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<link rel="canonical" href="https://www.hello-algo.com/chapter_hashing/hash_map/">
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@ -25,7 +25,7 @@
<title>6.1.   哈希表 - Hello 算法</title>
<title>7.1.   哈希表 - Hello 算法</title>
@ -79,7 +79,7 @@
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跳转至
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<div class="md-header__topic" data-md-component="header-topic">
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6.1. &nbsp; 哈希表
7.1. &nbsp; 哈希表
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2.4. &nbsp; 权衡时间与空间
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2.5. &nbsp; 小结
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6. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 散列表
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6.1. &nbsp; 哈希表
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6.1. &nbsp; 哈希表
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6.1.1. &nbsp; 哈希表效率
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7.1.1. &nbsp; 哈希表常用操作
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6.1.2. &nbsp; 哈希表常用操作
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7.1.2. &nbsp; 哈希函数
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6.1.3. &nbsp; 哈希函数
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6.1.4. &nbsp; 哈希冲突
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7.1.3. &nbsp; 哈希冲突
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6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
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@ -964,126 +996,7 @@
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7.1. &nbsp; 二叉树
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7.2. &nbsp; 二叉树遍历
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. &nbsp; 小结
7.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
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8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. &nbsp; 堆
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
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@ -1155,8 +1072,8 @@
<li class="md-nav__item">
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8.2. &nbsp; 建堆操作 *
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
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@ -1169,8 +1086,36 @@
<li class="md-nav__item">
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8.3. &nbsp; 小结
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.4. &nbsp; AVL 树 *
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<a href="../../chapter_tree/summary/" class="md-nav__link">
8.5. &nbsp; 小结
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. &nbsp; 图
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
</a>
</li>
@ -1244,8 +1187,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
</a>
</li>
@ -1258,22 +1201,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
9.4. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &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; 查找算法
10. &nbsp; &nbsp; 图
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10. &nbsp; &nbsp; 查找算法
10. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1333,8 +1262,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
<a href="../../chapter_graph/graph/" class="md-nav__link">
10.1. &nbsp; 图
</a>
</li>
@ -1347,8 +1276,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. &nbsp; 二分查找
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
</a>
</li>
@ -1361,8 +1290,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
</a>
</li>
@ -1375,7 +1304,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
<a href="../../chapter_graph/summary/" class="md-nav__link">
10.4. &nbsp; 小结
</a>
</li>
@ -1517,7 +1446,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
</li>
@ -1531,7 +1460,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
</a>
</li>
@ -1545,7 +1474,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
</li>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1613,8 +1546,36 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法（New）
<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>
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<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
</a>
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@ -1648,19 +1609,17 @@
<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_14">
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13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
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@ -1670,22 +1629,8 @@
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
</a>
</li>
@ -1718,6 +1663,77 @@
<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
14. &nbsp; &nbsp; 附录
<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_15">
<span class="md-nav__icon md-icon"></span>
14. &nbsp; &nbsp; 附录
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
</a>
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
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@ -1730,8 +1746,8 @@
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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参考文献
</label>
@ -1772,29 +1788,22 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#611" class="md-nav__link">
6.1.1. &nbsp; 哈希表效率
<a href="#711" class="md-nav__link">
7.1.1. &nbsp; 哈希表常用操作
</a>
</li>
<li class="md-nav__item">
<a href="#612" class="md-nav__link">
6.1.2. &nbsp; 哈希表常用操作
<a href="#712" class="md-nav__link">
7.1.2. &nbsp; 哈希函数
</a>
</li>
<li class="md-nav__item">
<a href="#613" class="md-nav__link">
6.1.3. &nbsp; 哈希函数
</a>
</li>
<li class="md-nav__item">
<a href="#614" class="md-nav__link">
6.1.4. &nbsp; 哈希冲突
<a href="#713" class="md-nav__link">
7.1.3. &nbsp; 哈希冲突
</a>
</li>
@ -1822,30 +1831,21 @@
<h1 id="61">6.1. &nbsp; 哈希表<a class="headerlink" href="#61" title="Permanent link">&para;</a></h1>
<h1 id="71">7.1. &nbsp; 哈希表<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
<p>哈希表通过建立「键 key」与「值 value」之间的映射，实现高效的元素查询。具体而言，我们向哈希表输入一个 key，则可以在 <span class="arithmatex">\(O(1)\)</span> 时间内获取对应的 value 。</p>
<p>以一个包含 <span class="arithmatex">\(n\)</span> 个学生的数据库为例，每个学生都有“姓名 <code>name</code>”和“学号 <code>id</code>”两项数据。假如我们希望实现查询功能，例如“输入一个学号，返回对应的姓名”，则可以采用哈希表来实现。</p>
<p><img alt="哈希表的抽象表示" src="../hash_map.assets/hash_map.png" /></p>
<p align="center"> Fig. 哈希表的抽象表示 </p>
<h2 id="611">6.1.1. &nbsp; 哈希表效率<a class="headerlink" href="#611" title="Permanent link">&para;</a></h2>
<p>除哈希表外，还可以使用以下数据结构来实现上述查询功能：</p>
<ol>
<li><strong>无序数组</strong>：每个元素为 <code>[学号, 姓名]</code> ；</li>
<li><strong>有序数组</strong>：将 <code>1.</code> 中的数组按照学号从小到大排序；</li>
<li><strong>链表</strong>：每个节点的值为 <code>[学号, 姓名]</code> ；</li>
<li><strong>二叉搜索树</strong>：每个节点的值为 <code>[学号, 姓名]</code> ，根据学号大小来构建树；</li>
</ol>
<p>各项操作的时间复杂度如下表所示（详解可见<a href="https://www.hello-algo.com/chapter_tree/binary_search_tree/">二叉搜索树章节</a>）。无论是查找元素还是增删元素，哈希表的时间复杂度都是 <span class="arithmatex">\(O(1)\)</span> ，全面胜出！</p>
<p>除哈希表外，我们还可以使用数组或链表实现查询功能，各项操作的时间复杂度如下表所示。</p>
<p>在哈希表中增删查改的时间复杂度都是 <span class="arithmatex">\(O(1)\)</span> ，全面胜出！因此，哈希表常用于对查找效率要求较高的场景。</p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>无序数组</th>
<th>有序数组</th>
<th>数组</th>
<th>链表</th>
<th>二叉搜索树</th>
<th>哈希表</th>
</tr>
</thead>
@ -1853,31 +1853,25 @@
<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">\(O(n)\)</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">\(O(1)\)</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">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
</tbody>
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<h2 id="612">6.1.2. &nbsp; 哈希表常用操作<a class="headerlink" href="#612" title="Permanent link">&para;</a></h2>
<h2 id="711">7.1.1. &nbsp; 哈希表常用操作<a class="headerlink" href="#711" title="Permanent link">&para;</a></h2>
<p>哈希表的基本操作包括 <strong>初始化、查询操作、添加与删除键值对</strong>。</p>
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<h2 id="613">6.1.3. &nbsp; 哈希函数<a class="headerlink" href="#613" title="Permanent link">&para;</a></h2>
<h2 id="712">7.1.2. &nbsp; 哈希函数<a class="headerlink" href="#712" title="Permanent link">&para;</a></h2>
<p>哈希表的底层实现为数组，同时可能包含链表、二叉树（红黑树）等数据结构，以提高查询性能（将在下节讨论）。</p>
<p>首先考虑最简单的情况，<strong>仅使用一个数组来实现哈希表</strong>。通常，我们将数组中的每个空位称为「桶 Bucket」，用于存储键值对。</p>
<p>我们将键值对 key, value 封装成一个类 <code>Entry</code> ，并将所有 <code>Entry</code> 放入数组中。这样，数组中的每个 <code>Entry</code> 都具有唯一的索引。为了建立 key 和索引之间的映射关系，我们需要使用「哈希函数 Hash Function」。</p>
@ -3072,7 +3066,7 @@ f(x) = x \% 100
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<h2 id="614">6.1.4. &nbsp; 哈希冲突<a class="headerlink" href="#614" title="Permanent link">&para;</a></h2>
<h2 id="713">7.1.3. &nbsp; 哈希冲突<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
<p>细心的你可能已经注意到，<strong>在某些情况下，哈希函数 <span class="arithmatex">\(f(x) = x % 100\)</span> 可能无法正常工作</strong>。具体来说，当输入的 key 后两位相同时，哈希函数的计算结果也会相同，从而指向同一个 value 。例如，查询学号为 <span class="arithmatex">\(12836\)</span> 和 <span class="arithmatex">\(20336\)</span> 的两个学生时，我们得到：</p>
<div class="arithmatex">\[
f(12836) = f(20336) = 36
@ -3164,7 +3158,7 @@ f(12836) = f(20336) = 36
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