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<title>13.2.   全排列问题（New） - Hello 算法</title>
<title>12.2.   全排列问题（New） - Hello 算法</title>
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13.2. &nbsp; 全排列问题（New）
12.2. &nbsp; 全排列问题（New）
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<li class="md-nav__item">
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6.1. &nbsp; 二分查找
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6.1. &nbsp; 哈希表
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<li class="md-nav__item">
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6.2. &nbsp; 二分查找边界
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6.2. &nbsp; 哈希冲突处理
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<li class="md-nav__item">
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7.1. &nbsp; 哈希表
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7.1. &nbsp; 二叉树
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<li class="md-nav__item">
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7.2. &nbsp; 哈希冲突处理
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7.2. &nbsp; 二叉树遍历
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
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7.3. &nbsp; 二叉树数组表示
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7.4. &nbsp; 二叉搜索树
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7.5. &nbsp; AVL 树 *
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8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
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<li class="md-nav__item">
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8.2. &nbsp; 二叉树遍历
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8.2. &nbsp; 建堆操作 *
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<li class="md-nav__item">
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
8.3. &nbsp; 二叉树数组表示
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8.4. &nbsp; 二叉搜索树
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8.5. &nbsp; AVL 树 *
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8.6. &nbsp; 小结
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8.3. &nbsp; 小结
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9.1. &nbsp; 图
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<li class="md-nav__item">
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9.2. &nbsp; 建堆操作 *
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<li class="md-nav__item">
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9.3. &nbsp; 图的遍历
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10. &nbsp; &nbsp; 图
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<li class="md-nav__item">
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10.1. &nbsp; 图
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10.1. &nbsp; 二分查找（New）
</a>
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<li class="md-nav__item">
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10.2. &nbsp; 二分查找边界（New）
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10.3. &nbsp; 图的遍历
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10.3. &nbsp; 哈希优化策略
</a>
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<li class="md-nav__item">
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10.4. &nbsp; 小结
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
10.4. &nbsp; 重识搜索算法
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10.5. &nbsp; 小结
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12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
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<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
12.2. &nbsp; 哈希优化策略
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12.3. &nbsp; 小结
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13. &nbsp; &nbsp; 回溯算法
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13.1. &nbsp; 回溯算法（New）
12.1. &nbsp; 回溯算法（New）
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<label class="md-nav__link md-nav__link--active" for="__toc">
13.2. &nbsp; 全排列问题（New）
12.2. &nbsp; 全排列问题（New）
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13.2. &nbsp; 全排列问题（New）
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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13.2.1. &nbsp; 无重复的情况
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13.2.3. &nbsp; 复杂度分析
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12.2.3. &nbsp; 复杂度分析
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13.3. &nbsp; N 皇后问题（New）
12.3. &nbsp; N 皇后问题（New）
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13.1. &nbsp; 编程环境安装
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13.2. &nbsp; 一起参与创作
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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
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
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13.2.1. &nbsp; 无重复的情况
<a href="#1221" class="md-nav__link">
12.2.1. &nbsp; 无重复的情况
</a>
</li>
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13.2.2. &nbsp; 考虑重复的情况
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12.2.2. &nbsp; 考虑重复的情况
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13.2.3. &nbsp; 复杂度分析
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12.2.3. &nbsp; 复杂度分析
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<h1 id="132">13.2. &nbsp; 全排列问题<a class="headerlink" href="#132" title="Permanent link">&para;</a></h1>
<h1 id="122">12.2. &nbsp; 全排列问题<a class="headerlink" href="#122" 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>如下表所示，列举了几个示例数组和其对应的所有排列。</p>
<div class="center-table">
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</tbody>
</table>
</div>
<h2 id="1321">13.2.1. &nbsp; 无重复的情况<a class="headerlink" href="#1321" title="Permanent link">&para;</a></h2>
<h2 id="1221">12.2.1. &nbsp; 无重复的情况<a class="headerlink" href="#1221" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">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>
</div>
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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="1322">13.2.2. &nbsp; 考虑重复的情况<a class="headerlink" href="#1322" title="Permanent link">&para;</a></h2>
<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>
<div class="admonition question">
<p class="admonition-title">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>
</div>
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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="1323">13.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#1323" title="Permanent link">&para;</a></h2>
<h2 id="1223">12.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#1223" 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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13.3. &nbsp; N 皇后问题（New）
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