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<title>7.2.   哈希冲突处理 - Hello 算法</title>
<title>6.2.   哈希冲突处理 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
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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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6. &nbsp; &nbsp; 二分查找
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6. &nbsp; &nbsp; 二分查找
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6.1. &nbsp; 二分查找
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6.2. &nbsp; 二分查找边界
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
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6. &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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7.1. &nbsp; 哈希表
6.1. &nbsp; 哈希表
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<label class="md-nav__link md-nav__link--active" for="__toc">
7.2. &nbsp; 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
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<a href="./" class="md-nav__link md-nav__link--active">
7.2. &nbsp; 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
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7.2.1. &nbsp; 哈希表扩容
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6.2.1. &nbsp; 哈希表扩容
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</li>
<li class="md-nav__item">
<a href="#722" class="md-nav__link">
7.2.2. &nbsp; 链式地址
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6.2.2. &nbsp; 链式地址
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<li class="md-nav__item">
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6.2.3. &nbsp; 开放寻址
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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. &nbsp; &nbsp; 树
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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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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
7.3. &nbsp; 二叉树数组表示
</a>
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<li class="md-nav__item">
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7.4. &nbsp; 二叉搜索树
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.5. &nbsp; AVL 树 *
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7.6. &nbsp; 小结
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8. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 堆
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8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
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<li class="md-nav__item">
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8.2. &nbsp; 二叉树遍历
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8.2. &nbsp; 建堆操作 *
</a>
</li>
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<li class="md-nav__item">
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8.3. &nbsp; 二叉树数组表示
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
8.4. &nbsp; 二叉搜索树
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
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8.6. &nbsp; 小结
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8.3. &nbsp; 小结
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9. &nbsp; &nbsp; 堆
9. &nbsp; &nbsp; 图
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<li class="md-nav__item">
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9.1. &nbsp; 堆
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9.1. &nbsp; 图
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<li class="md-nav__item">
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9.2. &nbsp; 建堆操作 *
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9.2. &nbsp; 图基础操作
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<li class="md-nav__item">
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9.3. &nbsp; 小结
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9.3. &nbsp; 图的遍历
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9.4. &nbsp; 小结
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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10. &nbsp; &nbsp; 图
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<li class="md-nav__item">
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10.1. &nbsp; 图
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10.1. &nbsp; 二分查找（New）
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<li class="md-nav__item">
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10.2. &nbsp; 图基础操作
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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>
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<li class="md-nav__item">
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10.4. &nbsp; 小结
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10.4. &nbsp; 重识搜索算法
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10.5. &nbsp; 小结
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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
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12.1. &nbsp; 搜索算法（New）
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12.1. &nbsp; 回溯算法（New）
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<li class="md-nav__item">
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12.2. &nbsp; 哈希优化策略
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12.2. &nbsp; 全排列问题（New）
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12.3. &nbsp; 小结
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12.3. &nbsp; N 皇后问题（New）
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13. &nbsp; &nbsp; 回溯算法
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13.1. &nbsp; 回溯算法（New）
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13.1. &nbsp; 编程环境安装
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13.2. &nbsp; 全排列问题（New）
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
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13.2. &nbsp; 一起参与创作
</a>
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14.1. &nbsp; 编程环境安装
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14.2. &nbsp; 一起参与创作
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参考文献
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#721" class="md-nav__link">
7.2.1. &nbsp; 哈希表扩容
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6.2.1. &nbsp; 哈希表扩容
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</li>
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7.2.2. &nbsp; 链式地址
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6.2.2. &nbsp; 链式地址
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<h1 id="72">7.2. &nbsp; 哈希冲突<a class="headerlink" href="#72" title="Permanent link">&para;</a></h1>
<h1 id="62">6.2. &nbsp; 哈希冲突<a class="headerlink" href="#62" title="Permanent link">&para;</a></h1>
<p>在理想情况下，哈希函数应为每个输入生成唯一的输出，实现 key 和 value 的一一对应。然而实际上，向哈希函数输入不同的 key 却产生相同输出的情况是存在的，这种现象被称为「哈希冲突 Hash Collision」。哈希冲突可能导致查询结果错误，从而严重影响哈希表的可用性。</p>
<p>那么，为何会出现哈希冲突呢？从本质上看，由于哈希函数的输入空间通常远大于输出空间，因此多个输入产生相同输出的情况是不可避免的。例如，若输入空间为全体整数，而输出空间为固定大小的数组，则必然有多个整数映射至同一数组索引。</p>
<p>为了减轻哈希冲突，一方面，<strong>可以通过扩大哈希表容量来降低冲突概率</strong>。极端情况下，当输入空间和输出空间大小相等时，哈希表等同于数组，每个 key 都对应唯一的数组索引，可谓“大力出奇迹”。</p>
<p>另一方面，<strong>可以考虑优化哈希表的表示以缓解哈希冲突</strong>，常用方法包括「链式地址 Separate Chaining」和「开放寻址 Open Addressing」。</p>
<h2 id="721">7.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#721" title="Permanent link">&para;</a></h2>
<h2 id="621">6.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
<p>哈希函数的最后一步通常是对桶数量 <span class="arithmatex">\(n\)</span> 取余，作用是将哈希值映射到桶索引范围，从而将 key 放入对应的桶中。当哈希表容量越大（即 <span class="arithmatex">\(n\)</span> 越大）时，多个 key 被分配到同一个桶中的概率就越低，冲突就越少。</p>
<p>因此，<strong>当哈希表内的冲突总体较为严重时，编程语言通常通过扩容哈希表来缓解冲突</strong>。类似于数组扩容，哈希表扩容需将所有键值对从原哈希表迁移至新哈希表，开销较大。</p>
<p>编程语言通常使用「负载因子 Load Factor」来衡量哈希冲突的严重程度，<strong>定义为哈希表中元素数量除以桶数量</strong>，常作为哈希表扩容的触发条件。在 Java 中，当负载因子 <span class="arithmatex">\(&gt; 0.75\)</span> 时，系统会将 HashMap 容量扩展为原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
<h2 id="722">7.2.2. &nbsp; 链式地址<a class="headerlink" href="#722" title="Permanent link">&para;</a></h2>
<h2 id="622">6.2.2. &nbsp; 链式地址<a class="headerlink" href="#622" title="Permanent link">&para;</a></h2>
<p>在原始哈希表中，每个桶仅能存储一个键值对。<strong>链式地址将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中</strong>。</p>
<p><img alt="链式地址" src="../hash_collision.assets/hash_collision_chaining.png" /></p>
<p align="center"> Fig. 链式地址 </p>
@@ -1993,7 +1954,7 @@
<li><strong>查询效率降低</strong>，因为需要线性遍历链表来查找对应元素；</li>
</ul>
<p>为了提高操作效率，<strong>可以将链表转换为「AVL 树」或「红黑树」</strong>，将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
<h2 id="723">7.2.3. &nbsp; 开放寻址<a class="headerlink" href="#723" title="Permanent link">&para;</a></h2>
<h2 id="623">6.2.3. &nbsp; 开放寻址<a class="headerlink" href="#623" title="Permanent link">&para;</a></h2>
<p>「开放寻址」方法不引入额外的数据结构，而是通过“多次探测”来解决哈希冲突，<strong>探测方主要包括线性探测、平方探测、多次哈希</strong>。</p>
<h3 id="_1">线性探测<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<p>「线性探测」采用固定步长的线性查找来解决哈希冲突。</p>
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上一页
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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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6.1. &nbsp; 哈希表
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
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6.2. &nbsp; 二分查找边界
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
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6. &nbsp; &nbsp; 散列表
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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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7.1.2. &nbsp; 哈希函数
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6.1.2. &nbsp; 哈希函数
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7.1.3. &nbsp; 哈希冲突
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6.1.3. &nbsp; 哈希冲突
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<li class="md-nav__item">
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7. &nbsp; &nbsp; 树
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
</a>
</li>
<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>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.6. &nbsp; 小结
</a>
</li>
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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>
</li>
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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>
</li>
<li class="md-nav__item">
<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>
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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>
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9.4. &nbsp; 小结
</a>
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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10. &nbsp; &nbsp; 搜索算法
</label>
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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">
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10.2. &nbsp; 图基础操作
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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">
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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>
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10.5. &nbsp; 小结
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12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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12.1. &nbsp; 搜索算法（New）
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12.1. &nbsp; 回溯算法（New）
</a>
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12.2. &nbsp; 全排列问题（New）
</a>
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12.3. &nbsp; 小结
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12.3. &nbsp; N 皇后问题（New）
</a>
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13.1. &nbsp; 回溯算法（New）
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13.1. &nbsp; 编程环境安装
</a>
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</a>
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
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13.2. &nbsp; 一起参与创作
</a>
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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>
</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>
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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
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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>
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<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>
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<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
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7. &nbsp; &nbsp; 散列表
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7.4. &nbsp; 二叉搜索树
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7.5. &nbsp; AVL 树 *
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8.1. &nbsp; 二叉树
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8.1. &nbsp; 堆
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8.2. &nbsp; 建堆操作 *
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<h1 id="73">7.3. &nbsp; 小结<a class="headerlink" href="#73" title="Permanent link">&para;</a></h1>
<h1 id="63">6.3. &nbsp; 小结<a class="headerlink" href="#63" title="Permanent link">&para;</a></h1>
<ul>
<li>哈希表能够在 <span class="arithmatex">\(O(1)\)</span> 时间内将键 key 映射到值 value，效率非常高。</li>
<li>常见的哈希表操作包括查询、添加与删除键值对、遍历键值对等。</li>
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