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<title>6.2.   哈希冲突处理 - Hello 算法</title>
<title>7.2.   哈希冲突处理 - Hello 算法</title>
@ -79,7 +79,7 @@
<div data-md-component="skip">
<a href="#62" class="md-skip">
<a href="#72" class="md-skip">
跳转至
</a>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
</span>
</div>
@ -441,8 +441,6 @@
<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. &nbsp; 权衡时间与空间
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 二分查找
<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
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6. &nbsp; &nbsp; 二分查找
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<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
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</li>
</ul>
</nav>
</li>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" checked>
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 散列表
<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
6. &nbsp; &nbsp; 散列表
7. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
7.1. &nbsp; 哈希表
</a>
</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
</a>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#621" class="md-nav__link">
6.2.1. &nbsp; 哈希表扩容
<a href="#721" class="md-nav__link">
7.2.1. &nbsp; 哈希表扩容
</a>
</li>
<li class="md-nav__item">
<a href="#622" class="md-nav__link">
6.2.2. &nbsp; 链式地址
<a href="#722" class="md-nav__link">
7.2.2. &nbsp; 链式地址
</a>
</li>
<li class="md-nav__item">
<a href="#623" class="md-nav__link">
6.2.3. &nbsp; 开放寻址
<a href="#723" class="md-nav__link">
7.2.3. &nbsp; 开放寻址
</a>
<nav class="md-nav" aria-label="6.2.3. &nbsp; 开放寻址">
<nav class="md-nav" aria-label="7.2.3. &nbsp; 开放寻址">
<ul class="md-nav__list">
<li class="md-nav__item">
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
6.3. &nbsp; 小结
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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7. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
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/binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. &nbsp; 小结
7.3. &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; 堆
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
</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">
8.1. &nbsp; 堆
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
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 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
8.5. &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; 图
9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
<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>
@ -1243,8 +1193,8 @@
<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>
@ -1257,8 +1207,8 @@
<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>
@ -1271,22 +1221,8 @@
<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>
@ -1329,14 +1265,14 @@
<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
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
10. &nbsp; &nbsp; 查找算法
10. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1346,8 +1282,8 @@
<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>
@ -1360,8 +1296,8 @@
<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>
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<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>
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<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>
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<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>
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<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>
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<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">
<label class="md-nav__title" for="__nav_13">
<span class="md-nav__icon md-icon"></span>
12. &nbsp; &nbsp; 回溯算法
12. &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_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>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
</a>
</li>
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<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
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<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>
<li class="md-nav__item">
<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>
@ -1731,6 +1683,77 @@
<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>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
</a>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_15">
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<label class="md-nav__title" for="__nav_16">
<span class="md-nav__icon md-icon"></span>
参考文献
</label>
@ -1785,25 +1808,25 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#621" class="md-nav__link">
6.2.1. &nbsp; 哈希表扩容
<a href="#721" class="md-nav__link">
7.2.1. &nbsp; 哈希表扩容
</a>
</li>
<li class="md-nav__item">
<a href="#622" class="md-nav__link">
6.2.2. &nbsp; 链式地址
<a href="#722" class="md-nav__link">
7.2.2. &nbsp; 链式地址
</a>
</li>
<li class="md-nav__item">
<a href="#623" class="md-nav__link">
6.2.3. &nbsp; 开放寻址
<a href="#723" class="md-nav__link">
7.2.3. &nbsp; 开放寻址
</a>
<nav class="md-nav" aria-label="6.2.3. &nbsp; 开放寻址">
<nav class="md-nav" aria-label="7.2.3. &nbsp; 开放寻址">
<ul class="md-nav__list">
<li class="md-nav__item">
@ -1848,16 +1871,16 @@
<h1 id="62">6.2. &nbsp; 哈希冲突<a class="headerlink" href="#62" title="Permanent link">&para;</a></h1>
<h1 id="72">7.2. &nbsp; 哈希冲突<a class="headerlink" href="#72" 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="621">6.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
<h2 id="721">7.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#721" 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="622">6.2.2. &nbsp; 链式地址<a class="headerlink" href="#622" title="Permanent link">&para;</a></h2>
<h2 id="722">7.2.2. &nbsp; 链式地址<a class="headerlink" href="#722" 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>
@ -1874,7 +1897,7 @@
<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="623">6.2.3. &nbsp; 开放寻址<a class="headerlink" href="#623" title="Permanent link">&para;</a></h2>
<h2 id="723">7.2.3. &nbsp; 开放寻址<a class="headerlink" href="#723" 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>
@ -1984,7 +2007,7 @@
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