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<title>12.3.   子集和问题 - Hello 算法</title>
<title>13.3.   子集和问题 - Hello 算法</title>
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12.3. &nbsp; 子集和问题
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12.3.1. &nbsp; 从全排列引出解法
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13.3.1. &nbsp; 从全排列引出解法
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12.3.2. &nbsp; 重复子集剪枝
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13.3.2. &nbsp; 重复子集剪枝
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12.3.3. &nbsp; 相等元素剪枝
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13.3.3. &nbsp; 相等元素剪枝
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<h1 id="123">12.3. &nbsp; 子集和问题<a class="headerlink" href="#123" title="Permanent link">&para;</a></h1>
<h1 id="133">13.3. &nbsp; 子集和问题<a class="headerlink" href="#133" title="Permanent link">&para;</a></h1>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。给定数组无重复元素，每个元素可以被选取多次。请以列表形式返回这些组合，列表中不应包含重复组合。</p>
</div>
<p>例如，输入集合 <span class="arithmatex">\(\{3, 4, 5\}\)</span> 和目标整数 <span class="arithmatex">\(9\)</span> ，由于集合中的数字可以被重复选取，因此解为 <span class="arithmatex">\(\{3, 3, 3\}, \{4, 5\}\)</span> 。请注意，子集是不区分元素顺序的，例如 <span class="arithmatex">\(\{4, 5\}\)</span> 和 <span class="arithmatex">\(\{5, 4\}\)</span> 是同一个子集。</p>
<h2 id="1231">12.3.1. &nbsp; 从全排列引出解法<a class="headerlink" href="#1231" title="Permanent link">&para;</a></h2>
<h2 id="1331">13.3.1. &nbsp; 从全排列引出解法<a class="headerlink" href="#1331" title="Permanent link">&para;</a></h2>
<p>类似于上节全排列问题的解法，我们可以把子集的生成过程想象成一系列选择的结果，并在选择过程中实时更新“元素和”，当元素和等于 <code>target</code> 时，就将子集记录至结果列表。</p>
<p>而与全排列问题不同的是，本题允许重复选取同一元素，因此无需借助 <code>selected</code> 布尔列表来记录元素是否已被选择。我们可以对全排列代码进行小幅修改，初步得到解题代码。</p>
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<p><img alt="子集搜索与越界剪枝" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
<p align="center"> Fig. 子集搜索与越界剪枝 </p>
<h2 id="1232">12.3.2. &nbsp; 重复子集剪枝<a class="headerlink" href="#1232" title="Permanent link">&para;</a></h2>
<h2 id="1332">13.3.2. &nbsp; 重复子集剪枝<a class="headerlink" href="#1332" title="Permanent link">&para;</a></h2>
<p>为了去除重复子集，<strong>一种直接的思路是对结果列表进行去重</strong>。但这个方法效率很低，因为：</p>
<ul>
<li>当数组元素较多，尤其是当 <code>target</code> 较大时，搜索过程会产生大量的重复子集。</li>
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<p><img alt="子集和 I 回溯过程" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
<p align="center"> Fig. 子集和 I 回溯过程 </p>
<h2 id="1233">12.3.3. &nbsp; 相等元素剪枝<a class="headerlink" href="#1233" title="Permanent link">&para;</a></h2>
<h2 id="1333">13.3.3. &nbsp; 相等元素剪枝<a class="headerlink" href="#1333" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。<strong>给定数组可能包含重复元素，每个元素只可被选择一次</strong>。请以列表形式返回这些组合，列表中不应包含重复组合。</p>
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