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<link rel="canonical" href="https://www.hello-algo.com/chapter_backtracking/backtracking_algorithm/">
<link rel="prev" href="../../chapter_searching/summary/">
<link rel="prev" href="../../chapter_sorting/summary/">
<link rel="next" href="../permutations_problem/">
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<title>13.1.   回溯算法（New） - Hello 算法</title>
<title>12.1.   回溯算法（New） - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
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13.1. &nbsp; 回溯算法（New）
12.1. &nbsp; 回溯算法（New）
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
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6.1. &nbsp; 哈希表
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search_edge/" class="md-nav__link">
6.2. &nbsp; 二分查找边界
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
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6.3. &nbsp; 小结
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7. &nbsp; &nbsp; 散列表
7. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
7.3. &nbsp; 二叉树数组表示
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<li class="md-nav__item">
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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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<li class="md-nav__item">
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8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
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</li>
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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 *
</a>
</li>
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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
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.4. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
</a>
</li>
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8.6. &nbsp; 小结
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8.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
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<li class="md-nav__item">
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9.1. &nbsp; 堆
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9.1. &nbsp; 图
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</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
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9.2. &nbsp; 图基础操作
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
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9.3. &nbsp; 图的遍历
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
10.1. &nbsp; 图
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10.1. &nbsp; 二分查找（New）
</a>
</li>
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<li class="md-nav__item">
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10.2. &nbsp; 图基础操作
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10.2. &nbsp; 二分查找边界（New）
</a>
</li>
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<li class="md-nav__item">
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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>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
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
</a>
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10.5. &nbsp; 小结
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</li>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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12.1. &nbsp; 搜索算法（New）
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</li>
<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
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
</a>
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13.1. &nbsp; 回溯算法（New）
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1311" class="md-nav__link">
13.1.1. &nbsp; 尝试与回退
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12.1.1. &nbsp; 尝试与回退
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12.1.2. &nbsp; 剪枝
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13.1.3. &nbsp; 常用术语
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12.1.3. &nbsp; 常用术语
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<li class="md-nav__item">
<a href="../permutations_problem/" class="md-nav__link">
13.2. &nbsp; 全排列问题（New）
12.2. &nbsp; 全排列问题（New）
</a>
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<li class="md-nav__item">
<a href="../n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
12.3. &nbsp; N 皇后问题（New）
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13. &nbsp; &nbsp; 附录
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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
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
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参考文献
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1311" class="md-nav__link">
13.1.1. &nbsp; 尝试与回退
<a href="#1211" class="md-nav__link">
12.1.1. &nbsp; 尝试与回退
</a>
</li>
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<a href="#1312" class="md-nav__link">
13.1.2. &nbsp; 剪枝
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12.1.2. &nbsp; 剪枝
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</li>
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<a href="#1313" class="md-nav__link">
13.1.3. &nbsp; 常用术语
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12.1.3. &nbsp; 常用术语
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</li>
<li class="md-nav__item">
<a href="#1314" class="md-nav__link">
13.1.4. &nbsp; 框架代码
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12.1.4. &nbsp; 框架代码
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12.1.5. &nbsp; 典型例题
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@@ -1955,7 +1916,7 @@
<h1 id="131">13.1. &nbsp; 回溯算法<a class="headerlink" href="#131" title="Permanent link">&para;</a></h1>
<h1 id="121">12.1. &nbsp; 回溯算法<a class="headerlink" href="#121" title="Permanent link">&para;</a></h1>
<p>「回溯算法 Backtracking Algorithm」是一种通过穷举来解决问题的方法，它的核心思想是从一个初始状态出发，暴力搜索所有可能的解决方案，当遇到正确的解则将其记录，直到找到解或者尝试了所有可能的选择都无法找到解为止。</p>
<p>回溯算法通常采用「深度优先搜索」来遍历解空间。在二叉树章节中，我们提到前序、中序和后序遍历都属于深度优先搜索。下面，我们从二叉树的前序遍历入手，逐步了解回溯算法的工作原理。</p>
<div class="admonition question">
@@ -2094,7 +2055,7 @@
<p><img alt="在前序遍历中搜索节点" src="../backtracking_algorithm.assets/preorder_find_nodes.png" /></p>
<p align="center"> Fig. 在前序遍历中搜索节点 </p>
<h2 id="1311">13.1.1. &nbsp; 尝试与回退<a class="headerlink" href="#1311" title="Permanent link">&para;</a></h2>
<h2 id="1211">12.1.1. &nbsp; 尝试与回退<a class="headerlink" href="#1211" title="Permanent link">&para;</a></h2>
<p><strong>之所以称之为回溯算法，是因为该算法在搜索解空间时会采用“尝试”与“回退”的策略</strong>。当算法在搜索过程中遇到某个状态无法继续前进或无法得到满足条件的解时，它会撤销上一步的选择，退回到之前的状态，并尝试其他可能的选择。</p>
<p>对于例题一，访问每个节点都代表一次“尝试”，而越过叶结点或返回父节点的 <code>return</code> 则表示“回退”。</p>
<p>值得说明的是，<strong>回退并不等价于函数返回</strong>。为解释这一点，我们对例题一稍作拓展。</p>
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</div>
</div>
</div>
<h2 id="1312">13.1.2. &nbsp; 剪枝<a class="headerlink" href="#1312" title="Permanent link">&para;</a></h2>
<h2 id="1212">12.1.2. &nbsp; 剪枝<a class="headerlink" href="#1212" title="Permanent link">&para;</a></h2>
<p>复杂的回溯问题通常包含一个或多个约束条件，<strong>约束条件通常可用于“剪枝”</strong>。</p>
<div class="admonition question">
<p class="admonition-title">例题三：在二叉树中搜索所有值为 <span class="arithmatex">\(7\)</span> 的节点，返回根节点到这些节点的路径，<strong>路径中不能包含值为 <span class="arithmatex">\(3\)</span> 的节点</strong></p>
@@ -2488,7 +2449,7 @@
<p><img alt="根据约束条件剪枝" src="../backtracking_algorithm.assets/preorder_find_constrained_paths.png" /></p>
<p align="center"> Fig. 根据约束条件剪枝 </p>
<h2 id="1313">13.1.3. &nbsp; 常用术语<a class="headerlink" href="#1313" title="Permanent link">&para;</a></h2>
<h2 id="1213">12.1.3. &nbsp; 常用术语<a class="headerlink" href="#1213" title="Permanent link">&para;</a></h2>
<p>为了更清晰地分析算法问题，我们总结一下回溯算法中常用术语的含义，并对照例题三给出对应示例。</p>
<table>
<thead>
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<p class="admonition-title">Tip</p>
<p>解、状态、约束条件等术语是通用的，适用于回溯算法、动态规划、贪心算法等。</p>
</div>
<h2 id="1314">13.1.4. &nbsp; 框架代码<a class="headerlink" href="#1314" title="Permanent link">&para;</a></h2>
<h2 id="1214">12.1.4. &nbsp; 框架代码<a class="headerlink" href="#1214" title="Permanent link">&para;</a></h2>
<p>回溯算法可用于解决许多搜索问题、约束满足问题和组合优化问题。为提升代码通用性，我们希望将回溯算法的“尝试、回退、剪枝”的主体框架提炼出来。</p>
<p>设 <code>state</code> 为问题的当前状态，<code>choices</code> 表示当前状态下可以做出的选择，则可得到以下回溯算法的框架代码。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="5:10"><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" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label></div>
@@ -3172,7 +3133,7 @@
</div>
</div>
<p>相较于基于前序遍历的实现代码，基于回溯算法框架的实现代码虽然显得啰嗦，但通用性更好。实际上，<strong>所有回溯问题都可以在该框架下解决</strong>。我们需要根据具体问题来定义 <code>state</code> 和 <code>choices</code> ，并实现框架中的各个方法。</p>
<h2 id="1315">13.1.5. &nbsp; 典型例题<a class="headerlink" href="#1315" title="Permanent link">&para;</a></h2>
<h2 id="1215">12.1.5. &nbsp; 典型例题<a class="headerlink" href="#1215" title="Permanent link">&para;</a></h2>
<p><strong>搜索问题</strong>：这类问题的目标是找到满足特定条件的解决方案。</p>
<ul>
<li>全排列问题：给定一个集合，求出其所有可能的排列组合。</li>
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上一页
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<title>13.3.   N 皇后问题（New） - Hello 算法</title>
<title>12.3.   N 皇后问题（New） - Hello 算法</title>
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
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<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search_edge/" class="md-nav__link">
6.2. &nbsp; 二分查找边界
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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
</a>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
</a>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
7.3. &nbsp; 二叉树数组表示
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.4. &nbsp; 二叉搜索树
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.5. &nbsp; AVL 树 *
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7.6. &nbsp; 小结
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8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
</a>
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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">
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8.4. &nbsp; 二叉搜索树
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
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<a href="../../chapter_tree/summary/" class="md-nav__link">
8.6. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
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<a href="../../chapter_heap/heap/" class="md-nav__link">
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9.1. &nbsp; 图
</a>
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
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9.2. &nbsp; 图基础操作
</a>
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<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
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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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<li class="md-nav__item">
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10.3. &nbsp; 图的遍历
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10.3. &nbsp; 哈希优化策略
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<li class="md-nav__item">
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10.4. &nbsp; 小结
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10.4. &nbsp; 重识搜索算法
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12. &nbsp; &nbsp; 搜索算法
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12.1. &nbsp; 搜索算法（New）
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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12.3. &nbsp; 小结
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13. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
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13.1. &nbsp; 回溯算法（New）
12.1. &nbsp; 回溯算法（New）
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<li class="md-nav__item">
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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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<label class="md-nav__link md-nav__link--active" for="__toc">
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12.3. &nbsp; N 皇后问题（New）
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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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12.3.1. &nbsp; 复杂度分析
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14. &nbsp; &nbsp; 附录
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12.3.1. &nbsp; 复杂度分析
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<h1 id="133-n">13.3. &nbsp; N 皇后问题<a class="headerlink" href="#133-n" title="Permanent link">&para;</a></h1>
<h1 id="123-n">12.3. &nbsp; N 皇后问题<a class="headerlink" href="#123-n" title="Permanent link">&para;</a></h1>
<div class="admonition question">
<p class="admonition-title">根据国际象棋的规则，皇后可以攻击与之处在同一行或同一列或同一斜线上的棋子。给定 <span class="arithmatex">\(n\)</span> 个皇后和一个 <span class="arithmatex">\(n \times n\)</span> 大小的棋盘，寻找使得所有皇后之间无法相互攻击的摆放方案。</p>
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</div>
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<h2 id="1331">13.3.1. &nbsp; 复杂度分析<a class="headerlink" href="#1331" title="Permanent link">&para;</a></h2>
<h2 id="1231">12.3.1. &nbsp; 复杂度分析<a class="headerlink" href="#1231" title="Permanent link">&para;</a></h2>
<p>逐行放置 <span class="arithmatex">\(n\)</span> 次，考虑列约束，则从第一行到最后一行分别有 <span class="arithmatex">\(n, n-1, \cdots, 2, 1\)</span> 个选择，<strong>因此时间复杂度为 <span class="arithmatex">\(O(n!)\)</span></strong> 。实际上，根据对角线约束的剪枝也能够大幅地缩小搜索空间，因而搜索效率往往优于以上时间复杂度。</p>
<p><code>state</code> 使用 <span class="arithmatex">\(O(n^2)\)</span> 空间，<code>cols</code> , <code>diags1</code> , <code>diags2</code> 皆使用 <span class="arithmatex">\(O(n)\)</span> 空间。最大递归深度为 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。因此，<strong>空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span></strong> 。</p>
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7.1. &nbsp; 二叉树
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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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8.5. &nbsp; AVL 树 *
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8.3. &nbsp; 小结
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9.2. &nbsp; 图基础操作
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10.2. &nbsp; 二分查找边界（New）
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10.3. &nbsp; 哈希优化策略
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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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12.2.1. &nbsp; 无重复的情况
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13.2.2. &nbsp; 考虑重复的情况
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<a href="#1321" class="md-nav__link">
13.2.1. &nbsp; 无重复的情况
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12.2.1. &nbsp; 无重复的情况
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</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
</a>
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@@ -1927,7 +1888,7 @@
<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">
@@ -1954,7 +1915,7 @@
</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>
@@ -2254,7 +2215,7 @@
<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>
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@@ -2578,7 +2539,7 @@
<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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