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<li class="md-nav__item"> <li class="md-nav__item">
<a href="#351" class="md-nav__link"> <a href="#351" class="md-nav__link">
3.5.1. &nbsp; 快速回顾 3.5.1. &nbsp; 知识回顾
</a> </a>
<nav class="md-nav" aria-label="3.5.1. &nbsp; 快速回顾"> <nav class="md-nav" aria-label="3.5.1. &nbsp; 知识回顾">
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<li class="md-nav__item"> <li class="md-nav__item">
<a href="#351" class="md-nav__link"> <a href="#351" class="md-nav__link">
3.5.1. &nbsp; 快速回顾 3.5.1. &nbsp; 知识回顾
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<ul class="md-nav__list"> <ul class="md-nav__list">
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<h1 id="35">3.5. &nbsp; 小结<a class="headerlink" href="#35" title="Permanent link">&para;</a></h1> <h1 id="35">3.5. &nbsp; 小结<a class="headerlink" href="#35" title="Permanent link">&para;</a></h1>
<h2 id="351">3.5.1. &nbsp; 快速回顾<a class="headerlink" href="#351" title="Permanent link">&para;</a></h2> <h2 id="351">3.5.1. &nbsp; 知识回顾<a class="headerlink" href="#351" title="Permanent link">&para;</a></h2>
<h3 id="_1">数据结构分类<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3> <h3 id="_1">数据结构分类<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<ul> <ul>
<li>数据结构可以从逻辑结构和物理结构两个角度进行分类。逻辑结构描述了数据元素之间的逻辑关系，而物理结构描述了数据在计算机内存中的存储方式。</li> <li>数据结构可以从逻辑结构和物理结构两个角度进行分类。逻辑结构描述了数据元素之间的逻辑关系，而物理结构描述了数据在计算机内存中的存储方式。</li>

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</ol> </ol>
<p>以学生数据 <code>key 学号 -&gt; value 姓名</code> 为例，我们可以设计如下哈希函数：</p> <p>以学生数据 <code>key 学号 -&gt; value 姓名</code> 为例，我们可以设计如下哈希函数：</p>
<div class="arithmatex">\[ <div class="arithmatex">\[
f(x) = x \% 100 f(x) = x \bmod {100}
\]</div> \]</div>
<p>其中 <span class="arithmatex">\(\bmod\)</span> 表示取余运算。</p>
<p><img alt="哈希函数工作原理" src="../hash_map.assets/hash_function.png" /></p> <p><img alt="哈希函数工作原理" src="../hash_map.assets/hash_function.png" /></p>
<p align="center"> Fig. 哈希函数工作原理 </p> <p align="center"> Fig. 哈希函数工作原理 </p>
@ -3129,7 +3130,7 @@ f(x) = x \% 100
</div> </div>
</div> </div>
<h2 id="713">7.1.3. &nbsp; 哈希冲突<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2> <h2 id="713">7.1.3. &nbsp; 哈希冲突<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
<p>细心的你可能已经注意到，<strong>在某些情况下，哈希函数 <span class="arithmatex">\(f(x) = x % 100\)</span> 可能无法正常工作</strong>。具体来说，当输入的 key 后两位相同时，哈希函数的计算结果也会相同，从而指向同一个 value 。例如，查询学号为 <span class="arithmatex">\(12836\)</span> 和 <span class="arithmatex">\(20336\)</span> 的两个学生时，我们得到：</p> <p>细心的你可能已经注意到，<strong>在某些情况下，哈希函数 <span class="arithmatex">\(f(x) = x \bmod 100\)</span> 可能无法正常工作</strong>。具体来说，当输入的 key 后两位相同时，哈希函数的计算结果也会相同，从而指向同一个 value 。例如，查询学号为 <span class="arithmatex">\(12836\)</span> 和 <span class="arithmatex">\(20336\)</span> 的两个学生时，我们得到：</p>
<div class="arithmatex">\[ <div class="arithmatex">\[
f(12836) = f(20336) = 36 f(12836) = f(20336) = 36
\]</div> \]</div>

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<p>下面来剖析代码实现。对于一个 <span class="arithmatex">\(d\)</span> 进制的数字 <span class="arithmatex">\(x\)</span> ，要获取其第 <span class="arithmatex">\(k\)</span> 位 <span class="arithmatex">\(x_k\)</span> ，可以使用以下计算公式：</p> <p>下面来剖析代码实现。对于一个 <span class="arithmatex">\(d\)</span> 进制的数字 <span class="arithmatex">\(x\)</span> ，要获取其第 <span class="arithmatex">\(k\)</span> 位 <span class="arithmatex">\(x_k\)</span> ，可以使用以下计算公式：</p>
<div class="arithmatex">\[ <div class="arithmatex">\[
x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \mod d x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
\]</div> \]</div>
<p>其中 <span class="arithmatex">\(\lfloor a \rfloor\)</span> 表示对浮点数 <span class="arithmatex">\(a\)</span> 向下取整，而 <span class="arithmatex">\(\mod d\)</span> 表示对 <span class="arithmatex">\(d\)</span> 取余。对于学号数据，<span class="arithmatex">\(d = 10\)</span> 且 <span class="arithmatex">\(k \in [1, 8]\)</span> 。</p> <p>其中 <span class="arithmatex">\(\lfloor a \rfloor\)</span> 表示对浮点数 <span class="arithmatex">\(a\)</span> 向下取整，而 <span class="arithmatex">\(\bmod \space d\)</span> 表示对 <span class="arithmatex">\(d\)</span> 取余。对于学号数据，<span class="arithmatex">\(d = 10\)</span> 且 <span class="arithmatex">\(k \in [1, 8]\)</span> 。</p>
<p>此外，我们需要小幅改动计数排序代码，使之可以根据数字的第 <span class="arithmatex">\(k\)</span> 位进行排序。</p> <p>此外，我们需要小幅改动计数排序代码，使之可以根据数字的第 <span class="arithmatex">\(k\)</span> 位进行排序。</p>
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