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第 8 章 &nbsp; 堆
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第 8 章 &nbsp; 堆
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8.1 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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9.2 &nbsp; 图基础操作
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9.3 &nbsp; 图的遍历
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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10.5 &nbsp; 重识搜索算法
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10.5.1 &nbsp; 暴力搜索
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10.5.2 &nbsp; 自适应搜索
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10.5.3 &nbsp; 搜索方法选取
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第 11 章 &nbsp; 排序
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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11.10 &nbsp; 基数排序
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11.11 &nbsp; 小结
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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第 13 章 &nbsp; 回溯
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13.4 &nbsp; N 皇后问题
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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<h1 id="105">10.5 &nbsp; 重识搜索算法<a class="headerlink" href="#105" title="Permanent link">&para;</a></h1>
<p><u>搜索算法（searching algorithm）</u>用于在数据结构（例如数组、链表、树或图）中搜索一个或一组满足特定条件的元素。</p>
<p>搜索算法可根据实现思路分为以下两类。</p>
<ul>
<li><strong>通过遍历数据结构来定位目标元素</strong>，例如数组、链表、树和图的遍历等。</li>
<li><strong>利用数据组织结构或数据包含的先验信息，实现高效元素查找</strong>，例如二分查找、哈希查找和二叉搜索树查找等。</li>
</ul>
<p>不难发现，这些知识点都已在前面的章节中介绍过，因此搜索算法对于我们来说并不陌生。在本节中，我们将从更加系统的视角切入，重新审视搜索算法。</p>
<h2 id="1051">10.5.1 &nbsp; 暴力搜索<a class="headerlink" href="#1051" title="Permanent link">&para;</a></h2>
<p>暴力搜索通过遍历数据结构的每个元素来定位目标元素。</p>
<ul>
<li>“线性搜索”适用于数组和链表等线性数据结构。它从数据结构的一端开始，逐个访问元素，直到找到目标元素或到达另一端仍没有找到目标元素为止。</li>
<li>“广度优先搜索”和“深度优先搜索”是图和树的两种遍历策略。广度优先搜索从初始节点开始逐层搜索，由近及远地访问各个节点。深度优先搜索从初始节点开始，沿着一条路径走到头，再回溯并尝试其他路径，直到遍历完整个数据结构。</li>
</ul>
<p>暴力搜索的优点是简单且通用性好，<strong>无须对数据做预处理和借助额外的数据结构</strong>。</p>
<p>然而，<strong>此类算法的时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ，其中 <span class="arithmatex">\(n\)</span> 为元素数量，因此在数据量较大的情况下性能较差。</p>
<h2 id="1052">10.5.2 &nbsp; 自适应搜索<a class="headerlink" href="#1052" title="Permanent link">&para;</a></h2>
<p>自适应搜索利用数据的特有属性（例如有序性）来优化搜索过程，从而更高效地定位目标元素。</p>
<ul>
<li>“二分查找”利用数据的有序性实现高效查找，仅适用于数组。</li>
<li>“哈希查找”利用哈希表将搜索数据和目标数据建立为键值对映射，从而实现查询操作。</li>
<li>“树查找”在特定的树结构（例如二叉搜索树）中，基于比较节点值来快速排除节点，从而定位目标元素。</li>
</ul>
<p>此类算法的优点是效率高，<strong>时间复杂度可达到 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span></strong> 。</p>
<p>然而，<strong>使用这些算法往往需要对数据进行预处理</strong>。例如，二分查找需要预先对数组进行排序，哈希查找和树查找都需要借助额外的数据结构，维护这些数据结构也需要额外的时间和空间开销。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>自适应搜索算法常被称为查找算法，<strong>主要用于在特定数据结构中快速检索目标元素</strong>。</p>
</div>
<h2 id="1053">10.5.3 &nbsp; 搜索方法选取<a class="headerlink" href="#1053" title="Permanent link">&para;</a></h2>
<p>给定大小为 <span class="arithmatex">\(n\)</span> 的一组数据，我们可以使用线性搜索、二分查找、树查找、哈希查找等多种方法从中搜索目标元素。各个方法的工作原理如图 10-11 所示。</p>
<p><a class="glightbox" href="../searching_algorithm_revisited.assets/searching_algorithms.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="多种搜索策略" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></a></p>
<p align="center"> 图 10-11 &nbsp; 多种搜索策略 </p>
<p>上述几种方法的操作效率与特性如表 10-1 所示。</p>
<p align="center"> 表 10-1 &nbsp; 查找算法效率对比 </p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>线性搜索</th>
<th>二分查找</th>
<th>树查找</th>
<th>哈希查找</th>
</tr>
</thead>
<tbody>
<tr>
<td>查找元素</td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>插入元素</td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>删除元素</td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>额外空间</td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>数据预处理</td>
<td>/</td>
<td>排序 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>建树 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>建哈希表 <span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>数据是否有序</td>
<td>无序</td>
<td>有序</td>
<td>有序</td>
<td>无序</td>
</tr>
</tbody>
</table>
</div>
<p>搜索算法的选择还取决于数据体量、搜索性能要求、数据查询与更新频率等。</p>
<p><strong>线性搜索</strong></p>
<ul>
<li>通用性较好，无须任何数据预处理操作。假如我们仅需查询一次数据，那么其他三种方法的数据预处理的时间比线性搜索的时间还要更长。</li>
<li>适用于体量较小的数据，此情况下时间复杂度对效率影响较小。</li>
<li>适用于数据更新频率较高的场景，因为该方法不需要对数据进行任何额外维护。</li>
</ul>
<p><strong>二分查找</strong></p>
<ul>
<li>适用于大数据量的情况，效率表现稳定，最差时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
<li>数据量不能过大，因为存储数组需要连续的内存空间。</li>
<li>不适用于高频增删数据的场景，因为维护有序数组的开销较大。</li>
</ul>
<p><strong>哈希查找</strong></p>
<ul>
<li>适合对查询性能要求很高的场景，平均时间复杂度为 <span class="arithmatex">\(O(1)\)</span> 。</li>
<li>不适合需要有序数据或范围查找的场景，因为哈希表无法维护数据的有序性。</li>
<li>对哈希函数和哈希冲突处理策略的依赖性较高，具有较大的性能劣化风险。</li>
<li>不适合数据量过大的情况，因为哈希表需要额外空间来最大程度地减少冲突，从而提供良好的查询性能。</li>
</ul>
<p><strong>树查找</strong></p>
<ul>
<li>适用于海量数据，因为树节点在内存中是分散存储的。</li>
<li>适合需要维护有序数据或范围查找的场景。</li>
<li>在持续增删节点的过程中，二叉搜索树可能产生倾斜，时间复杂度劣化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
<li>若使用 AVL 树或红黑树，则各项操作可在 <span class="arithmatex">\(O(\log n)\)</span> 效率下稳定运行，但维护树平衡的操作会增加额外的开销。</li>
</ul>
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