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<title>12.2.   全排列问题 - Hello 算法</title>
<title>13.2.   全排列问题 - Hello 算法</title>
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12.2. &nbsp; 全排列问题
13.2. &nbsp; 全排列问题
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12.2.1. &nbsp; 无重复的情况
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13.2.1. &nbsp; 无重复的情况
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12.2.2. &nbsp; 考虑重复的情况
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13.2.2. &nbsp; 考虑重复的情况
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12.2.3. &nbsp; 复杂度分析
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13.2.3. &nbsp; 复杂度分析
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<h1 id="122">12.2. &nbsp; 全排列问题<a class="headerlink" href="#122" title="Permanent link">&para;</a></h1>
<h1 id="132">13.2. &nbsp; 全排列问题<a class="headerlink" href="#132" title="Permanent link">&para;</a></h1>
<p>全排列问题是回溯算法的一个典型应用。它的定义是在给定一个集合（如一个数组或字符串）的情况下，找出这个集合中元素的所有可能的排列。</p>
<p>下表列举了几个示例数据，包括输入数组和对应的所有排列。</p>
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<h2 id="1221">12.2.1. &nbsp; 无重复的情况<a class="headerlink" href="#1221" title="Permanent link">&para;</a></h2>
<h2 id="1321">13.2.1. &nbsp; 无重复的情况<a class="headerlink" href="#1321" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>输入一个整数数组，数组中不包含重复元素，返回所有可能的排列。</p>
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<p><img alt="全排列剪枝示例" src="../permutations_problem.assets/permutations_i_pruning.png" /></p>
<p align="center"> Fig. 全排列剪枝示例 </p>
<h2 id="1222">12.2.2. &nbsp; 考虑重复的情况<a class="headerlink" href="#1222" title="Permanent link">&para;</a></h2>
<h2 id="1322">13.2.2. &nbsp; 考虑重复的情况<a class="headerlink" href="#1322" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>输入一个整数数组，<strong>数组中可能包含重复元素</strong>，返回所有不重复的排列。</p>
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<p><img alt="两种剪枝条件的作用范围" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
<p align="center"> Fig. 两种剪枝条件的作用范围 </p>
<h2 id="1223">12.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#1223" title="Permanent link">&para;</a></h2>
<h2 id="1323">13.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#1323" title="Permanent link">&para;</a></h2>
<p>假设元素两两之间互不相同，则 <span class="arithmatex">\(n\)</span> 个元素共有 <span class="arithmatex">\(n!\)</span> 种排列（阶乘）；在记录结果时，需要复制长度为 <span class="arithmatex">\(n\)</span> 的列表，使用 <span class="arithmatex">\(O(n)\)</span> 时间。因此，<strong>时间复杂度为 <span class="arithmatex">\(O(n!n)\)</span></strong> 。</p>
<p>最大递归深度为 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。<code>selected</code> 使用 <span class="arithmatex">\(O(n)\)</span> 空间。同一时刻最多共有 <span class="arithmatex">\(n\)</span> 个 <code>duplicated</code> ，使用 <span class="arithmatex">\(O(n^2)\)</span> 空间。因此，<strong>全排列 I 的空间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，全排列 II 的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span></strong> 。</p>
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