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<link rel="canonical" href="https://www.hello-algo.com/chapter_hashing/hash_map/">
<link rel="prev" href="../../chapter_binary_search/binary_search_edge/">
<link rel="prev" href="../../chapter_stack_and_queue/summary/">
<link rel="next" href="../hash_collision/">
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<title>7.1.   哈希表 - Hello 算法</title>
<title>6.1.   哈希表 - Hello 算法</title>
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<div data-md-component="skip">
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<a href="#61" class="md-skip">
跳转至
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<div class="md-header__topic" data-md-component="header-topic">
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7.1. &nbsp; 哈希表
6.1. &nbsp; 哈希表
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7. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 散列表
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<label class="md-nav__link md-nav__link--active" for="__toc">
7.1. &nbsp; 哈希表
6.1. &nbsp; 哈希表
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7.1. &nbsp; 哈希表
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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7.1.1. &nbsp; 哈希表常用操作
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6.1.1. &nbsp; 哈希表常用操作
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<a href="#712" class="md-nav__link">
7.1.2. &nbsp; 哈希函数
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6.1.2. &nbsp; 哈希函数
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</li>
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7.1.3. &nbsp; 哈希冲突
<a href="#613" class="md-nav__link">
6.1.3. &nbsp; 哈希冲突
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7.2. &nbsp; 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
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<li class="md-nav__item">
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7.3. &nbsp; 小结
6.3. &nbsp; 小结
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7. &nbsp; &nbsp; 树
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7.1. &nbsp; 二叉树
</a>
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
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<li class="md-nav__item">
<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 *
</a>
</li>
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<a href="../../chapter_tree/summary/" class="md-nav__link">
7.6. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
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
<a href="../../chapter_heap/heap/" class="md-nav__link">
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>
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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>
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<a href="../../chapter_tree/summary/" class="md-nav__link">
8.6. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
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
</label>
<ul class="md-nav__list" data-md-scrollfix>
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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>
<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
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
<span class="md-nav__icon md-icon"></span>
</label>
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
</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">
10.1. &nbsp; 图
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.5. &nbsp; 小结
</a>
</li>
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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
<span class="md-nav__icon md-icon"></span>
</label>
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12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<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 href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法（New）
</a>
</li>
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<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 href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
12.2. &nbsp; 全排列问题（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题（New）
</a>
</li>
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
13. &nbsp; &nbsp; 回溯算法
13. &nbsp; &nbsp; 附录
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13. &nbsp; &nbsp; 回溯算法
13. &nbsp; &nbsp; 附录
</label>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
13.2. &nbsp; 全排列问题（New）
</a>
</li>
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
</a>
</li>
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
14. &nbsp; &nbsp; 附录
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14. &nbsp; &nbsp; 附录
</label>
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<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
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参考文献
</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#711" class="md-nav__link">
7.1.1. &nbsp; 哈希表常用操作
<a href="#611" class="md-nav__link">
6.1.1. &nbsp; 哈希表常用操作
</a>
</li>
<li class="md-nav__item">
<a href="#712" class="md-nav__link">
7.1.2. &nbsp; 哈希函数
<a href="#612" class="md-nav__link">
6.1.2. &nbsp; 哈希函数
</a>
</li>
<li class="md-nav__item">
<a href="#713" class="md-nav__link">
7.1.3. &nbsp; 哈希冲突
<a href="#613" class="md-nav__link">
6.1.3. &nbsp; 哈希冲突
</a>
</li>
@ -1927,7 +1888,7 @@
<h1 id="71">7.1. &nbsp; 哈希表<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
<h1 id="61">6.1. &nbsp; 哈希表<a class="headerlink" href="#61" 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>
@ -1967,7 +1928,7 @@
</tbody>
</table>
</div>
<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>
<h2 id="611">6.1.1. &nbsp; 哈希表常用操作<a class="headerlink" href="#611" 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>
<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label></div>
<div class="tabbed-content">
@ -2293,7 +2254,7 @@
</div>
</div>
</div>
<h2 id="712">7.1.2. &nbsp; 哈希函数<a class="headerlink" href="#712" title="Permanent link">&para;</a></h2>
<h2 id="612">6.1.2. &nbsp; 哈希函数<a class="headerlink" href="#612" 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>
@ -3145,7 +3106,7 @@ f(x) = x \bmod {100}
</div>
</div>
</div>
<h2 id="713">7.1.3. &nbsp; 哈希冲突<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
<h2 id="613">6.1.3. &nbsp; 哈希冲突<a class="headerlink" href="#613" 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">\(f(x) = x \bmod 100\)</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
@ -3237,7 +3198,7 @@ f(12836) = f(20336) = 36
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