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写在前面
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引言
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引言
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算法无处不在
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算法是什么
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计算复杂度
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计算复杂度
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时间复杂度
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空间复杂度
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权衡时间与空间
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小结
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数据结构简介
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数据结构简介
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数据与内存
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数据结构分类
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小结
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数组与链表
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数组与链表
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链表（ ）
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列表（ ）
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小结
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栈与队列
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栈与队列
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栈（ ）
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< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
队列（ ）
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双向队列（ ）
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小结
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散列表
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散列表
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哈希表（ ）
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哈希冲突处理
< span class = "md-nav__icon md-icon" > < / span >
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哈希冲突处理
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目录
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6.2.1. 哈希表扩容
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< a href = "#622" class = "md-nav__link" >
6.2.2. 链式地址
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< a href = "#623" class = "md-nav__link" >
6.2.3. 开放寻址
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线性探测
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
多次哈希
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< / ul >
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< / li >
< li class = "md-nav__item" >
< a href = "../summary/" class = "md-nav__link" >
小结
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二叉树
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
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二叉树
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二叉树（ ）
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< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
二叉树遍历
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< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
二叉搜索树
< / a >
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< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
AVL 树 *
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< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
小结
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< label class = "md-nav__link" for = "__nav_9" id = "__nav_9_label" tabindex = "0" >
堆
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< / label >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
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堆
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< li class = "md-nav__item" >
< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
堆（ ）
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< / ul >
< / nav >
< / li >
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图
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< / label >
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< label class = "md-nav__title" for = "__nav_10" >
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图
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图（ ）
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
图基础操作
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< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
图的遍历
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查找算法
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查找算法
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/linear_search/" class = "md-nav__link" >
线性查找
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< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
二分查找
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "../../chapter_searching/hashing_search/" class = "md-nav__link" >
哈希查找
< / a >
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< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
小结
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排序算法
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排序算法
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排序简介
< / a >
< / li >
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
冒泡排序
< / a >
< / li >
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< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
插入排序
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< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
快速排序
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< / li >
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< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
归并排序
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小结
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< a href = "../../chapter_reference/" > 参考文献< / a >
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参考文献
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目录
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6.2.1. 哈希表扩容
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< a href = "#622" class = "md-nav__link" >
6.2.2. 链式地址
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< a href = "#623" class = "md-nav__link" >
6.2.3. 开放寻址
< / a >
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< ul class = "md-nav__list" >
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< a href = "#_1" class = "md-nav__link" >
线性探测
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#_2" class = "md-nav__link" >
多次哈希
< / a >
< / li >
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< / nav >
< / li >
< / ul >
< / nav >
< / div >
< / div >
< / div >
< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
< a href = "https://github.com/krahets/hello-algo/tree/main/docs/chapter_hashing/hash_collision.md" title = "编辑此页" class = "md-content__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
< / a >
< h1 id = "62" > 6.2. 哈希冲突< a class = "headerlink" href = "#62" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 理想情况下，哈希函数应该为每个输入产生唯一的输出，使得 key 和 value 一一对应。而实际上，往往存在向哈希函数输入不同的 key 而产生相同输出的情况，这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误， < / p >
< p > 那么，为什么会出现哈希冲突呢？本质上看，< strong > 由于哈希函数的输入空间往往远大于输出空间< / strong > ，因此不可避免地会出现多个输入产生相同输出的情况，即为哈希冲突。比如，输入空间是全体整数，输出空间是一个固定大小的桶（数组）的索引范围，那么必定会有多个整数同时映射到一个桶索引。< / p >
< p > 为了缓解哈希冲突，一方面，我们可以通过「哈希表扩容」来减小冲突概率。极端情况下，当输入空间和输出空间大小相等时，哈希表就等价于数组了，可谓“大力出奇迹”。< / p >
< p > 另一方面，< strong > 考虑通过优化数据结构以缓解哈希冲突< / strong > ，常见的方法有「链式地址」和「开放寻址」。< / p >
< h2 id = "621" > 6.2.1. 哈希表扩容< a class = "headerlink" href = "#621" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 「负载因子 Load Factor」定义为 < strong > 哈希表中元素数量除以桶槽数量（即数组大小）< / strong > ，代表哈希冲突的严重程度。< / p >
< p > < strong > 负载因子常用作哈希表扩容的触发条件< / strong > 。比如在 Java 中，当负载因子 < span class = "arithmatex" > \(> 0.75\)< / span > 时则触发扩容，将 HashMap 大小扩充至原先的 < span class = "arithmatex" > \(2\)< / span > 倍。< / p >
< p > 与数组扩容类似，< strong > 哈希表扩容操作的开销很大< / strong > ，因为需要将所有键值对从原哈希表依次移动至新哈希表。< / p >
< h2 id = "622" > 6.2.2. 链式地址< a class = "headerlink" href = "#622" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 在原始哈希表中，桶内的每个地址只能存储一个元素（即键值对）。< strong > 考虑将单个元素转化成一个链表，将所有冲突元素都存储在一个链表中< / strong > 。< / p >
< p > < img alt = "hash_collision_chaining" src = "../hash_collision.assets/hash_collision_chaining.png" / > < / p >
< p > 链式地址下，哈希表操作方法为：< / p >
< ul >
< li > < strong > 查询元素< / strong > ：先将 key 输入到哈希函数得到桶内索引，即可访问链表头结点，再通过遍历链表查找对应 value 。< / li >
< li > < strong > 添加元素< / strong > ：先通过哈希函数访问链表头部，再将结点（即键值对）添加到链表头部即可。< / li >
< li > < strong > 删除元素< / strong > ：同样先根据哈希函数结果访问链表头部，再遍历链表查找对应结点，删除之即可。< / li >
< / ul >
< p > 链式地址虽然解决了哈希冲突问题，但仍存在局限性，包括：< / p >
< ul >
< li > < strong > 占用空间变大< / strong > ，因为链表或二叉树包含结点指针，相比于数组更加耗费内存空间；< / li >
< 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 > 为了缓解时间效率问题，< strong > 可以把「链表」转化为「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. 开放寻址< a class = "headerlink" href = "#623" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 「开放寻址」不引入额外数据结构，而是通过“多次探测”来解决哈希冲突。根据探测方法的不同，主要分为 < strong > 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" > ¶ < / a > < / h3 >
< p > 「线性探测」使用固定步长的线性查找来解决哈希冲突。< / p >
< p > < strong > 插入元素< / strong > ：如果出现哈希冲突，则从冲突位置向后线性遍历（步长一般取 1 ），直到找到一个空位，则将元素插入到该空位中。< / p >
< p > < strong > 查找元素< / strong > ：若出现哈希冲突，则使用相同步长执行线性查找，会遇到两种情况：< / p >
< ol >
< li > 找到对应元素，返回 value 即可；< / li >
< li > 若遇到空位，则说明查找键值对不在哈希表中；< / li >
< / ol >
< p > < img alt = "hash_collision_linear_probing" src = "../hash_collision.assets/hash_collision_linear_probing.png" / > < / p >
< p > 线性探测存在以下缺陷：< / p >
< ul >
< li > < strong > 不能直接删除元素< / strong > 。删除元素会导致桶内出现一个空位，在查找其他元素时，该空位有可能导致程序认为元素不存在（即上述第 < code > 2.< / code > 种情况）。因此需要借助一个标志位来标记删除元素。< / li >
< li > < strong > 容易产生聚集< / strong > 。桶内被占用的连续位置越长，这些连续位置发生哈希冲突的可能性越大，从而进一步促进这一位置的“聚堆生长”，最终导致增删查改操作效率的劣化。< / li >
< / ul >
< h3 id = "_2" > 多次哈希< a class = "headerlink" href = "#_2" title = "Permanent link" > ¶ < / a > < / h3 >
< p > 顾名思义，「多次哈希」的思路是使用多个哈希函数 < span class = "arithmatex" > \(f_1(x)\)< / span > , < span class = "arithmatex" > \(f_2(x)\)< / span > , < span class = "arithmatex" > \(f_3(x)\)< / span > , < span class = "arithmatex" > \(\cdots\)< / span > 进行探测。< / p >
< p > < strong > 插入元素< / strong > ：若哈希函数 < span class = "arithmatex" > \(f_1(x)\)< / span > 出现冲突，则尝试 < span class = "arithmatex" > \(f_2(x)\)< / span > ，以此类推……直到找到空位后插入元素。< / p >
< p > < strong > 查找元素< / strong > ：以相同的哈希函数顺序查找，存在两种情况：< / p >
< ol >
< li > 找到目标元素，则返回之；< / li >
< li > 到空位或已尝试所有哈希函数，说明哈希表中无此元素；< / li >
< / ol >
< p > 相比于「线性探测」，「多次哈希」方法更不容易产生聚集，代价是多个哈希函数增加了额外计算量。< / p >
< div class = "admonition note" >
< p class = "admonition-title" > 工业界方案< / p >
< p > Java 采用「链式地址」。在 JDK 1.8 之后， < / p >
< p > Python 采用「开放寻址」。字典 dict 使用伪随机数进行探测。 < / p >
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