deploy

chapter_dynamic_programming/dp_problem_features/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2064,6 +2066,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>
@@ -2264,9 +2280,8 @@
 <div class="arithmatex">\[
 dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 \]</div>
-<p>这便可以引出「最优子结构」的含义：<strong>原问题的最优解是从子问题的最优解构建得来的</strong>。对于本题，我们从两个子问题最优解 <span class="arithmatex">\(dp[i-1]\)</span> , <span class="arithmatex">\(dp[i-2]\)</span> 中挑选出较优的那一个，并用它构建出原问题 <span class="arithmatex">\(dp[i]\)</span> 的最优解。</p>
-<p>相较于分治问题，动态规划问题的解也是由其子问题的解构成的。不同的是，<strong>动态规划中子问题的解不仅揭示了问题的局部最优解，而且还通过特定的递推关系链接起来，共同构建出原问题的全局最优解</strong>。</p>
-<p>那么，上节的爬楼梯题目有没有最优子结构呢？它要求解的是方案数量，看似是一个计数问题，但如果换一种问法：求解最大方案数量。我们意外地发现，<strong>虽然题目修改前后是等价的，但最优子结构浮现出来了</strong>：第 <span class="arithmatex">\(n\)</span> 阶最大方案数量等于第 <span class="arithmatex">\(n-1\)</span> 阶和第 <span class="arithmatex">\(n-2\)</span> 阶最大方案数量之和。所以说，最优子结构的是一个比较宽泛的概念，在不同问题中会有不同的含义。</p>
+<p>这便可以引出「最优子结构」的含义：<strong>原问题的最优解是从子问题的最优解构建得来的</strong>。本题显然具有最优子结构：我们从两个子问题最优解 <span class="arithmatex">\(dp[i-1]\)</span> , <span class="arithmatex">\(dp[i-2]\)</span> 中挑选出较优的那一个，并用它构建出原问题 <span class="arithmatex">\(dp[i]\)</span> 的最优解。</p>
+<p>那么，上节的爬楼梯题目有没有最优子结构呢？它要求解的是方案数量，看似是一个计数问题，但如果换一种问法：求解最大方案数量。我们意外地发现，<strong>虽然题目修改前后是等价的，但最优子结构浮现出来了</strong>：第 <span class="arithmatex">\(n\)</span> 阶最大方案数量等于第 <span class="arithmatex">\(n-1\)</span> 阶和第 <span class="arithmatex">\(n-2\)</span> 阶最大方案数量之和。所以说，最优子结构的解释方式比较灵活，在不同问题中会有不同的含义。</p>
 <p>根据以上状态转移方程，以及初始状态 <span class="arithmatex">\(dp[1] = cost[1]\)</span> , <span class="arithmatex">\(dp[2] = cost[2]\)</span> ，我们可以得出动态规划解题代码。</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:11"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label></div>
 <div class="tabbed-content">
@@ -2620,7 +2635,7 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 <p>给定一个共有 <span class="arithmatex">\(n\)</span> 阶的楼梯，你每步可以上 <span class="arithmatex">\(1\)</span> 阶或者 <span class="arithmatex">\(2\)</span> 阶。<strong>规定当爬到第 <span class="arithmatex">\(i\)</span> 阶时，系统自动会给第 <span class="arithmatex">\(2i\)</span> 阶上放上障碍物，之后所有轮都不允许跳到第 <span class="arithmatex">\(2i\)</span> 阶上</strong>。例如，前两轮分别跳到了第 <span class="arithmatex">\(2, 3\)</span> 阶上，则之后就不能跳到第 <span class="arithmatex">\(4, 6\)</span> 阶上。请问有多少种方案可以爬到楼顶。</p>
 </div>
 <p>在这个问题中，下次跳跃依赖于过去所有的状态，因为每一次跳跃都会在更高的阶梯上设置障碍，并影响未来的跳跃。对于这类问题，动态规划往往难以解决，或是因为计算复杂度过高而难以应用。</p>
-<p>实际上，许多组合优化问题（例如著名的旅行商问题）都不满足无后效性。对于这类问题，我们通常会选择使用其他方法，例如启发式搜索、遗传算法、强化学习等，从而降低时间复杂度，在有限时间内得到能够接受的局部最优解。</p>
+<p>实际上，许多复杂的组合优化问题（例如著名的旅行商问题）都不满足无后效性。对于这类问题，我们通常会选择使用其他方法，例如启发式搜索、遗传算法、强化学习等，从而降低时间复杂度，在有限时间内得到能够接受的局部最优解。</p>
 
 
 

chapter_dynamic_programming/dp_solution_pipeline/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2085,6 +2087,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>

chapter_dynamic_programming/edit_distance_problem/index.html

@@ -18,7 +18,7 @@
         <link rel="prev" href="../unbounded_knapsack_problem/">
       
       
-        <link rel="next" href="../../chapter_appendix/installation/">
+        <link rel="next" href="../summary/">
       
       <link rel="icon" href="../../assets/images/favicon.png">
       <meta name="generator" content="mkdocs-1.4.2, mkdocs-material-9.1.11">
@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2026,6 +2028,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>
@@ -2210,13 +2226,13 @@
 <li>若 <span class="arithmatex">\(s[n-1]\)</span> 和 <span class="arithmatex">\(t[m-1]\)</span> 相同，我们可以直接跳过它们，接下来考虑 <span class="arithmatex">\(s[n-2]\)</span> 和 <span class="arithmatex">\(t[m-2]\)</span> ;</li>
 <li>若 <span class="arithmatex">\(s[n-1]\)</span> 和 <span class="arithmatex">\(t[m-1]\)</span> 不同，我们需要对 <span class="arithmatex">\(s\)</span> 进行一次编辑（插入、删除、替换），使得两字符串尾部的字符相同，从而可以跳过它们，考虑规模更小的问题；</li>
 </ul>
-<p>也就是说，我们在字符串 <span class="arithmatex">\(s\)</span> 中进行的每一轮决策（编辑操作），都会使得 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> 中剩余的待匹配字符发生变化。因此，状态定义为当前在 <span class="arithmatex">\(s\)</span> , <span class="arithmatex">\(t\)</span> 中考虑的第 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> 个字符，记为 <span class="arithmatex">\([i, j]\)</span> 。</p>
+<p>也就是说，我们在字符串 <span class="arithmatex">\(s\)</span> 中进行的每一轮决策（编辑操作），都会使得 <span class="arithmatex">\(s\)</span> 和 <span class="arithmatex">\(t\)</span> 中剩余的待匹配字符发生变化。因此，状态为当前在 <span class="arithmatex">\(s\)</span> , <span class="arithmatex">\(t\)</span> 中考虑的第 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> 个字符，记为 <span class="arithmatex">\([i, j]\)</span> 。</p>
 <p>状态 <span class="arithmatex">\([i, j]\)</span> 对应的子问题：<strong>将 <span class="arithmatex">\(s\)</span> 的前 <span class="arithmatex">\(i\)</span> 个字符更改为 <span class="arithmatex">\(t\)</span> 的前 <span class="arithmatex">\(j\)</span> 个字符所需的最少编辑步数</strong>。</p>
 <p>至此得到一个尺寸为 <span class="arithmatex">\((i+1) \times (j+1)\)</span> 的二维 <span class="arithmatex">\(dp\)</span> 表。</p>
 <p><strong>第二步：找出最优子结构，进而推导出状态转移方程</strong></p>
 <p>考虑子问题 <span class="arithmatex">\(dp[i, j]\)</span> ，其对应的两个字符串的尾部字符为 <span class="arithmatex">\(s[i-1]\)</span> 和 <span class="arithmatex">\(t[j-1]\)</span> ，可根据不同编辑操作分为三种情况：</p>
 <ol>
-<li>在 <span class="arithmatex">\(s\)</span> 尾部添加 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i, j-1]\)</span> ；</li>
+<li>在 <span class="arithmatex">\(s[i-1]\)</span> 之后添加 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i, j-1]\)</span> ；</li>
 <li>删除 <span class="arithmatex">\(s[i-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i-1, j]\)</span> ；</li>
 <li>将 <span class="arithmatex">\(s[i-1]\)</span> 替换为 <span class="arithmatex">\(t[j-1]\)</span> ，则剩余子问题 <span class="arithmatex">\(dp[i-1, j-1]\)</span> ；</li>
 </ol>
@@ -2617,13 +2633,13 @@ dp[i, j] = dp[i-1, j-1]
         
         
           
-          <a href="../../chapter_appendix/installation/" class="md-footer__link md-footer__link--next" aria-label="下一页: 14.1. &amp;nbsp; 编程环境安装" rel="next">
+          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 13.7. &amp;nbsp; 小结（New）" rel="next">
             <div class="md-footer__title">
               <span class="md-footer__direction">
                 下一页
               </span>
               <div class="md-ellipsis">
-                14.1. &nbsp; 编程环境安装
+                13.7. &nbsp; 小结（New）
               </div>
             </div>
             <div class="md-footer__button md-icon">

chapter_dynamic_programming/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2016,6 +2018,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>

chapter_dynamic_programming/intro_to_dynamic_programming/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2071,6 +2073,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>
@@ -2439,7 +2455,8 @@ dp[i] = dp[i-1] + dp[i-2]
 <p align="center"> Fig. 方案数量递推关系 </p>
 
 <p>也就是说，在爬楼梯问题中，<strong>各个子问题之间不是相互独立的，原问题的解可以由子问题的解构成</strong>。</p>
-<p>我们可以基于此递推公式写出暴力搜索代码：以 <span class="arithmatex">\(dp[n]\)</span> 为起始点，<strong>从顶至底地将一个较大问题拆解为两个较小问题的和</strong>，直至到达最小子问题 <span class="arithmatex">\(dp[1]\)</span> 和 <span class="arithmatex">\(dp[2]\)</span> 时返回。其中，最小子问题的解 <span class="arithmatex">\(dp[1] = 1\)</span> , <span class="arithmatex">\(dp[2] = 2\)</span> 是已知的，代表爬到第 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> 阶分别有 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> 种方案。</p>
+<p>我们可以基于此递推公式写出暴力搜索代码：以 <span class="arithmatex">\(dp[n]\)</span> 为起始点，<strong>从顶至底地将一个较大问题拆解为两个较小问题的和</strong>，直至到达最小子问题 <span class="arithmatex">\(dp[1]\)</span> 和 <span class="arithmatex">\(dp[2]\)</span> 时返回。</p>
+<p>请注意，最小子问题的解 <span class="arithmatex">\(dp[1] = 1\)</span> , <span class="arithmatex">\(dp[2] = 2\)</span> 是已知的，代表爬到第 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> 阶分别有 <span class="arithmatex">\(1\)</span> , <span class="arithmatex">\(2\)</span> 种方案。</p>
 <p>观察以下代码，它和标准回溯代码都属于深度优先搜索，但更加简洁。</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:11"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label></div>
 <div class="tabbed-content">
@@ -2914,7 +2931,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </div>
 </div>
 <p><strong>我们将这种空间优化技巧称为「状态压缩」</strong>。在许多动态规划问题中，当前状态仅与前面有限个状态有关，不必保存所有的历史状态，这时我们可以应用状态压缩，只保留必要的状态，通过“降维”来节省内存空间。</p>
-<p>总的看来，子问题分解是一种通用的算法思路，在分治算法、动态规划、回溯算法中各有特点：</p>
+<p>总的看来，<strong>子问题分解是一种通用的算法思路，在分治、动态规划、回溯中各有特点</strong>：</p>
 <ul>
 <li>分治算法将原问题划分为几个独立的子问题，然后递归解决子问题，最后合并子问题的解得到原问题的解。例如，归并排序将长数组不断划分为两个短子数组，再将排序好的子数组合并为排序好的长数组。</li>
 <li>动态规划也是将原问题分解为多个子问题，但与分治算法的主要区别是，<strong>动态规划中的子问题往往不是相互独立的</strong>，原问题的解依赖于子问题的解，而子问题的解又依赖于更小的子问题的解。因此，动态规划通常会引入记忆化，保存已经解决的子问题的解，避免重复计算。</li>

chapter_dynamic_programming/knapsack_problem/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2071,6 +2073,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>
@@ -2711,7 +2727,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 <p><strong>最后考虑状态压缩</strong>。以上代码中的数组 <code>dp</code> 占用 <span class="arithmatex">\(O(n \times cap)\)</span> 空间。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> 。代码省略，有兴趣的同学可以自行实现。</p>
-<p>那么，我们是否可以仅用一个数组实现状态压缩呢？观察可知，每个状态都是由左上方或正上方的格子转移过来的。假设只有一个数组，当遍历到第 <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，<strong>为了避免左方区域的格子在状态转移中被覆盖，应该采取倒序遍历</strong>。</p>
+<p>那么，我们是否可以仅用一个数组实现状态压缩呢？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，当遍历到第 <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，<strong>为了避免左方区域的格子在状态转移中被覆盖，应该采取倒序遍历</strong>。</p>
 <p>以下动画展示了在单个数组下从第 <span class="arithmatex">\(i=1\)</span> 行转换至第 <span class="arithmatex">\(i=2\)</span> 行的过程。建议你思考一下正序遍历和倒序遍历的区别。</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:6"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">&lt;1&gt;</label><label for="__tabbed_5_2">&lt;2&gt;</label><label for="__tabbed_5_3">&lt;3&gt;</label><label for="__tabbed_5_4">&lt;4&gt;</label><label for="__tabbed_5_5">&lt;5&gt;</label><label for="__tabbed_5_6">&lt;6&gt;</label></div>
 <div class="tabbed-content">

chapter_dynamic_programming/unbounded_knapsack_problem/index.html

@@ -1834,7 +1834,7 @@
   
     <li class="md-nav__item">
       <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-        12.3. &nbsp; 子集和问题（New）
+        12.3. &nbsp; 子集和问题
       </a>
     </li>
   
@@ -1909,6 +1909,8 @@
           
         
           
+        
+          
         
       
       
@@ -2071,6 +2073,20 @@
 
             
           
+            
+              
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../summary/" class="md-nav__link">
+        13.7. &nbsp; 小结（New）
+      </a>
+    </li>
+  
+
+            
+          
         </ul>
       </nav>
     </li>
@@ -2367,7 +2383,25 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">unbounded_knapsack</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDP</span><span class="p">}</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 完全背包：动态规划 */</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</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="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">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</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">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</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="c1">// 状态转移</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</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="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</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="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-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选物品 i</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</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="n">c</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="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">                </span><span class="c1">// 不选和选物品 i 这两种方案的较大值</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</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="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</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="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</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>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">];</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -2490,7 +2524,25 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">unbounded_knapsack</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">unboundedKnapsackDPComp</span><span class="p">}</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 完全背包：状态压缩后的动态规划 */</span>
+<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</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="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">cap</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="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</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="c1">// 状态转移</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</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">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-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选物品 i</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">                </span><span class="n">dp</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="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></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-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">                </span><span class="c1">// 不选和选物品 i 这两种方案的较大值</span>
+<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">                </span><span class="n">dp</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="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</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="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</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>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">cap</span><span class="p">];</span>
+<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -2635,7 +2687,30 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDP</span><span class="p">}</span>
+<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* 零钱兑换：动态规划 */</span>
+<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</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">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// 状态转移：首行首列</span>
+<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
+<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="c1">// 状态转移：其余行列</span>
+<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选硬币 i</span>
+<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</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="n">a</span><span class="p">];</span>
+<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></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-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">                </span><span class="c1">// 不选和选硬币 i 这两种方案的较小值</span>
+<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</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="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
+<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">]</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -2792,7 +2867,28 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeDPComp</span><span class="p">}</span>
+<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* 零钱兑换：状态压缩后的动态规划 */</span>
+<a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
+<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Fill</span><span class="p">(</span><span class="n">dp</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
+<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
+<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="c1">// 状态转移</span>
+<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选硬币 i</span>
+<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
+<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></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-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">                </span><span class="c1">// 不选和选硬币 i 这两种方案的较小值</span>
+<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
+<a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-40-20" name="__codelineno-40-20" href="#__codelineno-40-20"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-40-21" name="__codelineno-40-21" href="#__codelineno-40-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-40-22" name="__codelineno-40-22" href="#__codelineno-40-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -2915,7 +3011,29 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDP</span><span class="p">}</span>
+<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="cm">/* 零钱兑换 II：动态规划 */</span>
+<a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</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">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
+<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="c1">// 初始化首列</span>
+<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="c1">// 状态转移</span>
+<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选硬币 i</span>
+<a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</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="n">a</span><span class="p">];</span>
+<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></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-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">                </span><span class="c1">// 不选和选硬币 i 这两种方案之和</span>
+<a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</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="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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>
+<a id="__codelineno-51-19" name="__codelineno-51-19" href="#__codelineno-51-19"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-51-20" name="__codelineno-51-20" href="#__codelineno-51-20"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-51-21" name="__codelineno-51-21" href="#__codelineno-51-21"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-51-22" name="__codelineno-51-22" href="#__codelineno-51-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">];</span>
+<a id="__codelineno-51-23" name="__codelineno-51-23" href="#__codelineno-51-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -3018,7 +3136,26 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">coin_change_ii</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">coinChangeIIDPComp</span><span class="p">}</span>
+<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* 零钱兑换 II：状态压缩后的动态规划 */</span>
+<a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
+<a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="c1">// 初始化 dp 表</span>
+<a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
+<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
+<a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">    </span><span class="c1">// 状态转移</span>
+<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</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">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">                </span><span class="c1">// 若超过背包容量，则不选硬币 i</span>
+<a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
+<a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></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-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="w">                </span><span class="c1">// 不选和选硬币 i 这两种方案之和</span>
+<a id="__codelineno-62-15" name="__codelineno-62-15" href="#__codelineno-62-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</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>
+<a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">            </span><span class="p">}</span>
+<a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="w">        </span><span class="p">}</span>
+<a id="__codelineno-62-18" name="__codelineno-62-18" href="#__codelineno-62-18"></a><span class="w">    </span><span class="p">}</span>
+<a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">];</span>
+<a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">