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<h1 id="131">13.1 &nbsp; 回溯算法<a class="headerlink" href="#131" 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。接下来，我们利用前序遍历构造一个回溯问题，逐步了解回溯算法的工作原理。</p>
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<h1 id="134-n">13.4 &nbsp; N 皇后问题<a class="headerlink" href="#134-n" title="Permanent link">&para;</a></h1>
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<p class="admonition-title">Question</p>
@@ -4031,11 +4063,73 @@
<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">\(n\)</span>、<span class="arithmatex">\(n-1\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</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>数组 <code>state</code> 使用 <span class="arithmatex">\(O(n^2)\)</span> 空间，数组 <code>cols</code>、<code>diags1</code> 和 <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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<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>表 13-2 列举了几个示例数据，包括输入数组和对应的所有排列。</p>
@@ -4394,11 +4426,73 @@
<p><img alt="两种剪枝条件的作用范围" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
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