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<p>状态 <span class="arithmatex">\([i, c]\)</span> 对应的子问题为:<strong>前 <span class="arithmatex">\(i\)</span> 个物品在剩余容量为 <span class="arithmatex">\(c\)</span> 的背包中的最大价值</strong>,记为 <span class="arithmatex">\(dp[i, c]\)</span> 。</p>
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<p>待求解的是 <span class="arithmatex">\(dp[n, cap]\)</span> ,因此需要一个尺寸为 <span class="arithmatex">\((n+1) \times (cap+1)\)</span> 的二维 <span class="arithmatex">\(dp\)</span> 表。</p>
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<p><strong>第二步:找出最优子结构,进而推导出状态转移方程</strong></p>
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<p>当我们做出物品 <span class="arithmatex">\(i\)</span> 的决策后,剩余的是前 <span class="arithmatex">\(i-1\)</span> 个物品的决策。因此,状态转移分为两种情况:</p>
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<p>当我们做出物品 <span class="arithmatex">\(i\)</span> 的决策后,剩余的是前 <span class="arithmatex">\(i-1\)</span> 个物品的决策,可分为以下两种情况。</p>
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<ul>
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<li><strong>不放入物品 <span class="arithmatex">\(i\)</span></strong> :背包容量不变,状态转移至 <span class="arithmatex">\([i-1, c]\)</span> 。</li>
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<li><strong>放入物品 <span class="arithmatex">\(i\)</span></strong> :背包容量减小 <span class="arithmatex">\(wgt[i-1]\)</span> ,价值增加 <span class="arithmatex">\(val[i-1]\)</span> ,状态转移至 <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span> 。</li>
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<li><strong>不放入物品 <span class="arithmatex">\(i\)</span></strong> :背包容量不变,状态变化为 <span class="arithmatex">\([i-1, c]\)</span> 。</li>
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<li><strong>放入物品 <span class="arithmatex">\(i\)</span></strong> :背包容量减小 <span class="arithmatex">\(wgt[i-1]\)</span> ,价值增加 <span class="arithmatex">\(val[i-1]\)</span> ,状态变化为 <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span> 。</li>
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</ul>
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<p>上述的状态转移向我们揭示了本题的最优子结构:<strong>最大价值 <span class="arithmatex">\(dp[i, c]\)</span> 等于不放入物品 <span class="arithmatex">\(i\)</span> 和放入物品 <span class="arithmatex">\(i\)</span> 两种方案中的价值更大的那一个</strong>。由此可推出状态转移方程:</p>
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<p>上述分析向我们揭示了本题的最优子结构:<strong>最大价值 <span class="arithmatex">\(dp[i, c]\)</span> 等于不放入物品 <span class="arithmatex">\(i\)</span> 和放入物品 <span class="arithmatex">\(i\)</span> 两种方案中的价值更大的那一个</strong>。由此可推出状态转移方程:</p>
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<div class="arithmatex">\[
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dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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\]</div>
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||||
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||||
<changefreq>daily</changefreq>
|
||||
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||||
<url>
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||||
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||||
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||||
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||||
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<loc>https://www.hello-algo.com/chapter_hashing/summary/</loc>
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||||
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<loc>https://www.hello-algo.com/chapter_heap/build_heap/</loc>
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||||
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||||
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||||
<loc>https://www.hello-algo.com/chapter_heap/summary/</loc>
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||||
<loc>https://www.hello-algo.com/chapter_introduction/algorithms_are_everywhere/</loc>
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<loc>https://www.hello-algo.com/chapter_introduction/summary/</loc>
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<loc>https://www.hello-algo.com/chapter_preface/about_the_book/</loc>
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<loc>https://www.hello-algo.com/chapter_preface/summary/</loc>
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<loc>https://www.hello-algo.com/chapter_searching/binary_search_edge/</loc>
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<loc>https://www.hello-algo.com/chapter_searching/summary/</loc>
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<loc>https://www.hello-algo.com/chapter_sorting/bubble_sort/</loc>
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<url>
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<loc>https://www.hello-algo.com/chapter_sorting/heap_sort/</loc>
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<loc>https://www.hello-algo.com/chapter_sorting/quick_sort/</loc>
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|
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|
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|
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|
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||||
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|
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|
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|
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<loc>https://www.hello-algo.com/chapter_tree/summary/</loc>
|
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|
||||
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|
||||
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|
||||
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|
||||
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|
BIN
sitemap.xml.gz
BIN
sitemap.xml.gz
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Loading…
Reference in New Issue
Block a user