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6.4. 小结
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< a href = "../../chapter_preface/" > 0. 前言< / a >
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0. 前言
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0.1. 关于本书
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0.2. 如何使用本书
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0.3. 小结
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1. 初识算法
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1.1. 算法无处不在
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1.2. 算法是什么
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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4. 数组与链表
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4.1. 数组
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5.1. 栈
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5.4. 小结
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6. 散列表
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6.1. 哈希表（ ）
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6.2. 哈希冲突（ ）
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6.4. 小结
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7. 树
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7.1. 二叉树
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7.2. 二叉树遍历
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7.3. 二叉树数组表示
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7.4. 二叉搜索树
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7.5. AVL 树 *
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7.6. 小结
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8.2. 建堆操作
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8.3. Top-K 问题（ ）
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8.4. 小结
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9. 图
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9.1. 图
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9.2. 图基础操作
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9.3. 图的遍历
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9.4. 小结
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11.2. 选择排序
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11.3. 冒泡排序
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11.4. 插入排序
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11.5. 快速排序
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11.6. 归并排序
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11.7. 堆排序
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11.8. 桶排序
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11.9. 计数排序
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11.10. 基数排序
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11.11. 小结
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12.3. 汉诺塔问题（ ）
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< a href = "../../chapter_backtracking/" > 13. 回溯< / a >
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13.1. 回溯算法
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13.2. 全排列问题
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13.3. 子集和问题
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13.4. N 皇后问题
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13.5. 小结
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14. 动态规划
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14.1. 初探动态规划（ ）
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< a href = "../../chapter_dynamic_programming/dp_problem_features/" class = "md-nav__link" >
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14.2. DP 问题特性（ ）
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14.3. DP 解题思路（ ）
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14.4. 0-1 背包问题（ ）
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14.5. 完全背包问题（ ）
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14.6. 编辑距离问题（ ）
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14.7. 小结（ ）
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< label class = "md-nav__link" for = "__nav_16" id = "__nav_16_label" tabindex = "0" >
15. 附录
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15.1. 编程环境安装
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< a href = "../../chapter_appendix/contribution/" class = "md-nav__link" >
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15.2. 一起参与创作
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6.4.1. Q & A
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< h1 id = "64" > 6.4. 小结< a class = "headerlink" href = "#64" title = "Permanent link" > ¶ < / a > < / h1 >
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< ul >
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< li > 输入 < code > key< / code > ，哈希表能够在 < span class = "arithmatex" > \(O(1)\)< / span > 时间内查询到 < code > value< / code > ，效率非常高。< / li >
< li > 常见的哈希表操作包括查询、添加键值对、删除键值对和遍历哈希表等。< / li >
< li > 哈希函数将 < code > key< / code > 映射为数组索引，从而访问对应桶并获取 < code > value< / code > 。< / li >
< li > 两个不同的 < code > key< / code > 可能在经过哈希函数后得到相同的数组索引，导致查询结果出错，这种现象被称为哈希冲突。< / li >
< li > 哈希表容量越大，哈希冲突的概率就越低。因此可以通过扩容哈希表来缓解哈希冲突。与数组扩容类似，哈希表扩容操作的开销很大。< / li >
< li > 负载因子定义为哈希表中元素数量除以桶数量，反映了哈希冲突的严重程度，常用作触发哈希表扩容的条件。< / li >
< li > 链式地址通过将单个元素转化为链表，将所有冲突元素存储在同一个链表中。然而，链表过长会降低查询效率，可以进一步将链表转换为红黑树来提高效率。< / li >
< li > 开放寻址通过多次探测来处理哈希冲突。线性探测使用固定步长，缺点是不能删除元素，且容易产生聚集。多次哈希使用多个哈希函数进行探测，相较线性探测更不易产生聚集，但多个哈希函数增加了计算量。< / li >
< li > 不同编程语言采取了不同的哈希表实现。例如， < code > HashMap< / code > 使用链式地址，而 Python 的 < code > Dict< / code > 采用开放寻址。< / li >
< li > 在哈希表中，我们希望哈希算法具有确定性、高效率和均匀分布的特点。在密码学中，哈希算法还应该具备抗碰撞性和雪崩效应。< / li >
< li > 哈希算法通常采用大质数作为模数，以最大化地保证哈希值的均匀分布，减少哈希冲突。< / li >
< li > 常见的哈希算法包括 MD5, SHA-1, SHA-2, SHA3 等。MD5 常用语校验文件完整性， < / li >
< li > 编程语言通常会为数据类型提供内置哈希算法，用于计算哈希表中的桶索引。通常情况下，只有不可变对象是可哈希的。< / li >
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< h2 id = "641-q-a" > 6.4.1. Q & A< a class = "headerlink" href = "#641-q-a" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "admonition question" >
< p class = "admonition-title" > 哈希表的时间复杂度为什么不是 < span class = "arithmatex" > \(O(n)\)< / span > ？ < / p >
< p > 当哈希冲突比较严重时，哈希表的时间复杂度会退化至 < span class = "arithmatex" > \(O(n)\)< / span > 。当哈希函数设计的比较好、容量设置比较合理、冲突比较平均时，时间复杂度是 < span class = "arithmatex" > \(O(1)\)< / span > 。我们使用编程语言内置的哈希表时，通常认为时间复杂度是 < span class = "arithmatex" > \(O(1)\)< / span > 。< / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 为什么不使用哈希函数 < span class = "arithmatex" > \(f(x) = x\)< / span > 呢？这样就不会有冲突了< / p >
< p > 在 < span class = "arithmatex" > \(f(x) = x\)< / span > 哈希函数下，每个元素对应唯一的桶索引，这与数组等价。然而，输入空间通常远大于输出空间（数组长度），因此哈希函数的最后一步往往是对数组长度取模。换句话说，哈希表的目标是将一个较大的状态空间映射到一个较小的空间，并提供 < span class = "arithmatex" > \(O(1)\)< / span > 的查询效率。< / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 哈希表底层实现是数组、链表、二叉树，但为什么效率可以比他们更高呢？< / p >
< p > 首先，哈希表的时间效率变高，但空间效率变低了。哈希表有相当一部分的内存是未使用的，< / p >
< p > 其次，只是在特定使用场景下时间效率变高了。如果一个功能能够在相同的时间复杂度下使用数组或链表实现，那么通常比哈希表更快。这是因为哈希函数计算需要开销，时间复杂度的常数项更大。< / p >
< p > 最后，哈希表的时间复杂度可能发生劣化。例如在链式地址中，我们采取在链表或红黑树中执行查找操作，仍然有退化至 < span class = "arithmatex" > \(O(n)\)< / span > 时间的风险。< / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 多次哈希有不能直接删除元素的缺陷吗？对于标记已删除的空间，这个空间还能再次使用吗？< / p >
< p > 多次哈希是开放寻址的一种，开放寻址法都有不能直接删除元素的缺陷，需要通过标记删除。被标记为已删除的空间是可以再次被使用的。当将新元素插入哈希表，并且通过哈希函数找到了被标记为已删除的位置时，该位置可以被新的元素使用。这样做既能保持哈希表的探测序列不变，又能保证哈希表的空间使用率。< / p >
< / div >
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< p class = "admonition-title" > 为什么在线性探测中，查找元素的时候会出现哈希冲突呢？< / p >
< p > 查找的时候通过哈希函数找到对应的桶和键值对，发现 < code > key< / code > 不匹配，这就代表有哈希冲突。因此，线性探测法会根据预先设定的步长依次向下查找，直至找到正确的键值对或无法找到跳出为止。< / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 为什么哈希表扩容能够缓解哈希冲突？< / p >
< p > 哈希函数的最后一步往往是对数组长度 < span class = "arithmatex" > \(n\)< / span > 取余，让输出值落入在数组索引范围；在扩容后，数组长度 < span class = "arithmatex" > \(n\)< / span > 发生变化，而 < code > key< / code > 对应的索引也可能发生变化。原先落在同一个桶的多个 < code > key< / code > ，在扩容后可能会被分配到多个桶中，从而实现哈希冲突的缓解。< / p >
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< script id = "__config" type = "application/json" > { "base" : "../.." , "features" : [ "content.action.edit" , "content.code.annotate" , "content.code.copy" , "content.tabs.link" , "content.tooltips" , "navigation.indexes" , "navigation.top" , "navigation.footer" , "navigation.tracking" , "search.highlight" , "search.share" , "search.suggest" , "toc.follow" ] , "search" : "../../assets/javascripts/workers/search.208ed371.min.js" , "translations" : { "clipboard.copied" : "\u5df2\u590d\u5236" , "clipboard.copy" : "\u590d\u5236" , "search.result.more.one" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.more.other" : "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.none" : "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.one" : "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.other" : "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c" , "search.result.placeholder" : "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22" , "search.result.term.missing" : "\u7f3a\u5c11" , "select.version" : "\u9009\u62e9\u5f53\u524d\u7248\u672c" } } < / script >
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< script src = "../../assets/javascripts/bundle.fac441b0.min.js" > < / script >
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< script src = "../../javascripts/mathjax.js" > < / script >
< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
< / body >
< / html >
