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2026-11-05 23:18:41 +08:00 · 2026-10-29 03:49:36 +08:00
parent 2e4fec7765
commit b67d4c595d
1348 changed files with 670900 additions and 9591 deletions
--- a/zh-hant/chapter_dynamic_programming/knapsack_problem/index.html
+++ b/zh-hant/chapter_dynamic_programming/knapsack_problem/index.html
@@ -32,6 +32,8 @@

          <link rel="alternate" href="/ru/chapter_dynamic_programming/knapsack_problem/" hreflang="ru">

+          <link rel="alternate" href="/ko/chapter_dynamic_programming/knapsack_problem/" hreflang="ko">
+


      <link rel="icon" href="../../assets/images/favicon.png">
@@ -230,6 +232,12 @@
            </a>
          </li>

+          <li class="md-select__item">
+            <a href="/ko/chapter_dynamic_programming/knapsack_problem/" hreflang="ko" class="md-select__link">
+              한국어
+            </a>
+          </li>
+
      </ul>
    </div>
  </div>
@@ -4739,7 +4747,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <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="w"> </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：暴力搜尋&quot;&quot;&quot;</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""0-1 背包：暴力搜尋"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># 若已選完所有物品或背包無剩餘容量，則返回價值 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>
@@ -4818,14 +4826,14 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <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">// 若超過背包容量，則只能選擇不放入背包</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-4-8" 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="o">&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>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// 計算不放入和放入物品 i 的最大價值</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="nx">no</span><span class="w"> </span><span class="o">:=</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-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="nx">yes</span><span class="w"> </span><span class="o">:=</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="o">-</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="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// 返回兩種方案中價值更大的那一個</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="nb">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -4991,7 +4999,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <details class="pythontutor">
 <summary>視覺化執行</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20i%3A%20int%2C%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20return%20max%28no%2C%20yes%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="視覺化執行" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20i%3A%20int%2C%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20return%20max%28no%2C%20yes%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20i%3A%20int%2C%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20return%20max%28no%2C%20yes%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt%2C%20val%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全螢幕觀看 &gt;</a></div></p>
 </details>
 <p>如圖 14-18 所示，由於每個物品都會產生不選和選兩條搜尋分支，因此時間複雜度為 <span class="arithmatex">\(O(2^n)\)</span> 。</p>
@@ -5008,7 +5016,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dfs_mem</span><span class="p">(</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></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-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;0-1 背包：記憶化搜尋&quot;&quot;&quot;</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""0-1 背包：記憶化搜尋"""</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># 若已選完所有物品或背包無剩餘容量，則返回價值 0</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></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-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="k">return</span> <span class="mi">0</span>
@@ -5110,14 +5118,14 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></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-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// 若超過背包容量，則只能選擇不放入背包</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></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-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></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="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></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-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// 計算不放入和放入物品 i 的最大價值</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="nx">no</span><span class="w"> </span><span class="o">:=</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-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="nx">yes</span><span class="w"> </span><span class="o">:=</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="o">-</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="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// 返回兩種方案中價值更大的那一個</span>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><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="p">=</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="nb">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><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="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></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-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="p">}</span>
 </code></pre></div>
@@ -5331,7 +5339,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <details class="pythontutor">
 <summary>視覺化執行</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20mem%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%A8%98%E9%8C%84%EF%BC%8C%E5%89%87%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%A8%98%E9%8C%84%E4%B8%A6%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no%2C%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="視覺化執行" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20mem%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%A8%98%E9%8C%84%EF%BC%8C%E5%89%87%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%A8%98%E9%8C%84%E4%B8%A6%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no%2C%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20mem%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%81%B8%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E7%84%A1%E5%89%A9%E9%A4%98%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%E5%83%B9%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%A8%98%E9%8C%84%EF%BC%8C%E5%89%87%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E5%8F%AA%E8%83%BD%E9%81%B8%E6%93%87%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20%23%20%E8%A8%88%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E5%83%B9%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20i%20-%201%2C%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%A8%98%E9%8C%84%E4%B8%A6%E8%BF%94%E5%9B%9E%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E4%B8%AD%E5%83%B9%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E5%80%8B%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no%2C%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt%2C%20val%2C%20mem%2C%20n%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全螢幕觀看 &gt;</a></div></p>
 </details>
 <p>圖 14-19 展示了在記憶化搜尋中被剪掉的搜尋分支。</p>
@@ -5344,7 +5352,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp</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">cap</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-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;0-1 背包：動態規劃&quot;&quot;&quot;</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""0-1 背包：動態規劃"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="c1"># 初始化 dp 表</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
@@ -5433,17 +5441,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="k">for</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="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="c1">// 狀態轉移</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="k">for</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">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="k">for</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">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></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-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></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="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">                </span><span class="c1">// 若超過背包容量，則不選物品 i</span>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">                </span><span class="nx">dp</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="p">=</span><span class="w"> </span><span class="nx">dp</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="nx">c</span><span class="p">]</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">                </span><span class="nx">dp</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="nx">dp</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="nx">c</span><span class="p">]</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">                </span><span class="c1">// 不選和選物品 i 這兩種方案的較大值</span>
-<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">                </span><span class="nx">dp</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="p">=</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="nb">float64</span><span class="p">(</span><span class="nx">dp</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="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">dp</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="nx">c</span><span class="o">-</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="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
+<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">                </span><span class="nx">dp</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="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</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="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</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="nx">c</span><span class="o">-</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="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5656,7 +5664,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <details class="pythontutor">
 <summary>視覺化執行</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281%2C%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E4%B8%8D%E9%81%B8%E7%89%A9%E5%93%81%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%B8%8D%E9%81%B8%E5%92%8C%E9%81%B8%E7%89%A9%E5%93%81%20i%20%E9%80%99%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E7%9A%84%E8%BC%83%E5%A4%A7%E5%80%BC%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D%2C%20dp%5Bi%20-%201%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20knapsack_dp%28wgt%2C%20val%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="視覺化執行" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281%2C%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E4%B8%8D%E9%81%B8%E7%89%A9%E5%93%81%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%B8%8D%E9%81%B8%E5%92%8C%E9%81%B8%E7%89%A9%E5%93%81%20i%20%E9%80%99%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E7%9A%84%E8%BC%83%E5%A4%A7%E5%80%BC%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D%2C%20dp%5Bi%20-%201%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20knapsack_dp%28wgt%2C%20val%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp%28wgt%3A%20list%5Bint%5D%2C%20val%3A%20list%5Bint%5D%2C%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281%2C%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%89%87%E4%B8%8D%E9%81%B8%E7%89%A9%E5%93%81%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%B8%8D%E9%81%B8%E5%92%8C%E9%81%B8%E7%89%A9%E5%93%81%20i%20%E9%80%99%E5%85%A9%E7%A8%AE%E6%96%B9%E6%A1%88%E7%9A%84%E8%BC%83%E5%A4%A7%E5%80%BC%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D%2C%20dp%5Bi%20-%201%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10%2C%2020%2C%2030%2C%2040%2C%2050%5D%0A%20%20%20%20val%20%3D%20%5B50%2C%20120%2C%20150%2C%20210%2C%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20knapsack_dp%28wgt%2C%20val%2C%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E9%81%8E%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E5%83%B9%E5%80%BC%E7%82%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全螢幕觀看 &gt;</a></div></p>
 </details>
 <p>如圖 14-20 所示，時間複雜度和空間複雜度都由陣列 <code>dp</code> 大小決定，即 <span class="arithmatex">\(O(n \times cap)\)</span> 。</p>
@@ -5745,7 +5753,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp_comp</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">cap</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-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;0-1 背包：空間最佳化後的動態規劃&quot;&quot;&quot;</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""0-1 背包：空間最佳化後的動態規劃"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># 初始化 dp 表</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
@@ -5837,7 +5845,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></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="o">&lt;=</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">                </span><span class="c1">// 不選和選物品 i 這兩種方案的較大值</span>
-<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="nb">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="o">-</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="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
+<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">                </span><span class="nx">dp</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="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="o">-</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="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">    </span><span class="p">}</span>
@@ -6019,7 +6027,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 <details class="pythontutor">
 <summary>視覺化執行</summary>
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 </details>

@@ -6130,6 +6138,8 @@ aria-label="頁脚"
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