deploy

chapter_dynamic_programming/knapsack_problem/index.html

@@ -3414,14 +3414,14 @@
 <p>在本节中，我们先来求解最常见的 0-1 背包问题。</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>给定 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span>、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。每个物品只能选择一次，问在不超过背包容量下能放入物品的最大价值。</p>
+<p>给定 <span class="arithmatex">\(n\)</span> 个物品，第 <span class="arithmatex">\(i\)</span> 个物品的重量为 <span class="arithmatex">\(wgt[i-1]\)</span>、价值为 <span class="arithmatex">\(val[i-1]\)</span> ，和一个容量为 <span class="arithmatex">\(cap\)</span> 的背包。每个物品只能选择一次，问在限定背包容量下能放入物品的最大价值。</p>
 </div>
 <p>观察图 14-17 ，由于物品编号 <span class="arithmatex">\(i\)</span> 从 <span class="arithmatex">\(1\)</span> 开始计数，数组索引从 <span class="arithmatex">\(0\)</span> 开始计数，因此物品 <span class="arithmatex">\(i\)</span> 对应重量 <span class="arithmatex">\(wgt[i-1]\)</span> 和价值 <span class="arithmatex">\(val[i-1]\)</span> 。</p>
 <p><a class="glightbox" href="../knapsack_problem.assets/knapsack_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的示例数据" class="animation-figure" src="../knapsack_problem.assets/knapsack_example.png" /></a></p>
 <p align="center"> 图 14-17 &nbsp; 0-1 背包的示例数据 </p>
 
-<p>我们可以将 0-1 背包问题看作是一个由 <span class="arithmatex">\(n\)</span> 轮决策组成的过程，每个物体都有不放入和放入两种决策，因此该问题是满足决策树模型的。</p>
-<p>该问题的目标是求解“在限定背包容量下的最大价值”，因此较大概率是个动态规划问题。</p>
+<p>我们可以将 0-1 背包问题看作一个由 <span class="arithmatex">\(n\)</span> 轮决策组成的过程，对于每个物体都有不放入和放入两种决策，因此该问题满足决策树模型。</p>
+<p>该问题的目标是求解“在限定背包容量下能放入物品的最大价值”，因此较大概率是一个动态规划问题。</p>
 <p><strong>第一步：思考每轮的决策，定义状态，从而得到 <span class="arithmatex">\(dp\)</span> 表</strong></p>
 <p>对于每个物品来说，不放入背包，背包容量不变；放入背包，背包容量减小。由此可得状态定义：当前物品编号 <span class="arithmatex">\(i\)</span> 和剩余背包容量 <span class="arithmatex">\(c\)</span> ，记为 <span class="arithmatex">\([i, c]\)</span> 。</p>
 <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>
@@ -3430,9 +3430,9 @@
 <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>
 <ul>
 <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>
-<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>
+<li><strong>放入物品 <span class="arithmatex">\(i\)</span></strong> ：背包容量减少 <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>
 </ul>
-<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>
+<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> 两种方案中价值更大的那一个</strong>。由此可推导出状态转移方程：</p>
 <div class="arithmatex">\[
 dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 \]</div>
@@ -3447,17 +3447,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <li><strong>递归参数</strong>：状态 <span class="arithmatex">\([i, c]\)</span> 。</li>
 <li><strong>返回值</strong>：子问题的解 <span class="arithmatex">\(dp[i, c]\)</span> 。</li>
 <li><strong>终止条件</strong>：当物品编号越界 <span class="arithmatex">\(i = 0\)</span> 或背包剩余容量为 <span class="arithmatex">\(0\)</span> 时，终止递归并返回价值 <span class="arithmatex">\(0\)</span> 。</li>
-<li><strong>剪枝</strong>：若当前物品重量超出背包剩余容量，则只能不放入背包。</li>
+<li><strong>剪枝</strong>：若当前物品重量超出背包剩余容量，则只能选择不放入背包。</li>
 </ul>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;0-1 背包：暴力搜索&quot;&quot;&quot;</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="mi">0</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">return</span> <span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># 计算不放入和放入物品 i 的最大价值</span>
@@ -3470,11 +3470,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3489,11 +3489,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3508,11 +3508,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">KnapsackDFS</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3527,11 +3527,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3546,11 +3546,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span> <span class="nf">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">Int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>    <span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>    <span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="p">==</span> <span class="mi">0</span> <span class="o">||</span> <span class="n">c</span> <span class="p">==</span> <span class="mi">0</span> <span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>        <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>    <span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a>    <span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a>    <span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span> <span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>        <span class="k">return</span> <span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="n">val</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a>    <span class="p">}</span>
@@ -3565,11 +3565,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3589,11 +3589,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">c</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
@@ -3608,11 +3608,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="p">}</span>
@@ -3627,11 +3627,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span> <span class="nf">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span>: <span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span>: <span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">i</span>: <span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span>: <span class="kt">usize</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3646,11 +3646,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 0-1 背包：暴力搜索 */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3665,11 +3665,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.zig</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// 0-1 背包：暴力搜索</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">or</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="p">}</span>
@@ -3685,12 +3685,12 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <p>如图 14-18 所示，由于每个物品都会产生不选和选两条搜索分支，因此时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> 。</p>
 <p>观察递归树，容易发现其中存在重叠子问题，例如 <span class="arithmatex">\(dp[1, 10]\)</span> 等。而当物品较多、背包容量较大，尤其是相同重量的物品较多时，重叠子问题的数量将会大幅增多。</p>
-<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的暴力搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs.png" /></a></p>
-<p align="center"> 图 14-18 &nbsp; 0-1 背包的暴力搜索递归树 </p>
+<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包问题的暴力搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs.png" /></a></p>
+<p align="center"> 图 14-18 &nbsp; 0-1 背包问题的暴力搜索递归树 </p>
 
 <h3 id="2">2. &nbsp; 方法二：记忆化搜索<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
 <p>为了保证重叠子问题只被计算一次，我们借助记忆列表 <code>mem</code> 来记录子问题的解，其中 <code>mem[i][c]</code> 对应 <span class="arithmatex">\(dp[i, c]\)</span> 。</p>
-<p>引入记忆化之后，<strong>时间复杂度取决于子问题数量</strong>，也就是 <span class="arithmatex">\(O(n \times cap)\)</span> 。</p>
+<p>引入记忆化之后，<strong>时间复杂度取决于子问题数量</strong>，也就是 <span class="arithmatex">\(O(n \times cap)\)</span> 。实现代码如下：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -3698,13 +3698,13 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a>    <span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">mem</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">int</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;0-1 背包：记忆化搜索&quot;&quot;&quot;</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>    <span class="c1"># 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>    <span class="c1"># 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a>        <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>    <span class="c1"># 若已有记录，则直接返回</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>    <span class="k">if</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a>        <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>    <span class="c1"># 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>    <span class="c1"># 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a>        <span class="k">return</span> <span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a>    <span class="c1"># 计算不放入和放入物品 i 的最大价值</span>
@@ -3718,7 +3718,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3726,7 +3726,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3742,7 +3742,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3750,7 +3750,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3766,7 +3766,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3774,7 +3774,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">KnapsackDFSMem</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3790,7 +3790,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3798,7 +3798,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3814,7 +3814,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span> <span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">mem</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[[</span><span class="nb">Int</span><span class="p">]],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">Int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="p">==</span> <span class="mi">0</span> <span class="o">||</span> <span class="n">c</span> <span class="p">==</span> <span class="mi">0</span> <span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a>        <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>    <span class="p">}</span>
@@ -3822,7 +3822,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a>    <span class="k">if</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span> <span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a>        <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a>    <span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a>    <span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a>    <span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span> <span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a>        <span class="k">return</span> <span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="n">val</span><span class="p">,</span> <span class="n">mem</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a>    <span class="p">}</span>
@@ -3838,7 +3838,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3846,7 +3846,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3869,7 +3869,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="nx">c</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="p">}</span>
@@ -3877,7 +3877,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="p">}</span>
@@ -3900,7 +3900,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">,</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">  </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">  </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">  </span><span class="p">}</span>
@@ -3908,7 +3908,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">  </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">  </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">  </span><span class="p">}</span>
@@ -3924,7 +3924,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="k">fn</span> <span class="nf">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span>: <span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span>: <span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">mem</span>: <span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">i</span>: <span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span>: <span class="kt">usize</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</span> <span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3932,7 +3932,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3948,7 +3948,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 0-1 背包：记忆化搜索 */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3956,7 +3956,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3972,7 +3972,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.zig</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 0-1 背包：记忆化搜索</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="o">:</span><span class="w"> </span><span class="n">anytype</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无容量，则返回价值 0</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// 若已选完所有物品或背包无剩余容量，则返回价值 0</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">or</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="p">}</span>
@@ -3980,7 +3980,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能不放入背包</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// 若超过背包容量，则只能选择不放入背包</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="p">}</span>
@@ -3995,12 +3995,12 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p>图 14-19 展示了在记忆化递归中被剪掉的搜索分支。</p>
-<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs_mem.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的记忆化搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></a></p>
-<p align="center"> 图 14-19 &nbsp; 0-1 背包的记忆化搜索递归树 </p>
+<p>图 14-19 展示了在记忆化搜索中被剪掉的搜索分支。</p>
+<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dfs_mem.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包问题的记忆化搜索递归树" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></a></p>
+<p align="center"> 图 14-19 &nbsp; 0-1 背包问题的记忆化搜索递归树 </p>
 
 <h3 id="3">3. &nbsp; 方法三：动态规划<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
-<p>动态规划实质上就是在状态转移中填充 <span class="arithmatex">\(dp\)</span> 表的过程，代码如下所示。</p>
+<p>动态规划实质上就是在状态转移中填充 <span class="arithmatex">\(dp\)</span> 表的过程，代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4294,7 +4294,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label><label for="__tabbed_4_10">&lt;10&gt;</label><label for="__tabbed_4_11">&lt;11&gt;</label><label for="__tabbed_4_12">&lt;12&gt;</label><label for="__tabbed_4_13">&lt;13&gt;</label><label for="__tabbed_4_14">&lt;14&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包的动态规划过程" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></a></p>
+<p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="0-1 背包问题的动态规划过程" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></a></p>
 </div>
 <div class="tabbed-block">
 <p><a class="glightbox" href="../knapsack_problem.assets/knapsack_dp_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="knapsack_dp_step2" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step2.png" /></a></p>
@@ -4337,11 +4337,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p align="center"> 图 14-20 &nbsp; 0-1 背包的动态规划过程 </p>
+<p align="center"> 图 14-20 &nbsp; 0-1 背包问题的动态规划过程 </p>
 
 <h3 id="4">4. &nbsp; 空间优化<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
-<p>由于每个状态都只与其上一行的状态有关，因此我们可以使用两个数组滚动前进，将空间复杂度从 <span class="arithmatex">\(O(n^2)\)</span> 将低至 <span class="arithmatex">\(O(n)\)</span> 。</p>
-<p>进一步思考，我们是否可以仅用一个数组实现空间优化呢？观察可知，每个状态都是由正上方或左上方的格子转移过来的。假设只有一个数组，当开始遍历第 <span class="arithmatex">\(i\)</span> 行时，该数组存储的仍然是第 <span class="arithmatex">\(i-1\)</span> 行的状态。</p>
+<p>由于每个状态都只与其上一行的状态有关，因此我们可以使用两个数组滚动前进，将空间复杂度从 <span class="arithmatex">\(O(n^2)\)</span> 降至 <span class="arithmatex">\(O(n)\)</span> 。</p>
+<p>进一步思考，我们能否仅用一个数组实现空间优化呢？观察可知，每个状态都是由正上方或左上方的格子转移过来的。假设只有一个数组，当开始遍历第 <span class="arithmatex">\(i\)</span> 行时，该数组存储的仍然是第 <span class="arithmatex">\(i-1\)</span> 行的状态。</p>
 <ul>
 <li>如果采取正序遍历，那么遍历到 <span class="arithmatex">\(dp[i, j]\)</span> 时，左上方 <span class="arithmatex">\(dp[i-1, 1]\)</span> ~ <span class="arithmatex">\(dp[i-1, j-1]\)</span> 值可能已经被覆盖，此时就无法得到正确的状态转移结果。</li>
 <li>如果采取倒序遍历，则不会发生覆盖问题，状态转移可以正确进行。</li>
@@ -4371,7 +4371,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <p align="center"> 图 14-21 &nbsp; 0-1 背包的空间优化后的动态规划过程 </p>
 
-<p>在代码实现中，我们仅需将数组 <code>dp</code> 的第一维 <span class="arithmatex">\(i\)</span> 直接删除，并且把内循环更改为倒序遍历即可。</p>
+<p>在代码实现中，我们仅需将数组 <code>dp</code> 的第一维 <span class="arithmatex">\(i\)</span> 直接删除，并且把内循环更改为倒序遍历即可：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">