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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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<div class="md-header__topic" data-md-component="header-topic">
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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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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 二分查找
6. &nbsp; &nbsp; 散列表
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6. &nbsp; &nbsp; 散列表
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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
</a>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
7. &nbsp; &nbsp; 树
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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
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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
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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
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.4. &nbsp; 二叉搜索树
</a>
</li>
<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. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 堆
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8. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 堆
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
8.6. &nbsp; 小结
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8.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 堆
9. &nbsp; &nbsp; 图
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9. &nbsp; &nbsp; 堆
9. &nbsp; &nbsp; 图
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. &nbsp; 图
</a>
</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 *
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
</a>
</li>
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9.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
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">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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>
</li>
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<a href="../../chapter_searching/summary/" class="md-nav__link">
10.5. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
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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. &nbsp; &nbsp; 搜索算法
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<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
</a>
</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
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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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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">
13.3. &nbsp; N 皇后问题（New）
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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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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13.3.1. &nbsp; 复杂度分析
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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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14.1. &nbsp; 编程环境安装
13.1. &nbsp; 编程环境安装
</a>
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<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
13.2. &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">
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13.3.1. &nbsp; 复杂度分析
<a href="#1231" class="md-nav__link">
12.3.1. &nbsp; 复杂度分析
</a>
</li>
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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>
<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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