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12.1.5. &nbsp; 典型例题
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<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>
<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，我们将从前序遍历入手，逐步了解回溯算法的工作原理。</p>
<div class="admonition question">
<p class="admonition-title">例题一</p>
<p>给定一个二叉树，搜索并记录所有值为 <span class="arithmatex">\(7\)</span> 的节点，返回节点列表。</p>
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<li>旅行商问题：在一个图中，从一个点出发，访问所有其他点恰好一次后返回起点，求最短路径。</li>
<li>最大团问题：给定一个无向图，找到最大的完全子图，即子图中的任意两个顶点之间都有边相连。</li>
</ul>
<p>请注意，回溯算法通常不是解决组合优化问题的最优方法。0-1 背包问题通常使用动态规划解决；旅行商是一个 NP-Hard 问题，常见的算法有遗传算法和蚁群算法等；最大团问题是图轮中的一个经典 NP-Hard 问题，通常用贪心算法等启发式算法来解决。</p>
<p>在接下来的章节中，我们将一起探讨几个经典的回溯算法问题：全排列、子集和、<span class="arithmatex">\(n\)</span> 皇后。</p>
<p>请注意，回溯算法通常不是解决组合优化问题的最优方法。0-1 背包问题通常使用动态规划解决；旅行商是一个 NP-Hard 问题，常用解决方法有遗传算法和蚁群算法等；最大团问题是图轮中的一个经典 NP-Hard 问题，通常用贪心算法等启发式算法来解决。</p>
<h2 id="1216">12.1.6. &nbsp; 优势与局限性<a class="headerlink" href="#1216" title="Permanent link">&para;</a></h2>
<p>回溯算法本质上是一种深度优先搜索算法，它尝试所有可能的解决方案直到找到满足条件的解。这种方法的优势在于它能够找到所有可能的解决方案，而且在合理的剪枝操作下，具有很高的效率。</p>
<p>然而，在处理大规模或者复杂问题时，<strong>回溯算法的运行效率可能难以接受</strong>。这是因为在最坏的情况下，回溯算法需要遍历解空间的所有可能解。例如，求解 <span class="arithmatex">\(n\)</span> 皇后问题的时间复杂度可以达到 <span class="arithmatex">\(O(n!)\)</span> 。回溯算法的空间复杂度也可能较高。因为在每一次递归调用时，都需要保存当前的状态（例如选择路径、用于剪枝的辅助变量等），对于深度很大的递归，空间需求可能会变得非常大。</p>
<p>即便如此，<strong>回溯算法仍然是某些搜索问题和约束满足问题的最佳解决方案</strong>。对于这些问题，由于我们无法预测哪些选择可生成有效的解，因此我们必须对所有可能的选择进行遍历。在这种情况下，<strong>关键是如何进行效率优化</strong>：</p>
<ul>
<li>上文介绍过的剪枝是一种常用的优化方法。它可以避免搜索那些肯定不会产生有效解的路径，从而节省时间和空间。</li>
<li>另一个常用的优化方法是加入「启发式搜索 Heuristic Search」策略，它在搜索过程中引入一些策略或者估计值，从而优先搜索最有可能产生有效解的路径。</li>
</ul>

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12.3.1. &nbsp; 参考全排列解法
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12.3.1. &nbsp; 参考全排列解法
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12.3.3. &nbsp; 考虑相等元素
12.3.3. &nbsp; 相等元素剪枝
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<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。给定数组无重复元素，每个元素可以被选取多次。请以列表形式返回这些组合，列表中不应包含重复组合。</p>
</div>
<p>例如，输入集合 <span class="arithmatex">\(\{3, 4, 5\}\)</span> 和目标整数 <span class="arithmatex">\(9\)</span> ，由于集合中的数字可以被重复选取，因此解为 <span class="arithmatex">\(\{3, 3, 3\}, \{4, 5\}\)</span> 。请注意，子集是不区分元素顺序的，例如 <span class="arithmatex">\(\{4, 5\}\)</span> 和 <span class="arithmatex">\(\{5, 4\}\)</span> 是同一个子集。</p>
<h2 id="1231">12.3.1. &nbsp; 参考全排列解法<a class="headerlink" href="#1231" title="Permanent link">&para;</a></h2>
<p>从回溯算法的角度看，我们可以把子集的生成过程想象成一系列选择的结果，并在选择过程中实时更新“元素和”，当元素和等于 <code>target</code> 时，就将子集记录至结果列表。</p>
<p>与上节全排列问题不同的是，本题允许重复选取同一元素，因此无需借助 <code>selected</code> 布尔列表来记录元素是否已被选择。我们可以对全排列代码进行小幅修改，初步得到解题代码。</p>
<h2 id="1231">12.3.1. &nbsp; 从全排列引出解法<a class="headerlink" href="#1231" title="Permanent link">&para;</a></h2>
<p>类似于上节全排列问题的解法，我们可以把子集的生成过程想象成一系列选择的结果，并在选择过程中实时更新“元素和”，当元素和等于 <code>target</code> 时，就将子集记录至结果列表。</p>
<p>而与全排列问题不同的是，本题允许重复选取同一元素，因此无需借助 <code>selected</code> 布尔列表来记录元素是否已被选择。我们可以对全排列代码进行小幅修改，初步得到解题代码。</p>
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<p><img alt="子集和 I 回溯过程" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
<p align="center"> Fig. 子集和 I 回溯过程 </p>
<h2 id="1233">12.3.3. &nbsp; 考虑相等元素<a class="headerlink" href="#1233" title="Permanent link">&para;</a></h2>
<h2 id="1233">12.3.3. &nbsp; 相等元素剪枝<a class="headerlink" href="#1233" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>给定一个正整数数组 <code>nums</code> 和一个目标正整数 <code>target</code> ，请找出所有可能的组合，使得组合中的元素和等于 <code>target</code> 。<strong>给定数组可能包含重复元素，每个元素只可被选择一次</strong>。请以列表形式返回这些组合，列表中不应包含重复组合。</p>

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<h2 align="center"> 《 Hello 算法 》</h2>
<p align="center"> 在动画与代码中掌握数据结构与算法 </p>
<p align="center"> 动画图解、一键运行的数据结构与算法教程 </p>
<p align="center">
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