff\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u53d6\u51fa\u3002
     
    \u201c\u76d8\u5b50\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u6bd4\u55bb\uff0c\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u79cd\u5143\u7d20\uff08\u4f8b\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u6570\u636e\u7ed3\u6784\u3002
     
    \u6211\u4eec\u5c06\u8fd9\u4e00\u645e\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u300c\u6808\u9876\u300d\uff0c\u5c06\u5e95\u90e8\u79f0\u4e3a\u300c\u6808\u5e95\u300d\uff0c\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u6808\u300d\uff0c\u5c06\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u6808\u300d\u3002
     
     
     Fig. \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 
    "},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1. \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"
    \u6808\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\uff0c\u6b64\u5904\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 push , pop , peek \u4e3a\u4f8b\u3002
     \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) 
    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5b9e\u73b0\u597d\u7684\u6808\u7c7b\u3002 \u67d0\u4e9b\u8bed\u8a00\u5e76\u672a\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u4f46\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u628a\u8be5\u8bed\u8a00\u7684\u300c\u6570\u7ec4\u300d\u6216\u300c\u94fe\u8868\u300d\u770b\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u901a\u8fc7\u201c\u8111\u8865\u201d\u6765\u5c4f\u853d\u65e0\u5173\u64cd\u4f5c\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig stack.java
    /* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
     stack.cpp
    /* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
     stack.py
    \"\"\" \u521d\u59cb\u5316\u6808 \"\"\"\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: List[int] = []\n\"\"\" \u5143\u7d20\u5165\u6808 \"\"\"\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\npeek: int = stack[-1]\n\"\"\" \u5143\u7d20\u51fa\u6808 \"\"\"\npop: int = stack.pop()\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nsize: int = len(stack)\n\"\"\" \u5224\u65ad\u662f\u5426\u4e3a\u7a7a \"\"\"\nis_empty: bool = len(stack) == 0\n
     stack_test.go
    /* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
     stack.js
    /* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
     stack.ts
    /* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
     stack.c
    \n
     stack.cs
    /* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new ();\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count()==0;\n
     stack.swift
    /* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
     stack.zig
    \n
    "},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2. \u00a0 \u6808\u7684\u5b9e\u73b0","text":"
    \u4e3a\u4e86\u66f4\u52a0\u6e05\u6670\u5730\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u81ea\u5df1\u52a8\u624b\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002
     
    \u6808\u89c4\u5b9a\u5143\u7d20\u662f\u5148\u5165\u540e\u51fa\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u6216\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64 \u6808\u53ef\u88ab\u770b\u4f5c\u662f\u4e00\u79cd\u53d7\u7ea6\u675f\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u8a00\u4e4b\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u4e4b\u5bf9\u5916\u7684\u8868\u73b0\u903b\u8f91\u7b26\u5408\u6808\u7684\u89c4\u5b9a\u5373\u53ef\u3002
    "},{"location":"chapter_stack_and_queue/stack/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"
    \u4f7f\u7528\u300c\u94fe\u8868\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u5c06\u94fe\u8868\u7684\u5934\u7ed3\u70b9\u770b\u4f5c\u6808\u9876\uff0c\u5c06\u5c3e\u7ed3\u70b9\u770b\u4f5c\u6808\u5e95\u3002
     
    \u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5230\u94fe\u8868\u5934\u90e8\u5373\u53ef\uff0c\u8fd9\u79cd\u7ed3\u70b9\u6dfb\u52a0\u65b9\u5f0f\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u5219\u5c06\u5934\u7ed3\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002
     LinkedListStackpush()pop() 
     
     
     
    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig linkedlist_stack.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode stackPeek;  // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new EmptyStackException();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate:\nListNode* stackTop; // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize;        // \u6808\u7684\u957f\u5ea6\npublic:\nLinkedListStack() {\nstackTop = nullptr;\nstkSize = 0;\n}\n~LinkedListStack() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u7ed3\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(stackTop);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nListNode* node = new ListNode(num);\nnode->next = stackTop;\nstackTop = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint num = top();\nListNode *tmp = stackTop;\nstackTop = stackTop->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nstkSize--;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (size() == 0)\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stackTop->val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode* node = stackTop;\nvector<int> res(size());\nfor (int i = res.size() - 1; i >= 0; i--) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
     linkedlist_stack.py
    class LinkedListStack:\n\"\"\" \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 \"\"\"\ndef __init__(self):\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__peek: Optional[ListNode] = None\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn not self.__peek\ndef push(self, val: int) -> None:\n\"\"\" \u5165\u6808 \"\"\"\nnode = ListNode(val)\nnode.next = self.__peek\nself.__peek = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\" \u51fa\u6808 \"\"\"\nnum: int = self.peek()\nself.__peek = self.__peek.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\n# \u5224\u7a7a\u5904\u7406\nif not self.__peek: return None\nreturn self.__peek.val\ndef to_list(self) -> List[int]:\n\"\"\" \u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\narr: List[int] = []\nnode = self.__peek\nwhile node:\narr.append(node.val)\nnode = node.next\narr.reverse()\nreturn arr\n
     linkedlist_stack.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\nreturn &linkedListStack{\ndata: list.New(),\n}\n}\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\ns.data.PushBack(value)\n}\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\nreturn s.data\n}\n
     linkedlist_stack.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n#stackPeek;     // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\n#stkSize = 0;   // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.#stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num) {\nconst node = new ListNode(num);\nnode.next = this.#stackPeek;\nthis.#stackPeek = node;\nthis.#stkSize++;\n}\n/* \u51fa\u6808 */\npop() {\nconst num = this.peek();\nthis.#stackPeek = this.#stackPeek.next;\nthis.#stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek() {\nif (!this.#stackPeek)\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#stackPeek;\nconst res = new Array(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate stackPeek: ListNode | null; // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nconst node = new ListNode(num);\nnode.next = this.stackPeek;\nthis.stackPeek = node;\nthis.stkSize++;\n}\n/* \u51fa\u6808 */\npop(): number {\nconst num = this.peek();\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nthis.stackPeek = this.stackPeek.next;\nthis.stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek(): number {\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.stackPeek;\nconst res = new Array<number>(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.c
    [class]{linkedListStack}-[func]{}\n
     linkedlist_stack.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack\n{\nprivate ListNode? stackPeek;  // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack()\n{\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num)\n{\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop()\n{\nif (stackPeek == null)\nthrow new Exception();\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek()\n{\nif (size() == 0 || stackPeek == null)\nthrow new Exception();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nif (stackPeek == null)\nreturn Array.Empty<int>();\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.Length - 1; i >= 0; i--)\n{\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
     linkedlist_stack.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate var _peek: ListNode? // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate var _size = 0 // \u6808\u7684\u957f\u5ea6\ninit() {}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nlet node = ListNode(x: num)\nnode.next = _peek\n_peek = node\n_size += 1\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n_peek = _peek?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn _peek!.val\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = _peek\nvar res = Array(repeating: 0, count: _size)\nfor i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
     linkedlist_stack.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u7ed3\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.stack_top = null;\nself.stk_size = 0;\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stk_size;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack_top.?.val;\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\nnode.next = self.stack_top;\nself.stack_top = node;\nself.stk_size += 1;\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.peek();\nself.stack_top = self.stack_top.?.next;\nself.stk_size -= 1;\nreturn num;\n} // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.stack_top;\nvar res = try self.mem_allocator.alloc(T, self.size());\nstd.mem.set(T, res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[res.len - i - 1] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"
    \u4f7f\u7528\u300c\u6570\u7ec4\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u8003\u8651\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u5f53\u4f5c\u6808\u9876\u3002\u8fd9\u6837\u8bbe\u8ba1\u4e0b\uff0c\u300c\u5165\u6808\u300d\u4e0e\u300c\u51fa\u6808\u300d\u64cd\u4f5c\u5c31\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u300c\u6dfb\u52a0\u5143\u7d20\u300d\u4e0e\u300c\u5220\u9664\u5143\u7d20\u300d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002
     ArrayStackpush()pop() 
     
     
     
    \u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u662f\u6e90\u6e90\u4e0d\u65ad\u7684\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528\u652f\u6301\u52a8\u6001\u6269\u5bb9\u7684\u300c\u5217\u8868\u300d\uff0c\u8fd9\u6837\u5c31\u65e0\u9700\u81ea\u884c\u5b9e\u73b0\u6570\u7ec4\u6269\u5bb9\u4e86\u3002\u4ee5\u4e0b\u662f\u793a\u4f8b\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig array_stack.java
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate ArrayList<Integer> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new ArrayList<>();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn stack.remove(size() - 1);\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new EmptyStackException();\nreturn stack.get(size() - 1);\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic Object[] toArray() {\nreturn stack.toArray();\n}\n}\n
     array_stack.cpp
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate:\nvector<int> stack;\npublic:\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn stack.empty();\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nstack.push_back(num);\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint oldTop = top();\nstack.pop_back();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif(empty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stack.back();\n}\n/* \u8fd4\u56de Vector */\nvector<int> toVector() {\nreturn stack;\n}\n};\n
     array_stack.py
    class ArrayStack:\n\"\"\" \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 \"\"\"\ndef __init__(self) -> None:\n\"\"\" \u6784\u9020\u65b9\u6cd5 \"\"\"\nself.__stack: List[int] = []\ndef size(self) -> int:\n\"\"\" \u83b7\u53d6\u6808\u7684\u957f\u5ea6 \"\"\"\nreturn len(self.__stack)\ndef is_empty(self) -> bool:\n\"\"\" \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a \"\"\"\nreturn self.__stack == []\ndef push(self, item: int) -> None:\n\"\"\" \u5165\u6808 \"\"\"\nself.__stack.append(item)\ndef pop(self) -> int:\n\"\"\" \u51fa\u6808 \"\"\"\nassert not self.is_empty(), \"\u6808\u4e3a\u7a7a\"\nreturn self.__stack.pop()\ndef peek(self) -> int:\n\"\"\" \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \"\"\"\nassert not self.is_empty(), \"\u6808\u4e3a\u7a7a\"\nreturn self.__stack[-1]\ndef to_list(self) -> List[int]:\n\"\"\" \u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370 \"\"\"\nreturn self.__stack\n
     array_stack.go
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\ndata []int // \u6570\u636e\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\nreturn &arrayStack{\n// \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\ndata: make([]int, 0, 16),\n}\n}\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\nreturn len(s.data)\n}\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\nreturn s.size() == 0\n}\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n// \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\ns.data = append(s.data, v)\n}\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\nval := s.peek()\ns.data = s.data[:len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\nval := s.data[len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\nreturn s.data\n}\n
     array_stack.js
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n#stack;\nconstructor() {\nthis.#stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nthis.#stack.push(num);\n}\n/* \u51fa\u6808 */\npop() {\nif (this.empty())\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop() {\nif (this.empty())\nthrow new Error(\"\u6808\u4e3a\u7a7a\");\nreturn this.#stack[this.#stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.#stack;\n}\n};\n
     array_stack.ts
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate stack: number[];\nconstructor() {\nthis.stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nthis.stack.push(num);\n}\n/* \u51fa\u6808 */\npop(): number | undefined {\nif (this.empty())\nthrow new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop(): number | undefined {\nif (this.empty())\nthrow new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack[this.stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.stack;\n}\n};\n
     array_stack.c
    [class]{arrayStack}-[func]{}\n
     array_stack.cs
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack\n{\nprivate List<int> stack;\npublic ArrayStack()\n{\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size()\n{\nreturn stack.Count();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty()\n{\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num)\n{\nstack.Add(num);\n}\n/* \u51fa\u6808 */\npublic int pop()\n{\nif (isEmpty())\nthrow new Exception();\nvar val = peek();\nstack.RemoveAt(size() - 1);\nreturn val;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek()\n{\nif (isEmpty())\nthrow new Exception();\nreturn stack[size() - 1];\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray()\n{\nreturn stack.ToArray();\n}\n}\n
     array_stack.swift
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate var stack: [Int]\ninit() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = []\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nstack.count\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nstack.isEmpty\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nstack.append(num)\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.removeLast()\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.last!\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nstack\n}\n}\n
     array_stack.zig
    // \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack: ?std.ArrayList(T) = null,     // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) void {\nif (self.stack == null) {\nself.stack = std.ArrayList(T).init(allocator);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.stack == null) return;\nself.stack.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stack.?.items.len;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack.?.items[self.size() - 1];\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\ntry self.stack.?.append(num);\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.stack.?.pop();\nreturn num;\n} // \u8fd4\u56de ArrayList\npub fn toList(self: *Self) std.ArrayList(T) {\nreturn self.stack.?;\n}\n};\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":""},{"location":"chapter_stack_and_queue/stack/#_3","title":"\u652f\u6301\u64cd\u4f5c","text":"
    \u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\uff0c\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u7ecf\u8d85\u51fa\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002
    "},{"location":"chapter_stack_and_queue/stack/#_4","title":"\u65f6\u95f4\u6548\u7387","text":"
    \u5728\u6570\u7ec4\uff08\u5217\u8868\uff09\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u90fd\u662f\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u64cd\u4f5c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u6548\u7387\u5f88\u597d\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u5219\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u90a3\u4e48\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
     
    \u5728\u94fe\u8868\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u53d8\u6162\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u7ed3\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u800c\u6548\u7387\u4e0d\u5982\u6570\u7ec4\u3002\u8fdb\u4e00\u6b65\u5730\u601d\u8003\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u4e0d\u662f int \u800c\u662f\u7ed3\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u5c31\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u5347\u6548\u7387\u3002
     
    \u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff08\u4f8b\u5982 int , double \uff09\u65f6\uff0c\u5219\u7ed3\u8bba\u5982\u4e0b\uff1a
     
       
    • \u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u4f1a\u53d8\u6162\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64 \u603b\u4f53\u6548\u7387\u66f4\u9ad8\uff1b
      •  
    • \u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b \u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\uff1b
      •  
    "},{"location":"chapter_stack_and_queue/stack/#_5","title":"\u7a7a\u95f4\u6548\u7387","text":"
    \u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u7ed9\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u8fc7\u6211\u4eec\u7684\u9700\u6c42\u3002\u5e76\u4e14\u6269\u5bb9\u673a\u5236\u4e00\u822c\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u6bd4\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u6211\u4eec\u7684\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u6570\u7ec4\u5b9e\u73b0\u6808\u4f1a\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002
     
    \u5f53\u7136\uff0c\u7531\u4e8e\u7ed3\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64 \u94fe\u8868\u7ed3\u70b9\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u66f4\u5927\u3002
     
    \u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u7701\u5185\u5b58\uff0c\u9700\u8981 case-by-case \u5730\u5206\u6790\u3002
    "},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4. \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"
       
    • \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u5c06\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u300c\u540e\u9000\u300d\u64cd\u4f5c\u6765\u56de\u5230\u4e0a\u4e00\u9875\u9762\uff0c\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u5219\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
      •  
    • \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u5f53\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u5c31\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u6765\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\uff0c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u65f6\u51fa\u6808\u3002
      •  
    "},{"location":"chapter_stack_and_queue/summary/","title":"5.4. \u00a0 \u5c0f\u7ed3","text":"
       
    • \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u3002
      •  
    • \u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u66f4\u597d\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u6269\u5bb9\u65f6\u4f1a\u5bfc\u81f4\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u5bf9\u5730\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
      •  
    • \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u4f1a\u9020\u6210\u4e00\u5b9a\u7a7a\u95f4\u6d6a\u8d39\uff0c\u7136\u800c\u94fe\u8868\u7ed3\u70b9\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u5185\u5b58\u66f4\u5927\u3002
      •  
    • \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e24\u79cd\u5b9e\u73b0\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\uff0c\u4e0e\u4e0a\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u540c\u3002
      •  
    • \u53cc\u5411\u961f\u5217\u7684\u4e24\u7aef\u90fd\u53ef\u4ee5\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002
      •  
    "},{"location":"chapter_tree/avl_tree/","title":"7.4. \u00a0 AVL \u6811 *","text":"
    \u5728\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u7ae0\u8282\u4e2d\u63d0\u5230\uff0c\u5728\u8fdb\u884c\u591a\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002\u6b64\u65f6\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4f1a\u7531 \\(O(\\log n)\\) \u52a3\u5316\u81f3 \\(O(n)\\) \u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6267\u884c\u4e24\u6b65\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u8be5\u4e8c\u53c9\u641c\u7d22\u6811\u5c31\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002
     
     
     Fig. AVL \u6811\u5728\u5220\u9664\u7ed3\u70b9\u540e\u53d1\u751f\u9000\u5316 
     
    \u518d\u6bd4\u5982\uff0c\u5728\u4ee5\u4e0b\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u7ed3\u70b9\u540e\uff0c\u6811\u4e25\u91cd\u5411\u5de6\u504f\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u53d1\u751f\u52a3\u5316\u3002
     
     
     Fig. AVL \u6811\u5728\u63d2\u5165\u7ed3\u70b9\u540e\u53d1\u751f\u9000\u5316 
     
    G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u5176 1962 \u5e74\u53d1\u8868\u7684\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u4f7f\u5f97\u5728\u4e0d\u65ad\u6dfb\u52a0\u4e0e\u5220\u9664\u7ed3\u70b9\u540e\uff0cAVL \u6811\u4ecd\u7136\u4e0d\u4f1a\u53d1\u751f\u9000\u5316\uff0c\u8fdb\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u80fd\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002
     
    \u6362\u8a00\u4e4b\uff0c\u5728\u9891\u7e41\u589e\u5220\u67e5\u6539\u7684\u4f7f\u7528\u573a\u666f\u4e2d\uff0cAVL \u6811\u53ef\u59cb\u7ec8\u4fdd\u6301\u5f88\u9ad8\u7684\u6570\u636e\u589e\u5220\u67e5\u6539\u6548\u7387\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002
    "},{"location":"chapter_tree/avl_tree/#741-avl","title":"7.4.1. \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"
    \u300cAVL \u6811\u300d\u65e2\u662f\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u53c8\u662f\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u79cd\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u53c8\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u3002
    "},{"location":"chapter_tree/avl_tree/#_1","title":"\u7ed3\u70b9\u9ad8\u5ea6","text":"
    \u5728 AVL \u6811\u7684\u64cd\u4f5c\u4e2d\uff0c\u9700\u8981\u83b7\u53d6\u7ed3\u70b9\u300c\u9ad8\u5ea6 Height\u300d\uff0c\u6240\u4ee5\u7ed9 AVL \u6811\u7684\u7ed3\u70b9\u7c7b\u6dfb\u52a0 height \u53d8\u91cf\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\npublic int val;        // \u7ed3\u70b9\u503c\npublic int height;     // \u7ed3\u70b9\u9ad8\u5ea6\npublic TreeNode left;  // \u5de6\u5b50\u7ed3\u70b9\npublic TreeNode right; // \u53f3\u5b50\u7ed3\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nstruct TreeNode {\nint val{};          // \u7ed3\u70b9\u503c\nint height = 0;     // \u7ed3\u70b9\u9ad8\u5ea6\nTreeNode *left{};   // \u5de6\u5b50\u7ed3\u70b9\nTreeNode *right{};  // \u53f3\u5b50\u7ed3\u70b9\nTreeNode() = default;\nexplicit TreeNode(int x) : val(x){}\n};\n
     
    \"\"\" AVL \u6811\u7ed3\u70b9\u7c7b \"\"\"\nclass TreeNode:\ndef __init__(self, val: int):\nself.val: int = val                    # \u7ed3\u70b9\u503c\nself.height: int = 0                   # \u7ed3\u70b9\u9ad8\u5ea6\nself.left: Optional[TreeNode] = None   # \u5de6\u5b50\u7ed3\u70b9\u5f15\u7528\nself.right: Optional[TreeNode] = None  # \u53f3\u5b50\u7ed3\u70b9\u5f15\u7528\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\ntype TreeNode struct {\nVal    int       // \u7ed3\u70b9\u503c\nHeight int       // \u7ed3\u70b9\u9ad8\u5ea6\nLeft   *TreeNode // \u5de6\u5b50\u7ed3\u70b9\u5f15\u7528\nRight  *TreeNode // \u53f3\u5b50\u7ed3\u70b9\u5f15\u7528\n}\n
     
    class TreeNode {\nval; // \u7ed3\u70b9\u503c\nheight; //\u7ed3\u70b9\u9ad8\u5ea6\nleft; // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nconstructor(val, left, right, height) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n
     
    class TreeNode {\nval: number;            // \u7ed3\u70b9\u503c\nheight: number;         // \u7ed3\u70b9\u9ad8\u5ea6\nleft: TreeNode | null;  // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nright: TreeNode | null; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nconstructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height; this.left = left === undefined ? null : left; this.right = right === undefined ? null : right; }\n}\n
     
    \n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\npublic int val;          // \u7ed3\u70b9\u503c\npublic int height;       // \u7ed3\u70b9\u9ad8\u5ea6\npublic TreeNode? left;   // \u5de6\u5b50\u7ed3\u70b9\npublic TreeNode? right;  // \u53f3\u5b50\u7ed3\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
     
    /* AVL \u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar height: Int // \u7ed3\u70b9\u9ad8\u5ea6\nvar left: TreeNode? // \u5de6\u5b50\u7ed3\u70b9\nvar right: TreeNode? // \u53f3\u5b50\u7ed3\u70b9\ninit(x: Int) {\nval = x\nheight = 0\n}\n}\n
     
    \n
     
    \u300c\u7ed3\u70b9\u9ad8\u5ea6\u300d\u662f\u6700\u8fdc\u53f6\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u8d70\u8fc7\u7684\u300c\u8fb9\u300d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\uff0c\u53f6\u7ed3\u70b9\u7684\u9ad8\u5ea6\u4e3a 0 \uff0c\u7a7a\u7ed3\u70b9\u7684\u9ad8\u5ea6\u4e3a -1\u3002\u6211\u4eec\u5c01\u88c5\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u4e0e\u66f4\u65b0\u7ed3\u70b9\u7684\u9ad8\u5ea6\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
     avl_tree.cpp
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode* node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == nullptr ? -1 : node->height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode* node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode->height = max(height(node->left), height(node->right)) + 1;\n}\n
     avl_tree.py
    def height(self, node: Optional[TreeNode]) -> int:\n\"\"\" \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 \"\"\"\n# \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nif node is not None:\nreturn node.height\nreturn -1\ndef __update_height(self, node: Optional[TreeNode]):\n\"\"\" \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 \"\"\"\n# \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = max([self.height(node.left), self.height(node.right)]) + 1\n
     avl_tree.go
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nif node != nil {\nreturn node.Height\n}\nreturn -1\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\nlh := t.height(node.Left)\nrh := t.height(node.Right)\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif lh > rh {\nnode.Height = lh + 1\n} else {\nnode.Height = rh + 1\n}\n}\n
     avl_tree.js
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nheight(node) {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
     avl_tree.ts
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{height}\n[class]{aVLTree}-[func]{updateHeight}\n
     avl_tree.cs
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node)\n{\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node)\n{\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.Max(height(node.left), height(node.right)) + 1;\n}\n
     avl_tree.swift
    /* \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nnode == nil ? -1 : node!.height\n}\n/* \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
     avl_tree.zig
    // \u83b7\u53d6\u7ed3\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n_ = self;\n// \u7a7a\u7ed3\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u7ed3\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn if (node == null) -1 else node.?.height;\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n// \u7ed3\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.?.height = std.math.max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_2","title":"\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50","text":"
    \u7ed3\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 Balance Factor\u300d\u662f \u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u9ad8\u5ea6\uff0c\u5e76\u5b9a\u4e49\u7a7a\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a 0 \u3002\u540c\u6837\u5730\uff0c\u6211\u4eec\u5c06\u83b7\u53d6\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u4ee5\u4fbf\u540e\u7eed\u4f7f\u7528\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
     avl_tree.cpp
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode* node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == nullptr) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
     avl_tree.py
    def balance_factor(self, node: Optional[TreeNode]) -> int:\n\"\"\" \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 \"\"\"\n# \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node is None:\nreturn 0\n# \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.left) - self.height(node.right)\n
     avl_tree.go
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node == nil {\nreturn 0\n}\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn t.height(node.Left) - t.height(node.Right)\n}\n
     avl_tree.js
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
     avl_tree.ts
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{balanceFactor}\n
     avl_tree.cs
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node)\n{\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
     avl_tree.swift
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nguard let node = node else { return 0 }\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node: node.left) - height(node: node.right)\n}\n
     avl_tree.zig
    // \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n// \u7a7a\u7ed3\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u7ed3\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.?.left) - self.height(node.?.right);\n}\n
     
    Note
     
    \u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002
    "},{"location":"chapter_tree/avl_tree/#742-avl","title":"7.4.2. \u00a0 AVL \u6811\u65cb\u8f6c","text":"
    AVL \u6811\u7684\u72ec\u7279\u4e4b\u5904\u5728\u4e8e\u300c\u65cb\u8f6c Rotation\u300d\u7684\u64cd\u4f5c\uff0c\u5176\u53ef \u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u7ed3\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u8a00\u4e4b\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u53ef\u4ee5\u4f7f\u6811\u4fdd\u6301\u4e3a\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u3002
     
    \u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7684\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u7ed3\u70b9\u79f0\u4e3a\u300c\u5931\u8861\u7ed3\u70b9\u300d\u3002\u6839\u636e\u7ed3\u70b9\u7684\u5931\u8861\u60c5\u51b5\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a \u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u4e00\u8d77\u6765\u770b\u770b\u5b83\u4eec\u662f\u5982\u4f55\u64cd\u4f5c\u7684\u3002
    "},{"location":"chapter_tree/avl_tree/#case-1-","title":"Case 1 - \u53f3\u65cb","text":"
    \u5982\u4e0b\u56fe\u6240\u793a\uff08\u7ed3\u70b9\u4e0b\u65b9\u4e3a\u300c\u5e73\u8861\u56e0\u5b50\u300d\uff09\uff0c\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u7ed3\u70b9\u662f \u7ed3\u70b9 3\u3002\u6211\u4eec\u805a\u7126\u5728\u4ee5\u8be5\u5931\u8861\u7ed3\u70b9\u4e3a\u6839\u7ed3\u70b9\u7684\u5b50\u6811\u4e0a\uff0c\u5c06\u8be5\u7ed3\u70b9\u8bb0\u4e3a node \uff0c\u5c06\u5176\u5de6\u5b50\u7ed3\u70b9\u8bb0\u4e3a child \uff0c\u6267\u884c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u8be5\u5b50\u6811\u5df2\u7ecf\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4e3a\u4e8c\u53c9\u641c\u7d22\u6811\u3002
     <1><2><3><4> 
     
     
     
     
    \u8fdb\u800c\uff0c\u5982\u679c\u7ed3\u70b9 child \u672c\u8eab\u6709\u53f3\u5b50\u7ed3\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u53f3\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u5de6\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c 
     
    \u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u9700\u8981\u901a\u8fc7\u4fee\u6539\u7ed3\u70b9\u6307\u9488\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\nTreeNode child = node.left;\nTreeNode grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.cpp
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode* rightRotate(TreeNode* node) {\nTreeNode* child = node->left;\nTreeNode* grandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.py
    def __right_rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u53f3\u65cb\u64cd\u4f5c \"\"\"\nchild = node.left\ngrand_child = child.right\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node\nnode.left = grand_child\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n
     avl_tree.go
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\nchild := node.Left\ngrandChild := child.Right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.Right = node\nnode.Left = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.js
    /* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.ts
    /* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{rightRotate}\n
     avl_tree.cs
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node)\n{\nTreeNode? child = node.left;\nTreeNode? grandChild = child?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.swift
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.left\nlet grandChild = child?.right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild?.right = node\nnode?.left = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.zig
    // \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.left;\nvar grandChild = child.?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.?.right = node;\nnode.?.left = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#case-2-","title":"Case 2 - \u5de6\u65cb","text":"
    \u7c7b\u4f3c\u5730\uff0c\u5982\u679c\u5c06\u53d6\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u90a3\u4e48\u5219\u9700\u8981\u300c\u5de6\u65cb\u300d\u64cd\u4f5c\u3002
     
     
     Fig. \u5de6\u65cb\u64cd\u4f5c 
     
    \u540c\u7406\uff0c\u82e5\u7ed3\u70b9 child \u672c\u8eab\u6709\u5de6\u5b50\u7ed3\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u5de6\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u53f3\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c 
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u300c\u5de6\u65cb\u300d\u548c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u4e24\u8005\u5bf9\u5e94\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u6839\u636e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u5f88\u65b9\u4fbf\u5730\u4ece\u300c\u53f3\u65cb\u300d\u63a8\u5bfc\u51fa\u300c\u5de6\u65cb\u300d\u3002\u5177\u4f53\u5730\uff0c\u53ea\u9700\u5c06\u300c\u53f3\u65cb\u300d\u4ee3\u7801\u4e2d\u7684\u628a\u6240\u6709\u7684 left \u66ff\u6362\u4e3a right \u3001\u6240\u6709\u7684 right \u66ff\u6362\u4e3a left \uff0c\u5373\u53ef\u5f97\u5230\u300c\u5de6\u65cb\u300d\u4ee3\u7801\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\nTreeNode child = node.right;\nTreeNode grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.cpp
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode* leftRotate(TreeNode* node) {\nTreeNode* child = node->right;\nTreeNode* grandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.py
    def __left_rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u5de6\u65cb\u64cd\u4f5c \"\"\"\nchild = node.right\ngrand_child = child.left\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node\nnode.right = grand_child\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n
     avl_tree.go
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\nchild := node.Right\ngrandChild := child.Left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.Left = node\nnode.Right = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.js
    /* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.ts
    /* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{leftRotate}\n
     avl_tree.cs
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node)\n{\nTreeNode? child = node.right;\nTreeNode? grandChild = child?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
     avl_tree.swift
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.right\nlet grandChild = child?.left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild?.left = node\nnode?.right = grandChild\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child\n}\n
     avl_tree.zig
    // \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.right;\nvar grandChild = child.?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.?.left = node;\nnode.?.right = grandChild;\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#case-3-","title":"Case 3 - \u5148\u5de6\u540e\u53f3","text":"
    \u5bf9\u4e8e\u4e0b\u56fe\u7684\u5931\u8861\u7ed3\u70b9 3 \uff0c\u5355\u4e00\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u6b64\u65f6\u9700\u8981\u300c\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u300d\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u5de6\u65cb\u300d\uff0c\u518d\u5bf9 node \u6267\u884c\u300c\u53f3\u65cb\u300d\u3002
     
     
     Fig. \u5148\u5de6\u65cb\u540e\u53f3\u65cb 
    "},{"location":"chapter_tree/avl_tree/#case-4-","title":"Case 4 - \u5148\u53f3\u540e\u5de6","text":"
    \u540c\u7406\uff0c\u53d6\u4ee5\u4e0a\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\uff0c\u5219\u9700\u8981\u300c\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u300d\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u53f3\u65cb\u300d\uff0c\u7136\u540e\u5bf9 node \u6267\u884c\u300c\u5de6\u65cb\u300d\u3002
     
     
     Fig. \u5148\u53f3\u65cb\u540e\u5de6\u65cb 
    "},{"location":"chapter_tree/avl_tree/#_3","title":"\u65cb\u8f6c\u7684\u9009\u62e9","text":"
    \u4e0b\u56fe\u63cf\u8ff0\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0 Cases \u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u91c7\u7528 \u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u540e\u5de6\u3001\u5148\u5de6\u540e\u53f3 \u7684\u65cb\u8f6c\u64cd\u4f5c\u3002
     
     
     Fig. AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 
     
    \u5177\u4f53\u5730\uff0c\u5728\u4ee3\u7801\u4e2d\u4f7f\u7528 \u5931\u8861\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u3001\u8f83\u9ad8\u4e00\u4fa7\u5b50\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u6765\u786e\u5b9a\u5931\u8861\u7ed3\u70b9\u5c5e\u4e8e\u4e0a\u56fe\u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002
     \u5931\u8861\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u7ed3\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb 
    \u4e3a\u65b9\u4fbf\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6b64\u51fd\u6570\u6765\u65cb\u8f6c\u5404\u79cd\u5931\u8861\u60c5\u51b5\uff0c\u4f7f\u5931\u8861\u7ed3\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.cpp
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode* rotate(TreeNode* node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint _balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (_balanceFactor > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (_balanceFactor < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.py
    def __rotate(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 \"\"\"\n# \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nbalance_factor = self.balance_factor(node)\n# \u5de6\u504f\u6811\nif balance_factor > 1:\nif self.balance_factor(node.left) >= 0:\n# \u53f3\u65cb\nreturn self.__right_rotate(node)\nelse:\n# \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = self.__left_rotate(node.left)\nreturn self.__right_rotate(node)\n# \u53f3\u504f\u6811\nelif balance_factor < -1:\nif self.balance_factor(node.right) <= 0:\n# \u5de6\u65cb\nreturn self.__left_rotate(node)\nelse:\n# \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = self.__right_rotate(node.right)\nreturn self.__left_rotate(node)\n# \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n
     avl_tree.go
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n// Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\nbf := t.balanceFactor(node)\n// \u5de6\u504f\u6811\nif bf > 1 {\nif t.balanceFactor(node.Left) >= 0 {\n// \u53f3\u65cb\nreturn t.rightRotate(node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.Left = t.leftRotate(node.Left)\nreturn t.rightRotate(node)\n}\n}\n// \u53f3\u504f\u6811\nif bf < -1 {\nif t.balanceFactor(node.Right) <= 0 {\n// \u5de6\u65cb\nreturn t.leftRotate(node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.Right = t.rightRotate(node.Right)\nreturn t.leftRotate(node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
     avl_tree.js
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.#rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.#leftRotate(node.left);\nreturn this.#rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.#leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.#rightRotate(node.right);\nreturn this.#leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.ts
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.leftRotate(node.left);\nreturn this.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.rightRotate(node.right);\nreturn this.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{rotate}\n
     avl_tree.cs
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node)\n{\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactorInt = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactorInt > 1)\n{\nif (balanceFactor(node.left) >= 0)\n{\n// \u53f3\u65cb\nreturn rightRotate(node);\n}\nelse\n{\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node?.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactorInt < -1)\n{\nif (balanceFactor(node.right) <= 0)\n{\n// \u5de6\u65cb\nreturn leftRotate(node);\n}\nelse\n{\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node?.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
     avl_tree.swift
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balanceFactor = balanceFactor(node: node)\n// \u5de6\u504f\u6811\nif balanceFactor > 1 {\nif self.balanceFactor(node: node?.left) >= 0 {\n// \u53f3\u65cb\nreturn rightRotate(node: node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode?.left = leftRotate(node: node?.left)\nreturn rightRotate(node: node)\n}\n}\n// \u53f3\u504f\u6811\nif balanceFactor < -1 {\nif self.balanceFactor(node: node?.right) <= 0 {\n// \u5de6\u65cb\nreturn leftRotate(node: node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode?.right = rightRotate(node: node?.right)\nreturn leftRotate(node: node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
     avl_tree.zig
    // \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n// \u83b7\u53d6\u7ed3\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nvar balance_factor = self.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balance_factor > 1) {\nif (self.balanceFactor(node.?.left) >= 0) {\n// \u53f3\u65cb\nreturn self.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.?.left = self.leftRotate(node.?.left);\nreturn self.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balance_factor < -1) {\nif (self.balanceFactor(node.?.right) <= 0) {\n// \u5de6\u65cb\nreturn self.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.?.right = self.rightRotate(node.?.right);\nreturn self.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#743-avl","title":"7.4.3. \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#_4","title":"\u63d2\u5165\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u7684\u7ed3\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u4e3b\u4f53\u7c7b\u4f3c\u3002\u4e0d\u540c\u7684\u662f\uff0c\u5728\u63d2\u5165\u7ed3\u70b9\u540e\uff0c\u4ece\u8be5\u7ed3\u70b9\u5230\u6839\u7ed3\u70b9\u7684\u8def\u5f84\u4e0a\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u300c\u5931\u8861\u7ed3\u70b9\u300d\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8be5\u7ed3\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7ed3\u70b9\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode insert(int val) {\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.cpp
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode* insert(int val) {\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode* insertHelper(TreeNode* node, int val) {\nif (node == nullptr)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node->val)\nnode->left = insertHelper(node->left, val);\nelse if (val > node->val)\nnode->right = insertHelper(node->right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.py
    def insert(self, val) -> TreeNode:\n\"\"\" \u63d2\u5165\u7ed3\u70b9 \"\"\"\nself.__root = self.__insert_helper(self.__root, val)\nreturn self.__root\ndef __insert_helper(self, node: Optional[TreeNode], val: int) -> TreeNode:\n\"\"\" \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn TreeNode(val)\n# 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9\nif val < node.val:\nnode.left = self.__insert_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__insert_helper(node.right, val)\nelse:\n# \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
     avl_tree.go
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc (t *aVLTree) insert(val int) *TreeNode {\nt.root = t.insertHelper(t.root, val)\nreturn t.root\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn NewTreeNode(val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif val < node.Val {\nnode.Left = t.insertHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.insertHelper(node.Right, val)\n} else {\n// \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n
     avl_tree.js
    /* \u63d2\u5165\u7ed3\u70b9 */\ninsert(val) {\nthis.root = this.#insertHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val) node.left = this.#insertHelper(node.left, val);\nelse if (val > node.val) node.right = this.#insertHelper(node.right, val);\nelse return node; // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nthis.#updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.ts
    /* \u63d2\u5165\u7ed3\u70b9 */\ninsert(val: number): TreeNode {\nthis.root = this.insertHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val) {\nnode.left = this.insertHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.insertHelper(node.right, val);\n} else {\nreturn node; // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nthis.updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{insert}\n[class]{aVLTree}-[func]{insertHelper}\n
     avl_tree.cs
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode? insert(int val)\n{\nroot = insertHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val)\n{\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
     avl_tree.swift
    /* \u63d2\u5165\u7ed3\u70b9 */\n@discardableResult\nfunc insert(val: Int) -> TreeNode? {\nroot = insertHelper(node: root, val: val)\nreturn root\n}\n/* \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn TreeNode(x: val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9 */\nif val < node!.val {\nnode?.left = insertHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = insertHelper(node: node?.right, val: val)\n} else {\nreturn node // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nupdateHeight(node: node) // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n
     avl_tree.zig
    // \u63d2\u5165\u7ed3\u70b9\nfn insert(self: *Self, val: T) !?*inc.TreeNode(T) {\nself.root = try self.insertHelper(self.root, val);\nreturn self.root;\n}\n// \u9012\u5f52\u63d2\u5165\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) {\nvar tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\ntmp_node.init(val);\nreturn tmp_node;\n}\n// 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u7ed3\u70b9\nif (val < node.?.val) {\nnode.?.left = try self.insertHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = try self.insertHelper(node.?.right, val);\n} else {\nreturn node;            // \u91cd\u590d\u7ed3\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nself.updateHeight(node);    // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_5","title":"\u5220\u9664\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u603b\u4f53\u76f8\u540c\u3002\u7c7b\u4f3c\u5730\uff0c\u5728\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u4e5f\u9700\u8981\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u7ed3\u70b9\u6062\u590d\u5e73\u8861\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig avl_tree.java
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode remove(int val) {\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode child = node.left != null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode temp = getInOrderNext(node.right);\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode getInOrderNext(TreeNode node) {\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left != null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.cpp
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode* remove(int val) {\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode* removeHelper(TreeNode* node, int val) {\nif (node == nullptr)\nreturn nullptr;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val)\nnode->left = removeHelper(node->left, val);\nelse if (val > node->val)\nnode->right = removeHelper(node->right, val);\nelse {\nif (node->left == nullptr || node->right == nullptr) {\nTreeNode* child = node->left != nullptr ? node->left : node->right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == nullptr) {\ndelete node;\nreturn nullptr;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\ndelete node;\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode* temp = getInOrderNext(node->right);\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode* getInOrderNext(TreeNode* node) {\nif (node == nullptr)\nreturn node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node->left != nullptr) {\nnode = node->left;\n}\nreturn node;\n}\n
     avl_tree.py
    def remove(self, val: int) -> Optional[TreeNode]:\n\"\"\" \u5220\u9664\u7ed3\u70b9 \"\"\"\nself.__root = self.__remove_helper(self.__root, val)\nreturn self.__root \ndef __remove_helper(self, node: Optional[TreeNode], val: int) -> Optional[TreeNode]:\n\"\"\" \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 \"\"\"\nif node is None:\nreturn None\n# 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif val < node.val:\nnode.left = self.__remove_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__remove_helper(node.right, val)\nelse:\nif node.left is None or node.right is None:\nchild = node.left or node.right\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child is None:\nreturn None\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse:\nnode = child\nelse:  # \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\ntemp = self.__get_inorder_next(node.right)\nnode.right = self.__remove_helper(node.right, temp.val)\nnode.val = temp.val\n# \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\ndef __get_inorder_next(self, node: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 \"\"\"\nif node is None:\nreturn None\n# \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile node.left is not None:\nnode = node.left\nreturn node\n
     avl_tree.go
    /* \u5220\u9664\u7ed3\u70b9 */\nfunc (t *aVLTree) remove(val int) *TreeNode {\nroot := t.removeHelper(t.root, val)\nreturn root\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node.Val {\nnode.Left = t.removeHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.removeHelper(node.Right, val)\n} else {\nif node.Left == nil || node.Right == nil {\nchild := node.Left\nif node.Right != nil {\nchild = node.Right\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\ntemp := t.getInOrderNext(node.Right)\nnode.Right = t.removeHelper(node.Right, temp.Val)\nnode.Val = temp.Val\n}\n}\n// \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc (t *aVLTree) getInOrderNext(node *TreeNode) *TreeNode {\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nfor node.Left != nil {\nnode = node.Left\n}\nreturn node\n}\n
     avl_tree.js
    /* \u5220\u9664\u7ed3\u70b9 */\nremove(val) {\nthis.root = this.#removeHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) node.left = this.#removeHelper(node.left, val);\nelse if (val > node.val) node.right = this.#removeHelper(node.right, val);\nelse {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse node = child;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nconst temp = this.#getInOrderNext(node.right);\nnode.right = this.#removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.#updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\n#getInOrderNext(node) {\nif (node === null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left !== null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.ts
    /* \u5220\u9664\u7ed3\u70b9 */\nremove(val: number): TreeNode {\nthis.root = this.removeHelper(this.root, val);\nreturn this.root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) {\nnode.left = this.removeHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.removeHelper(node.right, val);\n} else {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) {\nreturn null;\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nconst temp = this.getInOrderNext(node.right);\nnode.right = this.removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.updateHeight(node); // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\ngetInOrderNext(node: TreeNode): TreeNode {\nif (node === null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left !== null) {\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.c
    [class]{aVLTree}-[func]{remove}\n[class]{aVLTree}-[func]{removeHelper}\n[class]{aVLTree}-[func]{getInOrderNext}\n
     avl_tree.cs
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode? remove(int val)\n{\nroot = removeHelper(root, val);\nreturn root;\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val)\n{\nif (node == null) return null;\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse\n{\nif (node.left == null || node.right == null)\n{\nTreeNode? child = node.left != null ? node.left : node.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n}\nelse\n{\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nTreeNode? temp = getInOrderNext(node.right);\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode? getInOrderNext(TreeNode? node)\n{\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.left != null)\n{\nnode = node.left;\n}\nreturn node;\n}\n
     avl_tree.swift
    /* \u5220\u9664\u7ed3\u70b9 */\n@discardableResult\nfunc remove(val: Int) -> TreeNode? {\nroot = removeHelper(node: root, val: val)\nreturn root\n}\n/* \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node!.val {\nnode?.left = removeHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = removeHelper(node: node?.right, val: val)\n} else {\nif node?.left == nil || node?.right == nil {\nlet child = node?.left != nil ? node?.left : node?.right\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\nnode = child\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nlet temp = getInOrderNext(node: node?.right)\nnode?.right = removeHelper(node: node?.right, val: temp!.val)\nnode?.val = temp!.val\n}\n}\nupdateHeight(node: node) // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc getInOrderNext(node: TreeNode?) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile node?.left != nil {\nnode = node?.left\n}\nreturn node\n}\n
     avl_tree.zig
    // \u5220\u9664\u7ed3\u70b9\nfn remove(self: *Self, val: T) ?*inc.TreeNode(T) {\nself.root = self.removeHelper(self.root, val);\nreturn self.root;\n}\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) return null;\n// 1. \u67e5\u627e\u7ed3\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif (val < node.?.val) {\nnode.?.left = self.removeHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = self.removeHelper(node.?.right, val);\n} else {\nif (node.?.left == null or node.?.right == null) {\nvar child = if (node.?.left != null) node.?.left else node.?.right;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null) {\nreturn null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n} else {\nnode = child;\n}\n} else {\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u7ed3\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u7ed3\u70b9\u66ff\u6362\u5f53\u524d\u7ed3\u70b9\nvar temp = self.getInOrderNext(node.?.right);\nnode.?.right = self.removeHelper(node.?.right, temp.?.val);\nnode.?.val = temp.?.val;\n}\n}\nself.updateHeight(node);    // \u66f4\u65b0\u7ed3\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u7ed3\u70b9\nreturn node;\n}\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09\nfn getInOrderNext(self: *Self, node_: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n_ = self;\nvar node = node_;\nif (node == null) return node;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node.?.left != null) {\nnode = node.?.left;\n}\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_6","title":"\u67e5\u627e\u7ed3\u70b9","text":"
    \u300cAVL \u6811\u300d\u7684\u7ed3\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
    "},{"location":"chapter_tree/avl_tree/#744-avl","title":"7.4.4. \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"
       
    • \u7ec4\u7ec7\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u573a\u666f\uff1b
      •  
    • \u7528\u4e8e\u5efa\u7acb\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\uff1b
      •  
     
    \u4e3a\u4ec0\u4e48\u7ea2\u9ed1\u6811\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\uff1f
     
    \u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u56e0\u6b64\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u66f4\u5c11\uff0c\u7ed3\u70b9\u589e\u5220\u64cd\u4f5c\u76f8\u6bd4 AVL \u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002
    "},{"location":"chapter_tree/binary_search_tree/","title":"7.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"
    \u300c\u4e8c\u53c9\u641c\u7d22\u6811 Binary Search Tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a
     
       
    1. \u5bf9\u4e8e\u6839\u7ed3\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u7684\u503c \\(<\\) \u6839\u7ed3\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u7ed3\u70b9\u7684\u503c\uff1b
      2.  
    2. \u4efb\u610f\u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u4e5f\u6ee1\u8db3\u6761\u4ef6 1. \uff1b
      3.  
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811 
    "},{"location":"chapter_tree/binary_search_tree/#731","title":"7.3.1. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_search_tree/#_1","title":"\u67e5\u627e\u7ed3\u70b9","text":"
    \u7ed9\u5b9a\u76ee\u6807\u7ed3\u70b9\u503c num \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u7ed3\u70b9 cur \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u7ed3\u70b9 root \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u7ed3\u70b9\u503c cur.val \u548c num \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb
     
       
    • \u82e5 cur.val < num \uff0c\u8bf4\u660e\u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.right \uff1b
      •  
    • \u82e5 cur.val > num \uff0c\u8bf4\u660e\u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.left \uff1b
      •  
    • \u82e5 cur.val = num \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u7ed3\u70b9\u5373\u53ef\uff1b
      •  
     <1><2><3><4> 
     
     
     
     
    \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u548c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u5982\u51fa\u4e00\u8f99\uff0c\u4e5f\u662f\u5728\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode search(int num) {\nTreeNode cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.cpp
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode* search(int num) {\nTreeNode* cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur->val > num) cur = cur->left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.py
    def search(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u67e5\u627e\u7ed3\u70b9 \"\"\"\ncur: Optional[TreeNode] = self.__root\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur is not None:\n# \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelif cur.val > num:\ncur = cur.left\n# \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse:\nbreak\nreturn cur\n
     binary_search_tree.go
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\nnode := bst.root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor node != nil {\nif node.Val < num {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nnode = node.Right\n} else if node.Val > num {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nnode = node.Left\n} else {\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn node\n}\n
     binary_search_tree.js
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunction search(num) {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.ts
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunction search(num: number): TreeNode | null {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val < num) {\ncur = cur.right; // \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else if (cur.val > num) {\ncur = cur.left; // \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\nbreak; // \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{search}\n
     binary_search_tree.cs
    /* \u67e5\u627e\u7ed3\u70b9 */\nTreeNode? search(int num)\n{\nTreeNode? cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
     binary_search_tree.swift
    /* \u67e5\u627e\u7ed3\u70b9 */\nfunc search(num: Int) -> TreeNode? {\nvar cur = root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if cur!.val > num {\ncur = cur?.left\n}\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur\n}\n
     binary_search_tree.zig
    // \u67e5\u627e\u7ed3\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\nvar cur = self.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u76ee\u6807\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else if (cur.?.val > num) {\ncur = cur.?.left;\n// \u627e\u5230\u76ee\u6807\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n} else {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u7ed3\u70b9\nreturn cur;\n}\n
    "},{"location":"chapter_tree/binary_search_tree/#_2","title":"\u63d2\u5165\u7ed3\u70b9","text":"
    \u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 num \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u7ed3\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u5206\u4e3a\u4e24\u6b65\uff1a
     
       
    1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u7c7b\u4f3c\uff0c\u6211\u4eec\u4ece\u6839\u7ed3\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u7ed3\u70b9\u503c\u548c num \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u7ed3\u70b9\uff08\u904d\u5386\u5230 \\(\\text{null}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\uff1b
      2.  
    2. \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u7ed3\u70b9\uff1a\u521d\u59cb\u5316\u7ed3\u70b9 num \uff0c\u5c06\u8be5\u7ed3\u70b9\u653e\u5230 \\(\\text{null}\\) \u7684\u4f4d\u7f6e \uff1b
      3.  
     
    \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u7ed3\u70b9\uff0c\u5426\u5219\u5c06\u4f1a\u8fdd\u80cc\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\u82e5\u5f85\u63d2\u5165\u7ed3\u70b9\u5728\u6811\u4e2d\u5df2\u7ecf\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\nreturn node;\n}\n
     binary_search_tree.cpp
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode* insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr) return nullptr;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num) return nullptr;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur->left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode* node = new TreeNode(num);\nif (pre->val < num) pre->right = node;\nelse pre->left = node;\nreturn node;\n}\n
     binary_search_tree.py
    def insert(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u63d2\u5165\u7ed3\u70b9 \"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.__root is None:\nreturn None\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\ncur, pre = self.__root, None\nwhile cur is not None:\n# \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur.val == num:\nreturn None\npre = cur\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u63d2\u5165\u7ed3\u70b9 val\nnode = TreeNode(num)\nif pre.val < num:\npre.right = node\nelse:\npre.left = node\nreturn node\n
     binary_search_tree.go
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) *TreeNode {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5f85\u63d2\u5165\u7ed3\u70b9\u4e4b\u524d\u7684\u7ed3\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nreturn nil\n}\npre = cur\nif cur.Val < num {\ncur = cur.Right\n} else {\ncur = cur.Left\n}\n}\n// \u63d2\u5165\u7ed3\u70b9\nnode := NewTreeNode(num)\nif pre.Val < num {\npre.Right = node\n} else {\npre.Left = node\n}\nreturn cur\n}\n
     binary_search_tree.js
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunction insert(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return null;\nlet cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\nreturn node;\n}\n
     binary_search_tree.ts
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunction insert(num: number): TreeNode | null {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn null;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val === num) {\nreturn null; // \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = new TreeNode(num);\nif (pre!.val < num) {\npre!.right = node;\n} else {\npre!.left = node;\n}\nreturn node;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{insert}\n
     binary_search_tree.cs
    /* \u63d2\u5165\u7ed3\u70b9 */\nTreeNode? insert(int num)\n{\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nTreeNode node = new TreeNode(num);\nif (pre != null)\n{\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\n}\nreturn node;\n}\n
     binary_search_tree.swift
    /* \u63d2\u5165\u7ed3\u70b9 */\nfunc insert(num: Int) -> TreeNode? {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn nil\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur!.val == num {\nreturn nil\n}\npre = cur\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nlet node = TreeNode(x: num)\nif pre!.val < num {\npre?.right = node\n} else {\npre?.left = node\n}\nreturn node\n}\n
     binary_search_tree.zig
    // \u63d2\u5165\u7ed3\u70b9\nfn insert(self: *Self, num: T) !?*inc.TreeNode(T) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return null;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u7ed3\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.?.val == num) return null;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u63d2\u5165\u7ed3\u70b9 val\nvar node = try self.mem_allocator.create(inc.TreeNode(T));\nnode.init(num);\nif (pre.?.val < num) {\npre.?.right = node;\n} else {\npre.?.left = node;\n}\nreturn node;\n}\n
     
    \u4e3a\u4e86\u63d2\u5165\u7ed3\u70b9\uff0c\u9700\u8981\u501f\u52a9 \u8f85\u52a9\u7ed3\u70b9 pre \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u7ed3\u70b9\uff0c\u8fd9\u6837\u5728\u904d\u5386\u5230 \\(\\text{null}\\) \u65f6\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u7ed3\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u7ed3\u70b9\u63d2\u5165\u64cd\u4f5c\u3002
     
    \u4e0e\u67e5\u627e\u7ed3\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u7ed3\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
    "},{"location":"chapter_tree/binary_search_tree/#_3","title":"\u5220\u9664\u7ed3\u70b9","text":"
    \u4e0e\u63d2\u5165\u7ed3\u70b9\u4e00\u6837\uff0c\u6211\u4eec\u9700\u8981\u5728\u5220\u9664\u64cd\u4f5c\u540e\u7ef4\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u7ed3\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u3002\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5728\u4e8c\u53c9\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u83b7\u53d6\u5f85\u5220\u9664\u7ed3\u70b9\u3002\u63a5\u4e0b\u6765\uff0c\u6839\u636e\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf\uff0c\u5220\u9664\u64cd\u4f5c\u9700\u8981\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 0\\) \u65f6\uff0c\u8868\u660e\u5f85\u5220\u9664\u7ed3\u70b9\u662f\u53f6\u7ed3\u70b9\uff0c\u76f4\u63a5\u5220\u9664\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u7ed3\u70b9\uff08\u5ea6\u4e3a 0\uff09 
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u7ed3\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u7ed3\u70b9\u5373\u53ef\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u7ed3\u70b9\uff08\u5ea6\u4e3a 1\uff09 
     
    \u5f53\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf \\(= 2\\) \u65f6\uff0c\u5220\u9664\u64cd\u4f5c\u5206\u4e3a\u4e09\u6b65\uff1a
     
       
    1. \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\u5728 \u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 \u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff0c\u8bb0\u4e3a nex \uff1b
      2.  
    2. \u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex \uff1b
      3.  
    3. \u4f7f\u7528 nex \u66ff\u6362\u5f85\u5220\u9664\u7ed3\u70b9\uff1b
      4.  
     <1><2><3><4> 
     
     
     
     
    \u5220\u9664\u7ed3\u70b9\u64cd\u4f5c\u4e5f\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u7ed3\u70b9 \\(O(\\log n)\\) \uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u7ed3\u70b9 \\(O(\\log n)\\) \u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_search_tree.java
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nTreeNode child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left == cur) pre.left = child;\nelse pre.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode nex = getInOrderNext(cur.right);\nint tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode getInOrderNext(TreeNode root) {\nif (root == null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left != null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.cpp
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode* remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr) return nullptr;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num) cur = cur->right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur->left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == nullptr) return nullptr;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur->left == nullptr || cur->right == nullptr) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u7ed3\u70b9\nTreeNode* child = cur->left != nullptr ? cur->left : cur->right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre->left == cur) pre->left = child;\nelse pre->right = child;\n// \u91ca\u653e\u5185\u5b58\ndelete cur;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode* nex = getInOrderNext(cur->right);\nint tmp = nex->val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex->val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur->val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode* getInOrderNext(TreeNode* root) {\nif (root == nullptr) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root->left != nullptr) {\nroot = root->left;\n}\nreturn root;\n}\n
     binary_search_tree.py
    def remove(self, num: int) -> Optional[TreeNode]:\n\"\"\" \u5220\u9664\u7ed3\u70b9 \"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.__root is None:\nreturn None\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\ncur, pre = self.__root, None\nwhile cur is not None:\n# \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur.val == num:\nbreak\npre = cur\nif cur.val < num:  # \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur.right\nelse:  # \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur.left\n# \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur is None:\nreturn None\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif cur.left is None or cur.right is None:\n# \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nchild = cur.left or cur.right\n# \u5220\u9664\u7ed3\u70b9 cur\nif pre.left == cur:\npre.left = child\nelse:\npre.right = child\n# \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse:\n# \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nnex: TreeNode = self.get_inorder_next(cur.right)\ntmp: int = nex.val\n# \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nself.remove(nex.val)\n# \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp\nreturn cur\ndef get_inorder_next(self, root: Optional[TreeNode]) -> Optional[TreeNode]:\n\"\"\" \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 \"\"\"\nif root is None:\nreturn root\n# \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile root.left is not None:\nroot = root.left\nreturn root\n
     binary_search_tree.go
    /* \u5220\u9664\u7ed3\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) *TreeNode {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5f85\u5220\u9664\u7ed3\u70b9\u4e4b\u524d\u7684\u7ed3\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nbreak\n}\npre = cur\nif cur.Val < num {\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\ncur = cur.Right\n} else {\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\ncur = cur.Left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u4e3a 0 \u6216 1\nif cur.Left == nil || cur.Right == nil {\nvar child *TreeNode = nil\n// \u53d6\u51fa\u5f85\u5220\u9664\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\nif cur.Left != nil {\nchild = cur.Left\n} else {\nchild = cur.Right\n}\n// \u5c06\u5b50\u7ed3\u70b9\u66ff\u6362\u4e3a\u5f85\u5220\u9664\u7ed3\u70b9\nif pre.Left == cur {\npre.Left = child\n} else {\npre.Right = child\n}\n// \u5b50\u7ed3\u70b9\u6570\u4e3a 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u7ed3\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nnext := bst.getInOrderNext(cur)\ntemp := next.Val\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 next\nbst.remove(next.Val)\n// \u5c06 next \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.Val = temp\n}\nreturn cur\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc (bst *binarySearchTree) getInOrderNext(node *TreeNode) *TreeNode {\nif node == nil {\nreturn node\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nfor node.Left != nil {\nnode = node.Left\n}\nreturn node\n}\n
     binary_search_tree.js
    /* \u5220\u9664\u7ed3\u70b9 */\nfunction remove(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return null;\nlet cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left === cur) pre.left = child;\nelse pre.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet nex = getInOrderNext(cur.right);\nlet tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunction getInOrderNext(root) {\nif (root === null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left !== null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.ts
    /* \u5220\u9664\u7ed3\u70b9 */\nfunction remove(num: number): TreeNode | null {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn null;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) {\nbreak;\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) {\nreturn null;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre!.left === cur) {\npre!.left = child;\n} else {\npre!.right = child;\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet next = getInOrderNext(cur.right);\nlet tmp = next!.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(next!.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunction getInOrderNext(root: TreeNode | null): TreeNode | null {\nif (root === null) {\nreturn null;\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left !== null) {\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.c
    [class]{binarySearchTree}-[func]{remove}\n[class]{binarySearchTree}-[func]{getInOrderNext}\n
     binary_search_tree.cs
    /* \u5220\u9664\u7ed3\u70b9 */\nTreeNode? remove(int num)\n{\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null) return null;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null)\n{\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null || pre == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null)\n{\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nTreeNode? child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.left == cur)\n{\npre.left = child;\n}\nelse\n{\npre.right = child;\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse\n{\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nTreeNode? nex = getInOrderNext(cur.right);\nif (nex != null)\n{\nint tmp = nex.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(nex.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.val = tmp;\n}\n}\nreturn cur;\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nTreeNode? getInOrderNext(TreeNode? root)\n{\nif (root == null) return root;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (root.left != null)\n{\nroot = root.left;\n}\nreturn root;\n}\n
     binary_search_tree.swift
    /* \u5220\u9664\u7ed3\u70b9 */\n@discardableResult\nfunc remove(num: Int) -> TreeNode? {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn nil\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur!.val == num {\nbreak\n}\npre = cur\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn nil\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif cur?.left == nil || cur?.right == nil {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nlet child = cur?.left != nil ? cur?.left : cur?.right\n// \u5220\u9664\u7ed3\u70b9 cur\nif pre?.left === cur {\npre?.left = child\n} else {\npre?.right = child\n}\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nlet nex = getInOrderNext(root: cur?.right)\nlet tmp = nex!.val\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\nremove(num: nex!.val)\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur?.val = tmp\n}\nreturn cur\n}\n/* \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09 */\nfunc getInOrderNext(root: TreeNode?) -> TreeNode? {\nvar root = root\nif root == nil {\nreturn root\n}\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile root?.left != nil {\nroot = root?.left\n}\nreturn root\n}\n
     binary_search_tree.zig
    // \u5220\u9664\u7ed3\u70b9\nfn remove(self: *Self, num: T) ?*inc.TreeNode(T) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return null;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.?.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u5f85\u5220\u9664\u7ed3\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return null;\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 0 or 1\nif (cur.?.left == null or cur.?.right == null) {\n// \u5f53\u5b50\u7ed3\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u7ed3\u70b9\nvar child = if (cur.?.left != null) cur.?.left else cur.?.right;\n// \u5220\u9664\u7ed3\u70b9 cur\nif (pre.?.left == cur) {\npre.?.left = child;\n} else {\npre.?.right = child;\n}\n// \u5b50\u7ed3\u70b9\u6570\u91cf = 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\nvar nex = self.getInOrderNext(cur.?.right);\nvar tmp = nex.?.val;\n// \u9012\u5f52\u5220\u9664\u7ed3\u70b9 nex\n_ = self.remove(nex.?.val);\n// \u5c06 nex \u7684\u503c\u590d\u5236\u7ed9 cur\ncur.?.val = tmp;\n}\nreturn cur;\n}\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u4e0b\u4e00\u4e2a\u7ed3\u70b9\uff08\u4ec5\u9002\u7528\u4e8e root \u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u60c5\u51b5\uff09\nfn getInOrderNext(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n_ = self;\nvar node_tmp = node;\nif (node_tmp == null) return null;\n// \u5faa\u73af\u8bbf\u95ee\u5de6\u5b50\u7ed3\u70b9\uff0c\u76f4\u5230\u53f6\u7ed3\u70b9\u65f6\u4e3a\u6700\u5c0f\u7ed3\u70b9\uff0c\u8df3\u51fa\nwhile (node_tmp.?.left != null) {\nnode_tmp = node_tmp.?.left;\n}\nreturn node_tmp;\n}\n
    "},{"location":"chapter_tree/binary_search_tree/#_4","title":"\u6392\u5e8f","text":"
    \u6211\u4eec\u77e5\u9053\uff0c\u300c\u4e2d\u5e8f\u904d\u5386\u300d\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u9075\u5faa\u201c\u5de6\u5b50\u7ed3\u70b9 \\(<\\) \u6839\u7ed3\u70b9 \\(<\\) \u53f3\u5b50\u7ed3\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u7ed3\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u6761\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002
     
    \u501f\u52a9\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u800c\u65e0\u9700\u989d\u5916\u6392\u5e8f\uff0c\u975e\u5e38\u9ad8\u6548\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 
    "},{"location":"chapter_tree/binary_search_tree/#732","title":"7.3.2. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"
    \u5047\u8bbe\u7ed9\u5b9a \\(n\\) \u4e2a\u6570\u5b57\uff0c\u6700\u5e38\u7528\u7684\u5b58\u50a8\u65b9\u5f0f\u662f\u300c\u6570\u7ec4\u300d\uff0c\u90a3\u4e48\u5bf9\u4e8e\u8fd9\u4e32\u4e71\u5e8f\u7684\u6570\u5b57\uff0c\u5e38\u89c1\u64cd\u4f5c\u7684\u6548\u7387\u4e3a\uff1a
     
       
    • \u67e5\u627e\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u65e0\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u6570\u7ec4\u6765\u786e\u5b9a\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u63d2\u5165\u5143\u7d20\uff1a\u53ea\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u5148\u67e5\u627e\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u518d\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u8be5\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20\uff1a\u9700\u8981\u904d\u5386\u6570\u7ec4\u6765\u786e\u5b9a\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
     
    \u4e3a\u4e86\u5f97\u5230\u5148\u9a8c\u4fe1\u606f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u9884\u5148\u5c06\u6570\u7ec4\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230\u4e00\u4e2a\u300c\u6392\u5e8f\u6570\u7ec4\u300d\uff0c\u6b64\u65f6\u64cd\u4f5c\u6548\u7387\u4e3a\uff1a
     
       
    • \u67e5\u627e\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u5e73\u5747\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff1b
      •  
    • \u63d2\u5165\u5143\u7d20\uff1a\u5148\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u518d\u63d2\u5165\u5230\u6307\u5b9a\u4f4d\u7f6e\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u5220\u9664\u5143\u7d20\uff1a\u5148\u67e5\u627e\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u518d\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u8be5\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
      •  
    • \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20\uff1a\u6570\u7ec4\u5934\u90e8\u548c\u5c3e\u90e8\u5143\u7d20\u5373\u662f\u6700\u5c0f\u548c\u6700\u5927\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
      •  
     
    \u89c2\u5bdf\u53d1\u73b0\uff0c\u65e0\u5e8f\u6570\u7ec4\u548c\u6709\u5e8f\u6570\u7ec4\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u504f\u79d1\u201d\u7684\uff0c\u5373\u6709\u7684\u5feb\u6709\u7684\u6162\uff1b\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5728\u6570\u636e\u91cf \\(n\\) \u5f88\u5927\u65f6\u6709\u5de8\u5927\u4f18\u52bf\u3002
     \u65e0\u5e8f\u6570\u7ec4 \u6709\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u6307\u5b9a\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \u83b7\u53d6\u6700\u5c0f / \u6700\u5927\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \\(O(\\log n)\\)"},{"location":"chapter_tree/binary_search_tree/#733","title":"7.3.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u9000\u5316","text":"
    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u662f\u201c\u5de6\u53f3\u5e73\u8861\u201d\u7684\uff08\u8be6\u89c1\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u7ae0\u8282\uff09\uff0c\u6b64\u65f6\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u7ed3\u70b9\u3002
     
    \u5982\u679c\u6211\u4eec\u52a8\u6001\u5730\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\uff0c\u5219\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u6b64\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u9000\u5316\u4e4b \\(O(n)\\) \u3002
     
    Note
     
    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5982\u4f55\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\uff0c\u4e5f\u662f\u4e00\u4e2a\u9700\u8981\u91cd\u8981\u8003\u8651\u7684\u95ee\u9898\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\u4e0e\u9000\u5316 
    "},{"location":"chapter_tree/binary_search_tree/#734","title":"7.3.4. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"
       
    • \u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u9ad8\u6548\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
      •  
    • \u5404\u79cd\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
      •  
    • \u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4fdd\u6301\u5176\u5df2\u6392\u5e8f\u3002
      •  
    "},{"location":"chapter_tree/binary_tree/","title":"7.1. \u00a0 \u4e8c\u53c9\u6811","text":"
    \u300c\u4e8c\u53c9\u6811 Binary Tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u7c7b\u4f3c\u4e8e\u94fe\u8868\uff0c\u4e8c\u53c9\u6811\u4e5f\u662f\u4ee5\u7ed3\u70b9\u4e3a\u5355\u4f4d\u5b58\u50a8\u7684\uff0c\u7ed3\u70b9\u5305\u542b\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u7ed3\u70b9\u503c\nTreeNode left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;          // \u7ed3\u70b9\u503c\nTreeNode *left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode *right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n
     
    \"\"\" \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b \"\"\"\nclass TreeNode:\ndef __init__(self, val: int):\nself.val: int = val                   # \u7ed3\u70b9\u503c\nself.left: Optional[TreeNode] = None  # \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nself.right: Optional[TreeNode] = None # \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal   int\nLeft  *TreeNode\nRight *TreeNode\n}\n/* \u7ed3\u70b9\u521d\u59cb\u5316\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\nreturn &TreeNode{\nLeft:  nil,\nRight: nil,\nVal:   v,\n}\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nfunction TreeNode(val, left, right) {\nthis.val = (val === undefined ? 0 : val); // \u7ed3\u70b9\u503c\nthis.left = (left === undefined ? null : left); // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nthis.right = (right === undefined ? null : right); // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nval: number;\nleft: TreeNode | null;\nright: TreeNode | null;\nconstructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val; // \u7ed3\u70b9\u503c\nthis.left = left === undefined ? null : left; // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nthis.right = right === undefined ? null : right; // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\n}\n}\n
     
    \n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nint val;          // \u7ed3\u70b9\u503c\nTreeNode? left;   // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode? right;  // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
     
    /* \u4e8c\u53c9\u6811\u7ed3\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u7ed3\u70b9\u503c\nvar left: TreeNode? // \u5de6\u5b50\u7ed3\u70b9\u6307\u9488\nvar right: TreeNode? // \u53f3\u5b50\u7ed3\u70b9\u6307\u9488\ninit(x: Int) {\nval = x\n}\n}\n
     
    \n
     
    \u7ed3\u70b9\u7684\u4e24\u4e2a\u6307\u9488\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u7ed3\u70b9 Left Child Node\u300d\u548c\u300c\u53f3\u5b50\u7ed3\u70b9 Right Child Node\u300d\uff0c\u5e76\u4e14\u79f0\u8be5\u7ed3\u70b9\u4e3a\u4e24\u4e2a\u5b50\u7ed3\u70b9\u7684\u300c\u7236\u7ed3\u70b9 Parent Node\u300d\u3002\u7ed9\u5b9a\u4e8c\u53c9\u6811\u67d0\u7ed3\u70b9\uff0c\u5c06\u5de6\u5b50\u7ed3\u70b9\u4ee5\u4e0b\u7684\u6811\u79f0\u4e3a\u8be5\u7ed3\u70b9\u7684\u300c\u5de6\u5b50\u6811 Left Subtree\u300d\uff0c\u53f3\u5b50\u6811\u540c\u7406\u3002
     
    \u9664\u4e86\u53f6\u7ed3\u70b9\u5916\uff0c\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u6709\u5b50\u7ed3\u70b9\u548c\u5b50\u6811\u3002\u4f8b\u5982\uff0c\u82e5\u5c06\u4e0b\u56fe\u7684\u300c\u7ed3\u70b9 2\u300d\u770b\u4f5c\u7236\u7ed3\u70b9\uff0c\u90a3\u4e48\u5176\u5de6\u5b50\u7ed3\u70b9\u548c\u53f3\u5b50\u7ed3\u70b9\u5206\u522b\u4e3a\u300c\u7ed3\u70b9 4\u300d\u548c\u300c\u7ed3\u70b9 5\u300d\uff0c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5206\u522b\u4e3a\u300c\u7ed3\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u7ed3\u70b9\u5f62\u6210\u7684\u6811\u300d\u548c\u300c\u7ed3\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u7ed3\u70b9\u5f62\u6210\u7684\u6811\u300d\u3002
     
     
     Fig. \u7236\u7ed3\u70b9\u3001\u5b50\u7ed3\u70b9\u3001\u5b50\u6811 
    "},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1. \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"
    \u4e8c\u53c9\u6811\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5efa\u8bae\u5c3d\u91cf\u7406\u89e3\u5e76\u8bb0\u4f4f\u3002\u540e\u7eed\u53ef\u80fd\u9057\u5fd8\uff0c\u53ef\u4ee5\u5728\u9700\u8981\u4f7f\u7528\u65f6\u56de\u6765\u67e5\u770b\u786e\u8ba4\u3002
     
       
    • \u300c\u6839\u7ed3\u70b9 Root Node\u300d\uff1a\u4e8c\u53c9\u6811\u6700\u9876\u5c42\u7684\u7ed3\u70b9\uff0c\u5176\u6ca1\u6709\u7236\u7ed3\u70b9\uff1b
      •  
    • \u300c\u53f6\u7ed3\u70b9 Leaf Node\u300d\uff1a\u6ca1\u6709\u5b50\u7ed3\u70b9\u7684\u7ed3\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u90fd\u6307\u5411 \\(\\text{null}\\) \uff1b
      •  
    • \u7ed3\u70b9\u6240\u5904\u300c\u5c42 Level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u4f9d\u6b21\u589e\u52a0\uff0c\u6839\u7ed3\u70b9\u6240\u5904\u5c42\u4e3a 1 \uff1b
      •  
    • \u7ed3\u70b9\u300c\u5ea6 Degree\u300d\uff1a\u7ed3\u70b9\u7684\u5b50\u7ed3\u70b9\u6570\u91cf\u3002\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u8303\u56f4\u662f 0, 1, 2 \uff1b
      •  
    • \u300c\u8fb9 Edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u7ed3\u70b9\u7684\u8fb9\uff0c\u5373\u7ed3\u70b9\u6307\u9488\uff1b
      •  
    • \u4e8c\u53c9\u6811\u300c\u9ad8\u5ea6\u300d\uff1a\u4e8c\u53c9\u6811\u4e2d\u6839\u7ed3\u70b9\u5230\u6700\u8fdc\u53f6\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
    • \u7ed3\u70b9\u300c\u6df1\u5ea6 Depth\u300d \uff1a\u6839\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
    • \u7ed3\u70b9\u300c\u9ad8\u5ea6 Height\u300d\uff1a\u6700\u8fdc\u53f6\u7ed3\u70b9\u5230\u8be5\u7ed3\u70b9\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\uff1b
      •  
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed 
     
    \u9ad8\u5ea6\u4e0e\u6df1\u5ea6\u7684\u5b9a\u4e49
     
    \u503c\u5f97\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u300c\u9ad8\u5ea6\u300d\u548c\u300c\u6df1\u5ea6\u300d\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u800c\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u7ed3\u70b9\u7684\u6570\u91cf\u201d\uff0c\u6b64\u65f6\u9ad8\u5ea6\u6216\u6df1\u5ea6\u90fd\u9700\u8981 + 1 \u3002
    "},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2. \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":"
    \u521d\u59cb\u5316\u4e8c\u53c9\u6811\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5148\u521d\u59cb\u5316\u7ed3\u70b9\uff0c\u518d\u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree.java
    // \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.cpp
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
     binary_tree.py
    \"\"\" \u521d\u59cb\u5316\u4e8c\u53c9\u6811 \"\"\"\n# \u521d\u59cb\u5316\u7ed3\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
     binary_tree.go
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
     binary_tree.js
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.ts
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.c
    \n
     binary_tree.cs
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u7ed3\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
     binary_tree.swift
    // \u521d\u59cb\u5316\u7ed3\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
     binary_tree.zig
    \n
     
    \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u90fd\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u5b9e\u73b0\u3002
     
     
     Fig. \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree.java
    TreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.cpp
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1->left = n2;\n
     binary_tree.py
    \"\"\" \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 \"\"\"\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2\n
     binary_tree.go
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u7ed3\u70b9 P\nn1.Left = n2\n
     binary_tree.js
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.ts
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.c
    \n
     binary_tree.cs
    /* \u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2;\n
     binary_tree.swift
    let P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u7ed3\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u7ed3\u70b9 P\nn1.left = n2\n
     binary_tree.zig
    \n
     
    Note
     
    \u63d2\u5165\u7ed3\u70b9\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u5220\u9664\u7ed3\u70b9\u5f80\u5f80\u610f\u5473\u7740\u5220\u9664\u4e86\u8be5\u7ed3\u70b9\u7684\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u90fd\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u5b9e\u73b0\u6709\u610f\u4e49\u7684\u64cd\u4f5c\u3002
    "},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3. \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#_1","title":"\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 Perfect Binary Tree\u300d\u7684\u6240\u6709\u5c42\u7684\u7ed3\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u6240\u6709\u7ed3\u70b9\u7684\u5ea6 = 2 \uff1b\u82e5\u6811\u9ad8\u5ea6 \\(= h\\) \uff0c\u5219\u7ed3\u70b9\u603b\u6570 \\(= 2^{h+1} - 1\\) \uff0c\u5448\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u7740\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u3002
     
    Tip
     
    \u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\uff0c\u8bf7\u6ce8\u610f\u4e0e\u5b8c\u6ee1\u4e8c\u53c9\u6811\u533a\u5206\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_2","title":"\u5b8c\u5168\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u5168\u4e8c\u53c9\u6811 Complete Binary Tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u7ed3\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u7ed3\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002
     
    \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u5982\u679c\u6309\u7167\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u6765\u5b58\u50a8\uff0c\u90a3\u4e48\u7a7a\u7ed3\u70b9 null \u4e00\u5b9a\u5168\u90e8\u51fa\u73b0\u5728\u5e8f\u5217\u7684\u5c3e\u90e8\uff0c\u56e0\u6b64\u6211\u4eec\u5c31\u53ef\u4ee5\u4e0d\u7528\u5b58\u50a8\u8fd9\u4e9b null \u4e86\u3002
     
     
     Fig. \u5b8c\u5168\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_3","title":"\u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"
    \u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 Full Binary Tree\u300d\u9664\u4e86\u53f6\u7ed3\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u7ed3\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u7ed3\u70b9\u3002
     
     
     Fig. \u5b8c\u6ee1\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#_4","title":"\u5e73\u8861\u4e8c\u53c9\u6811","text":"
    \u300c\u5e73\u8861\u4e8c\u53c9\u6811 Balanced Binary Tree\u300d\u4e2d\u4efb\u610f\u7ed3\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c \\(\\leq 1\\) \u3002
     
     
     Fig. \u5e73\u8861\u4e8c\u53c9\u6811 
    "},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4. \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"
    \u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u7684\u7ed3\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff1b\u800c\u5f53\u6240\u6709\u7ed3\u70b9\u90fd\u504f\u5411\u4e00\u8fb9\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u300c\u94fe\u8868\u300d\u3002
     
       
    • \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u201c\u6700\u4f73\u72b6\u6001\u201d\uff0c\u53ef\u4ee5\u5b8c\u5168\u53d1\u6325\u51fa\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\uff1b
      •  
    • \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \uff1b
      •  
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u4e0e\u6700\u4e8c\u53c9\u6811\u7684\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u5dee\u60c5\u51b5 
     
    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5728\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u7ed3\u70b9\u6570\u91cf\u3001\u7ed3\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u6216\u6781\u5c0f\u503c\u3002
     \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u7ed3\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u53f6\u7ed3\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u7ed3\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u6811\u7684\u7ed3\u70b9\u603b\u6570\u4e3a \\(n\\) \u65f6\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree/#715","title":"7.1.5. \u00a0 \u4e8c\u53c9\u6811\u8868\u793a\u65b9\u5f0f *","text":"
    \u6211\u4eec\u4e00\u822c\u4f7f\u7528\u4e8c\u53c9\u6811\u7684\u300c\u94fe\u8868\u8868\u793a\u300d\uff0c\u5373\u5b58\u50a8\u5355\u4f4d\u4e3a\u7ed3\u70b9 TreeNode \uff0c\u7ed3\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\uff08\u5f15\u7528\uff09\u76f8\u8fde\u63a5\u3002\u672c\u6587\u524d\u8ff0\u793a\u4f8b\u4ee3\u7801\u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u5728\u94fe\u8868\u8868\u793a\u4e0b\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002
     
    \u90a3\u80fd\u5426\u53ef\u4ee5\u7528\u300c\u6570\u7ec4\u8868\u793a\u300d\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002\u5148\u6765\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff0c\u5c06\u7ed3\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u7f16\u53f7\uff08\u4ece 0 \u5f00\u59cb\uff09\uff0c\u90a3\u4e48\u53ef\u4ee5\u63a8\u5bfc\u5f97\u51fa\u7236\u7ed3\u70b9\u7d22\u5f15\u4e0e\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u300c\u6620\u5c04\u516c\u5f0f\u300d\uff1a\u8bbe\u7ed3\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u7ed3\u70b9\u7684\u5de6\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \u3001\u53f3\u5b50\u7ed3\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002
     
    \u672c\u8d28\u4e0a\uff0c\u6620\u5c04\u516c\u5f0f\u7684\u4f5c\u7528\u5c31\u662f\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u5bf9\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u7684\u4efb\u610f\u7ed3\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u4f7f\u7528\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b50\u7ed3\u70b9\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\uff08\u5373\u6570\u7ec4\uff09\u6765\u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3002
     
     
     Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u7136\u800c\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u53ea\u662f\u4e2a\u4f8b\uff0c\u4e8c\u53c9\u6811\u4e2d\u95f4\u5c42\u5f80\u5f80\u5b58\u5728\u8bb8\u591a\u7a7a\u7ed3\u70b9\uff08\u5373 null \uff09\uff0c\u800c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b\u7a7a\u7ed3\u70b9\uff0c\u5e76\u4e14\u6211\u4eec\u65e0\u6cd5\u5355\u51ed\u5e8f\u5217\u6765\u731c\u6d4b\u7a7a\u7ed3\u70b9\u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u5373\u7406\u8bba\u4e0a\u5b58\u5728\u8bb8\u591a\u79cd\u4e8c\u53c9\u6811\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002\u663e\u7136\uff0c\u8fd9\u79cd\u60c5\u51b5\u65e0\u6cd5\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u4e8c\u53c9\u6811\u3002
     
     
     Fig. \u7ed9\u5b9a\u6570\u7ec4\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 
     
    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8003\u8651\u6309\u7167\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5f62\u5f0f\u6765\u8868\u793a\u6240\u6709\u4e8c\u53c9\u6811\uff0c\u5373\u5728\u5e8f\u5217\u4e2d\u4f7f\u7528\u7279\u6b8a\u7b26\u53f7\u6765\u663e\u5f0f\u5730\u8868\u793a\u201c\u7a7a\u4f4d\u201d\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5e8f\u5217\uff08\u6570\u7ec4\uff09\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig 
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4e3a\u4e86\u7b26\u5408\u6570\u636e\u7c7b\u578b\u4e3a int \uff0c\u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\n// \u8be5\u65b9\u6cd5\u7684\u4f7f\u7528\u524d\u63d0\u662f\u6ca1\u6709\u7ed3\u70b9\u7684\u503c = INT_MAX\nvector<int> tree = { 1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15 };\n
     
    \"\"\" \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a \"\"\"\n# \u76f4\u63a5\u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
     
    \n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
     
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n
     
    \n
     
     
     Fig. \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u56de\u987e\u300c\u5b8c\u5168\u4e8c\u53c9\u6811\u300d\u7684\u5b9a\u4e49\uff0c\u5176\u53ea\u6709\u6700\u5e95\u5c42\u6709\u7a7a\u7ed3\u70b9\uff0c\u5e76\u4e14\u6700\u5e95\u5c42\u7684\u7ed3\u70b9\u5c3d\u91cf\u9760\u5de6\uff0c\u56e0\u800c\u6240\u6709\u7a7a\u7ed3\u70b9\u90fd\u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002\u56e0\u4e3a\u6211\u4eec\u5148\u9a8c\u5730\u786e\u5b9a\u4e86\u7a7a\u4f4d\u7684\u4f4d\u7f6e\uff0c\u6240\u4ee5\u5728\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u201c\u7a7a\u4f4d\u201d\u3002\u56e0\u6b64\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002
     
     
     Fig. \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a 
     
    \u6570\u7ec4\u8868\u793a\u6709\u4e24\u4e2a\u4f18\u70b9\uff1a \u4e00\u662f\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u8282\u7701\u7a7a\u95f4\uff1b\u4e8c\u662f\u53ef\u4ee5\u968f\u673a\u8bbf\u95ee\u7ed3\u70b9\u3002\u7136\u800c\uff0c\u5f53\u4e8c\u53c9\u6811\u4e2d\u7684\u201c\u7a7a\u4f4d\u201d\u5f88\u591a\u65f6\uff0c\u6570\u7ec4\u4e2d\u53ea\u5305\u542b\u5f88\u5c11\u7ed3\u70b9\u7684\u6570\u636e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u5f88\u4f4e\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2. \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"
    \u4ece\u7269\u7406\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u904d\u5386\u65b9\u5f0f\u4e5f\u662f\u901a\u8fc7\u6307\u9488\uff08\u5373\u5f15\u7528\uff09\u9010\u4e2a\u904d\u5386\u7ed3\u70b9\u3002\u540c\u65f6\uff0c\u6811\u8fd8\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u5bfc\u81f4\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u4f7f\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002
     
    \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u5f0f\u6709\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1. \u00a0 \u5c42\u5e8f\u904d\u5386","text":"
    \u300c\u5c42\u5e8f\u904d\u5386 Level-Order Traversal\u300d\u4ece\u9876\u81f3\u5e95\u3001\u4e00\u5c42\u4e00\u5c42\u5730\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u5c42\u4e2d\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u7ed3\u70b9\u3002
     
    \u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u662f\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Traversal\u300d\uff0c\u5176\u4f53\u73b0\u7740\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5c42\u8fdb\u904d\u5386\u65b9\u5f0f\u3002
     
     
     Fig. \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 
    "},{"location":"chapter_tree/binary_tree_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"
    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u4e00\u822c\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u7684\u89c4\u5219\u662f\u201c\u5148\u8fdb\u5148\u51fa\u201d\uff0c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u89c4\u5219\u662f \u201d\u4e00\u5c42\u5c42\u5e73\u63a8\u201c \uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002
     JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree_bfs.java
    /* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nQueue<TreeNode> queue = new LinkedList<>() {{ add(root); }};\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<Integer> list = new ArrayList<>();\nwhile (!queue.isEmpty()) {\nTreeNode node = queue.poll();  // \u961f\u5217\u51fa\u961f\nlist.add(node.val);            // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null)\nqueue.offer(node.left);    // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right != null)\nqueue.offer(node.right);   // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.cpp
    /* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode* root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue<TreeNode*> queue;\nqueue.push(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvector<int> vec;\nwhile (!queue.empty()) {\nTreeNode* node = queue.front();\nqueue.pop();                 // \u961f\u5217\u51fa\u961f\nvec.push_back(node->val);    // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node->left != nullptr)\nqueue.push(node->left);  // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node->right != nullptr)\nqueue.push(node->right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn vec;\n}\n
     binary_tree_bfs.py
    def level_order(root: Optional[TreeNode]) -> List[int]:\n\"\"\" \u5c42\u5e8f\u904d\u5386 \"\"\"\n# \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue: Deque[TreeNode] = collections.deque()\nqueue.append(root)\n# \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nres: List[int] = []\nwhile queue:\nnode: TreeNode = queue.popleft() # \u961f\u5217\u51fa\u961f\nres.append(node.val)             # \u4fdd\u5b58\u7ed3\u70b9\u503c\nif node.left is not None:\nqueue.append(node.left)      # \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif node.right is not None:\nqueue.append(node.right)     # \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\nreturn res\n
     binary_tree_bfs.go
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []int {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nqueue := list.New()\nqueue.PushBack(root)\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nnums := make([]int, 0)\nfor queue.Len() > 0 {\n// \u961f\u5217\u51fa\u961f\nnode := queue.Remove(queue.Front()).(*TreeNode)\n// \u4fdd\u5b58\u7ed3\u70b9\u503c\nnums = append(nums, node.Val)\nif node.Left != nil {\n// \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nqueue.PushBack(node.Left)\n}\nif node.Right != nil {\n// \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\nqueue.PushBack(node.Right)\n}\n}\nreturn nums\n}\n
     binary_tree_bfs.js
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list = [];\nwhile (queue.length) {\nlet node = queue.shift();   // \u961f\u5217\u51fa\u961f\nlist.push(node.val);        // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left)\nqueue.push(node.left);  // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right)\nqueue.push(node.right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.ts
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list: number[] = [];\nwhile (queue.length) {\nlet node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left) {\nqueue.push(node.left); // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif (node.right) {\nqueue.push(node.right); // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\n}\nreturn list;\n}\n
     binary_tree_bfs.c
    [class]{}-[func]{levelOrder}\n
     binary_tree_bfs.cs
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode root)\n{\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nQueue<TreeNode> queue = new();\nqueue.Enqueue(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> list = new();\nwhile (queue.Count != 0)\n{\nTreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\nlist.Add(node.val);              // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null)\nqueue.Enqueue(node.left);    // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\nif (node.right != null)\nqueue.Enqueue(node.right);   // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\nreturn list;\n}\n
     binary_tree_bfs.swift
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nvar queue: [TreeNode] = [root]\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list: [Int] = []\nwhile !queue.isEmpty {\nlet node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\nlist.append(node.val) // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif let left = node.left {\nqueue.append(left) // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif let right = node.right {\nqueue.append(right) // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}\n}\nreturn list\n}\n
     binary_tree_bfs.zig
    // \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u7ed3\u70b9\nconst L = std.TailQueue(*inc.TreeNode(T));\nvar queue = L{};\nvar root_node = try mem_allocator.create(L.Node);\nroot_node.data = root;\nqueue.append(root_node); // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list = std.ArrayList(T).init(std.heap.page_allocator);\nwhile (queue.len > 0) {\nvar queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\nvar node = queue_node.data;\ntry list.append(node.val);              // \u4fdd\u5b58\u7ed3\u70b9\u503c\nif (node.left != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.left.?;\nqueue.append(tmp_node);             // \u5de6\u5b50\u7ed3\u70b9\u5165\u961f\n}\nif (node.right != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.right.?;\nqueue.append(tmp_node);             // \u53f3\u5b50\u7ed3\u70b9\u5165\u961f\n}        }\nreturn list;\n}\n
    "},{"location":"chapter_tree/binary_tree_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u7ed3\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7ed3\u70b9\u6570\u91cf\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5f53\u4e3a\u6ee1\u4e8c\u53c9\u6811\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\(\\frac{n + 1}{2}\\) \u4e2a\u7ed3\u70b9\uff0c\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2. \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"
    \u76f8\u5bf9\u5730\uff0c\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u5176\u4f53\u73b0\u7740\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\u3002
     
    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5de6\u4fa7\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7684\u793a\u610f\u56fe\uff0c\u53f3\u4e0a\u65b9\u662f\u5bf9\u5e94\u7684\u9012\u5f52\u5b9e\u73b0\u4ee3\u7801\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u4e2a\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u8d70\u7684\u8fc7\u7a0b\u4e2d\uff0c\u5728\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3002
     
     
     Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386 
     \u4f4d\u7f6e \u542b\u4e49 \u6b64\u5904\u8bbf\u95ee\u7ed3\u70b9\u65f6\u5bf9\u5e94 \u6a59\u8272\u5706\u5708\u5904 \u521a\u8fdb\u5165\u6b64\u7ed3\u70b9\uff0c\u5373\u5c06\u8bbf\u95ee\u8be5\u7ed3\u70b9\u7684\u5de6\u5b50\u6811 \u524d\u5e8f\u904d\u5386 Pre-Order Traversal \u84dd\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\uff0c\u5373\u5c06\u8bbf\u95ee\u53f3\u5b50\u6811 \u4e2d\u5e8f\u904d\u5386 In-Order Traversal \u7d2b\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff0c\u5373\u5c06\u8fd4\u56de \u540e\u5e8f\u904d\u5386 Post-Order Traversal"},{"location":"chapter_tree/binary_tree_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZig binary_tree_dfs.java
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.add(root.val);\n}\n
     binary_tree_dfs.cpp
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nvec.push_back(root->val);\npreOrder(root->left);\npreOrder(root->right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left);\nvec.push_back(root->val);\ninOrder(root->right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode* root) {\nif (root == nullptr) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root->left);\npostOrder(root->right);\nvec.push_back(root->val);\n}\n
     binary_tree_dfs.py
    def pre_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u524d\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nres.append(root.val)\npre_order(root=root.left)\npre_order(root=root.right)\ndef in_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u4e2d\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\nin_order(root=root.left)\nres.append(root.val)\nin_order(root=root.right)\ndef post_order(root: Optional[TreeNode]) -> None:\n\"\"\" \u540e\u5e8f\u904d\u5386 \"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npost_order(root=root.left)\npost_order(root=root.right)\nres.append(root.val)\n
     binary_tree_dfs.go
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nnums = append(nums, node.Val)\npreOrder(node.Left)\npreOrder(node.Right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.Left)\nnums = append(nums, node.Val)\ninOrder(node.Right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(node.Left)\npostOrder(node.Right)\nnums = append(nums, node.Val)\n}\n
     binary_tree_dfs.js
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
     binary_tree_dfs.ts
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
     binary_tree_dfs.c
    [class]{}-[func]{preOrder}\n[class]{}-[func]{inOrder}\n[class]{}-[func]{postOrder}\n
     binary_tree_dfs.cs
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.Add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.Add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? root)\n{\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.Add(root.val);\n}\n
     binary_tree_dfs.swift
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.append(root.val)\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root: root.left)\nlist.append(root.val)\ninOrder(root: root.right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\npostOrder(root: root.left)\npostOrder(root: root.right)\nlist.append(root.val)\n}\n
     binary_tree_dfs.zig
    // \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u7ed3\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\ntry list.append(root.?.val);\ntry preOrder(T, root.?.left);\ntry preOrder(T, root.?.right);\n}\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u7ed3\u70b9 -> \u53f3\u5b50\u6811\ntry inOrder(T, root.?.left);\ntry list.append(root.?.val);\ntry inOrder(T, root.?.right);\n}\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u7ed3\u70b9\ntry postOrder(T, root.?.left);\ntry postOrder(T, root.?.right);\ntry list.append(root.?.val);\n}\n
     
    Note
     
    \u4f7f\u7528\u5faa\u73af\u4e00\u6837\u53ef\u4ee5\u5b9e\u73b0\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\uff0c\u4f46\u4ee3\u7801\u76f8\u5bf9\u7e41\u7410\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002
    "},{"location":"chapter_tree/binary_tree_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"
    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u7ed3\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7ed3\u70b9\u6570\u91cf\u3002
     
    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5f53\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
    "},{"location":"chapter_tree/summary/","title":"7.5. \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#_1","title":"\u4e8c\u53c9\u6811","text":"
       
    • \u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u5305\u542b\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\uff0c\u5206\u522b\u6307\u5411\u5de6\u5b50\u7ed3\u70b9\u548c\u53f3\u5b50\u7ed3\u70b9\u3002
      •  
    • \u9009\u5b9a\u4e8c\u53c9\u6811\u4e2d\u67d0\u7ed3\u70b9\uff0c\u5c06\u5176\u5de6\uff08\u53f3\uff09\u5b50\u7ed3\u70b9\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002
      •  
    • \u4e8c\u53c9\u6811\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5305\u62ec\u6839\u7ed3\u70b9\u3001\u53f6\u7ed3\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u3001\u6df1\u5ea6\u7b49\u3002
      •  
    • \u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u7ed3\u70b9\u63d2\u5165\u3001\u7ed3\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002
      •  
    • \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u5305\u62ec\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u3001\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u72b6\u6001\uff0c\u94fe\u8868\u5219\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002
      •  
    • \u4e8c\u53c9\u6811\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u8868\u793a\uff0c\u5177\u4f53\u505a\u6cd5\u662f\u5c06\u7ed3\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u57fa\u4e8e\u7236\u7ed3\u70b9\u548c\u5b50\u7ed3\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5b9e\u73b0\u6307\u9488\u3002
      •  
    "},{"location":"chapter_tree/summary/#_2","title":"\u4e8c\u53c9\u6811\u904d\u5386","text":"
       
    • \u4e8c\u53c9\u6811\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5c42\u8fdb\u5f0f\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\u3002
      •  
    • \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f53\u73b0\u7740\u201c\u8d70\u5230\u5934\u3001\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002
      •  
    "},{"location":"chapter_tree/summary/#_3","title":"\u4e8c\u53c9\u641c\u7d22\u6811","text":"
       
    • \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u540e\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \uff0c\u56e0\u6b64\u5982\u4f55\u907f\u514d\u9000\u5316\u662f\u975e\u5e38\u91cd\u8981\u7684\u8bfe\u9898\u3002
      •  
    • AVL \u6811\u53c8\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5176\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u5f97\u5728\u4e0d\u65ad\u63d2\u5165\u4e0e\u5220\u9664\u7ed3\u70b9\u540e\uff0c\u4ecd\u7136\u53ef\u4ee5\u4fdd\u6301\u4e8c\u53c9\u6811\u7684\u5e73\u8861\uff08\u4e0d\u9000\u5316\uff09\u3002
      •  
    • AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u7ed3\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u6062\u590d\u5e73\u8861\u3002
      •  
    "}]}
\ No newline at end of file
diff --git a/sitemap.xml.gz b/sitemap.xml.gz
index 888c0af47703cede75fd3cb2739c77d17f3ced46..d88693c20099604dc987dcb5e38db7b07c37e418 100644
GIT binary patch
delta 15
WcmZo;ZDVDV@8;leOy0=WzytstSp&2H

delta 15
WcmZo;ZDVDV@8;l$2-wKhzytstgafz$