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<title>13.4.   0-1 背包问题（New） - Hello 算法</title>
<title>14.4.   0-1 背包问题（New） - Hello 算法</title>
@ -79,7 +79,7 @@
<div data-md-component="skip">
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跳转至
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13.4. &nbsp; 0-1 背包问题（New）
14.4. &nbsp; 0-1 背包问题（New）
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@ -1771,6 +1771,87 @@
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12. &nbsp; &nbsp; 分治
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12.1. &nbsp; 分治算法（New）
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@ -1806,7 +1887,7 @@
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12.1. &nbsp; 回溯算法
13.1. &nbsp; 回溯算法
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@ -1820,7 +1901,7 @@
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
12.2. &nbsp; 全排列问题
13.2. &nbsp; 全排列问题
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@ -1834,7 +1915,7 @@
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12.3. &nbsp; 子集和问题
13.3. &nbsp; 子集和问题
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12.4. &nbsp; N 皇后问题
13.4. &nbsp; N 皇后问题
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@ -1941,7 +2022,7 @@
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13.1. &nbsp; 初探动态规划（New）
14.1. &nbsp; 初探动态规划（New）
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@ -1955,7 +2036,7 @@
<li class="md-nav__item">
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13.2. &nbsp; DP 问题特性（New）
14.2. &nbsp; DP 问题特性（New）
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@ -1969,7 +2050,7 @@
<li class="md-nav__item">
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13.3. &nbsp; DP 解题思路（New）
14.3. &nbsp; DP 解题思路（New）
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@ -1992,12 +2073,12 @@
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13.4. &nbsp; 0-1 背包问题（New）
14.4. &nbsp; 0-1 背包问题（New）
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13.4. &nbsp; 0-1 背包问题（New）
14.4. &nbsp; 0-1 背包问题（New）
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@ -2016,22 +2097,22 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<a href="#1341" class="md-nav__link">
13.4.1. &nbsp; 方法一：暴力搜索
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14.4.1. &nbsp; 方法一：暴力搜索
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</li>
<li class="md-nav__item">
<a href="#1342" class="md-nav__link">
13.4.2. &nbsp; 方法二：记忆化搜索
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14.4.2. &nbsp; 方法二：记忆化搜索
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<a href="#1343" class="md-nav__link">
13.4.3. &nbsp; 方法三：动态规划
<a href="#1443" class="md-nav__link">
14.4.3. &nbsp; 方法三：动态规划
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@ -2052,7 +2133,7 @@
<li class="md-nav__item">
<a href="../unbounded_knapsack_problem/" class="md-nav__link">
13.5. &nbsp; 完全背包问题（New）
14.5. &nbsp; 完全背包问题（New）
</a>
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@ -2066,7 +2147,7 @@
<li class="md-nav__item">
<a href="../edit_distance_problem/" class="md-nav__link">
13.6. &nbsp; 编辑距离问题（New）
14.6. &nbsp; 编辑距离问题（New）
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@ -2080,76 +2161,7 @@
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14.1. &nbsp; 编程环境安装
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14.2. &nbsp; 一起参与创作
14.7. &nbsp; 小结（New）
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@ -2180,6 +2192,75 @@
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15.1. &nbsp; 编程环境安装
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@ -2192,8 +2273,8 @@
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参考文献
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@ -2234,22 +2315,22 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1341" class="md-nav__link">
13.4.1. &nbsp; 方法一：暴力搜索
<a href="#1441" class="md-nav__link">
14.4.1. &nbsp; 方法一：暴力搜索
</a>
</li>
<li class="md-nav__item">
<a href="#1342" class="md-nav__link">
13.4.2. &nbsp; 方法二：记忆化搜索
<a href="#1442" class="md-nav__link">
14.4.2. &nbsp; 方法二：记忆化搜索
</a>
</li>
<li class="md-nav__item">
<a href="#1343" class="md-nav__link">
13.4.3. &nbsp; 方法三：动态规划
<a href="#1443" class="md-nav__link">
14.4.3. &nbsp; 方法三：动态规划
</a>
</li>
@ -2277,7 +2358,7 @@
<h1 id="134-0-1">13.4. &nbsp; 0-1 背包问题<a class="headerlink" href="#134-0-1" title="Permanent link">&para;</a></h1>
<h1 id="144-0-1">14.4. &nbsp; 0-1 背包问题<a class="headerlink" href="#144-0-1" title="Permanent link">&para;</a></h1>
<p>背包问题是一个非常好的动态规划入门题目，是动态规划中最常见的问题形式。其具有很多变种，例如 0-1 背包问题、完全背包问题、多重背包问题等。</p>
<p>在本节中，我们先来学习基础的的 0-1 背包问题。</p>
<div class="admonition question">
@ -2312,7 +2393,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
<p class="admonition-title">Tip</p>
<p>完成以上三步后，我们可以直接实现从底至顶的动态规划解法。而为了展示本题包含的重叠子问题，本文也同时给出从顶至底的暴力搜索和记忆化搜索解法。</p>
</div>
<h2 id="1341">13.4.1. &nbsp; 方法一：暴力搜索<a class="headerlink" href="#1341" title="Permanent link">&para;</a></h2>
<h2 id="1441">14.4.1. &nbsp; 方法一：暴力搜索<a class="headerlink" href="#1441" title="Permanent link">&para;</a></h2>
<p>搜索代码包含以下要素：</p>
<ul>
<li><strong>递归参数</strong>：状态 <span class="arithmatex">\([i, c]\)</span> ；<strong>返回值</strong>：子问题的解 <span class="arithmatex">\(dp[i, c]\)</span> 。</li>
@ -2429,7 +2510,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
<p><img alt="0-1 背包的暴力搜索递归树" src="../knapsack_problem.assets/knapsack_dfs.png" /></p>
<p align="center"> Fig. 0-1 背包的暴力搜索递归树 </p>
<h2 id="1342">13.4.2. &nbsp; 方法二：记忆化搜索<a class="headerlink" href="#1342" title="Permanent link">&para;</a></h2>
<h2 id="1442">14.4.2. &nbsp; 方法二：记忆化搜索<a class="headerlink" href="#1442" title="Permanent link">&para;</a></h2>
<p>为了防止重复求解重叠子问题，我们借助一个记忆列表 <code>mem</code> 来记录子问题的解，其中 <code>mem[i][c]</code> 对应解 <span class="arithmatex">\(dp[i, c]\)</span> 。</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">
@ -2561,7 +2642,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
<p><img alt="0-1 背包的记忆化搜索递归树" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></p>
<p align="center"> Fig. 0-1 背包的记忆化搜索递归树 </p>
<h2 id="1343">13.4.3. &nbsp; 方法三：动态规划<a class="headerlink" href="#1343" title="Permanent link">&para;</a></h2>
<h2 id="1443">14.4.3. &nbsp; 方法三：动态规划<a class="headerlink" href="#1443" title="Permanent link">&para;</a></h2>
<p>动态规划解法本质上就是在状态转移中填充 <span class="arithmatex">\(dp\)</span> 表的过程，代码如下所示。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:11"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label></div>
<div class="tabbed-content">
@ -2943,7 +3024,7 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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@ -2952,20 +3033,20 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
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