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4.3. 列表
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4.4. 小结
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5.2. 队列
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6.3.2. 哈希算法的设计
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6.3.3. 常见哈希算法
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7. 树
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7.1. 二叉树
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7.4. 二叉搜索树
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7.6. 小结
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8.1. 堆
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8.2. 建堆操作
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9.2. 图基础操作
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10. 搜索
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10.2. 二分查找边界（ ）
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10.5. 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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11.5. 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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< / a >
< h1 id = "63" > 6.3. 哈希算法< a class = "headerlink" href = "#63" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 在上两节中，我们了解了哈希表的工作原理，以及哈希冲突的处理方法。然而，无论是开放寻址还是链地址法，< strong > 它们只能保证哈希表可以在发生冲突时正常工作，但无法减少哈希冲突的发生< / strong > 。< / p >
< p > 如果哈希冲突过于频繁，哈希表的性能则会急剧劣化。例如对于链地址哈希表，理想情况下键值对平均分布在各个桶中，达到最好的查询效率；最差情况下全部键值都被存储到同一个桶中，时间复杂度退化至 < span class = "arithmatex" > \(O(n)\)< / span > 。< / p >
< p > < img alt = "哈希冲突的最佳与最差情况" src = "../hash_algorithm.assets/hash_collision_best_worst_condition.png" / > < / p >
< p align = "center" > Fig. 哈希冲突的最佳与最差情况 < / p >
< p > < strong > 键值对的分布情况是由哈希函数决定的< / strong > 。回忆哈希函数的计算步骤，先计算哈希值，再对数组长度取模：< / p >
< div class = "highlight" > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "nv" > index< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > hash< span class = "o" > (< / span > key< span class = "o" > )< / span > < span class = "w" > < / span > %< span class = "w" > < / span > capacity
< / code > < / pre > < / div >
< p > 观察以上公式，当哈希表容量 < code > capacity< / code > 固定时，< strong > 哈希算法 < code > hash()< / code > 决定了输出值< / strong > ，进而决定了键值对在哈希表中的分布。因此，为了减小哈希冲突的发生概率，我们需要将注意力集中在哈希算法 < code > hash()< / code > 的设计上。< / p >
< h2 id = "631" > 6.3.1. 哈希算法的目标< a class = "headerlink" href = "#631" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 为了在编程语言中实现“既快又稳”的哈希表数据结构，哈希算法应包含以下特点：< / p >
< ul >
< li > < strong > 确定性< / strong > ：对于相同的输入，哈希算法应始终产生相同的输出。这样才能确保哈希表是可靠的。< / li >
< li > < strong > 效率高< / strong > ：计算哈希值的过程应该足够快。计算开销越小，哈希表的实用性越高。< / li >
< li > < strong > 均匀分布< / strong > ：哈希算法应使得键值对平均分布在哈希表中。分布越平均，哈希冲突的概率就越低。< / li >
< / ul >
< p > 实际上，哈希算法除了可以用于实现哈希表，还广泛应用于其他领域中，包括：< / p >
< ul >
< li > < strong > 密码存储< / strong > ：为了保护用户密码的安全，系统通常不会直接存储用户的明文密码，而是存储密码的哈希值。当用户输入密码时，系统会对输入的密码计算哈希值，然后与存储的哈希值进行比较。如果两者匹配，那么密码就被视为正确。< / li >
< li > < strong > 数据完整性检查< / strong > ：数据发送方可以计算数据的哈希值并将其一同发送；接收方可以重新计算接收到的数据的哈希值，并与接收到的哈希值进行比较。如果两者匹配，那么数据就被视为完整的。< / li >
< / ul >
< p > 对于密码学的相关应用，哈希算法需要满足更高的安全标准，以防止从哈希值推导出原始密码等逆向工程，包括：< / p >
< ul >
< li > < strong > 抗碰撞性< / strong > ：应当极其困难找到两个不同的输入，使得它们的哈希值相同。< / li >
< li > < strong > 雪崩效应< / strong > ：输入的微小变化应当导致输出的显著且不可预测的变化。< / li >
< / ul >
< p > 注意，< strong > “均匀分布”与“抗碰撞性”是两个独立的概念< / strong > ，满足均匀分布不一定满足抗碰撞性。例如，在随机输入 < code > key< / code > 下，哈希函数 < code > key % 100< / code > 可以产生均匀分布的输出。然而，该哈希算法过于简单，所有后两位相等的 < code > key< / code > 的输出都相同，因此我们可以很容易地从哈希值反推出可用的 < code > key< / code > ，从而破解密码。< / p >
< h2 id = "632" > 6.3.2. 哈希算法的设计< a class = "headerlink" href = "#632" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 哈希算法的设计是一个复杂且需要考虑许多因素的问题。然而，对于一些简单场景，我们也能设计一些简单的哈希算法，以字符串哈希为例：< / p >
< ul >
< li > < strong > 加法哈希< / strong > ：对输入的每个字符的 ASCII 码进行相加，将得到的总和作为哈希值。< / li >
< li > < strong > 乘法哈希< / strong > ：利用了乘法的不相关性，每轮乘以一个常数，将各个字符的 ASCII 码累积到哈希值中。< / li >
< li > < strong > 异或哈希< / strong > ：将输入数据的每个元素通过异或操作累积到一个哈希值中。< / li >
< li > < strong > 旋转哈希< / strong > ：将每个字符的 ASCII 码累积到一个哈希值中，每次累积之前都会对哈希值进行旋转操作。< / li >
< / ul >
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< div class = "highlight" > < span class = "filename" > simple_hash.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -[< / span > < span class = "n" > func< / span > < span class = "o" > ]< / span > < span class = "p" > {< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a >
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< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -[< / span > < span class = "n" > func< / span > < span class = "o" > ]< / span > < span class = "p" > {< / span > < span class = "n" > xorHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a >
< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -[< / span > < span class = "n" > func< / span > < span class = "o" > ]< / span > < span class = "p" > {< / span > < span class = "n" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "p" > [< / span > < span class = "k" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "p" > [< / span > < span class = "k" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > mulHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "p" > [< / span > < span class = "k" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > xorHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a >
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / a > < span class = "p" > [< / span > < span class = "k" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > add_hash< / span > < span class = "p" > (< / span > < span class = "n" > key< / span > < span class = "p" > :< / span > < span class = "nb" > str< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-2" name = "__codelineno-3-2" href = "#__codelineno-3-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 加法哈希" " " < / span >
< a id = "__codelineno-3-3" name = "__codelineno-3-3" href = "#__codelineno-3-3" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-3-4" name = "__codelineno-3-4" href = "#__codelineno-3-4" > < / a > < span class = "n" > modulus< / span > < span class = "o" > =< / span > < span class = "mi" > 1000000007< / span >
< a id = "__codelineno-3-5" name = "__codelineno-3-5" href = "#__codelineno-3-5" > < / a > < span class = "k" > for< / span > < span class = "n" > c< / span > < span class = "ow" > in< / span > < span class = "n" > key< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-6" name = "__codelineno-3-6" href = "#__codelineno-3-6" > < / a > < span class = "nb" > hash< / span > < span class = "o" > +=< / span > < span class = "nb" > ord< / span > < span class = "p" > (< / span > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-3-7" name = "__codelineno-3-7" href = "#__codelineno-3-7" > < / a > < span class = "k" > return< / span > < span class = "nb" > hash< / span > < span class = "o" > %< / span > < span class = "n" > modulus< / span >
< a id = "__codelineno-3-8" name = "__codelineno-3-8" href = "#__codelineno-3-8" > < / a >
< a id = "__codelineno-3-9" name = "__codelineno-3-9" href = "#__codelineno-3-9" > < / a > < span class = "k" > def< / span > < span class = "nf" > mul_hash< / span > < span class = "p" > (< / span > < span class = "n" > key< / span > < span class = "p" > :< / span > < span class = "nb" > str< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-10" name = "__codelineno-3-10" href = "#__codelineno-3-10" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 乘法哈希" " " < / span >
< a id = "__codelineno-3-11" name = "__codelineno-3-11" href = "#__codelineno-3-11" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-3-12" name = "__codelineno-3-12" href = "#__codelineno-3-12" > < / a > < span class = "n" > modulus< / span > < span class = "o" > =< / span > < span class = "mi" > 1000000007< / span >
< a id = "__codelineno-3-13" name = "__codelineno-3-13" href = "#__codelineno-3-13" > < / a > < span class = "k" > for< / span > < span class = "n" > c< / span > < span class = "ow" > in< / span > < span class = "n" > key< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-14" name = "__codelineno-3-14" href = "#__codelineno-3-14" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "mi" > 31< / span > < span class = "o" > *< / span > < span class = "nb" > hash< / span > < span class = "o" > +< / span > < span class = "nb" > ord< / span > < span class = "p" > (< / span > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-3-15" name = "__codelineno-3-15" href = "#__codelineno-3-15" > < / a > < span class = "k" > return< / span > < span class = "nb" > hash< / span > < span class = "o" > %< / span > < span class = "n" > modulus< / span >
< a id = "__codelineno-3-16" name = "__codelineno-3-16" href = "#__codelineno-3-16" > < / a >
< a id = "__codelineno-3-17" name = "__codelineno-3-17" href = "#__codelineno-3-17" > < / a > < span class = "k" > def< / span > < span class = "nf" > xor_hash< / span > < span class = "p" > (< / span > < span class = "n" > key< / span > < span class = "p" > :< / span > < span class = "nb" > str< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-18" name = "__codelineno-3-18" href = "#__codelineno-3-18" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 异或哈希" " " < / span >
< a id = "__codelineno-3-19" name = "__codelineno-3-19" href = "#__codelineno-3-19" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-3-20" name = "__codelineno-3-20" href = "#__codelineno-3-20" > < / a > < span class = "n" > modulus< / span > < span class = "o" > =< / span > < span class = "mi" > 1000000007< / span >
< a id = "__codelineno-3-21" name = "__codelineno-3-21" href = "#__codelineno-3-21" > < / a > < span class = "k" > for< / span > < span class = "n" > c< / span > < span class = "ow" > in< / span > < span class = "n" > key< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-22" name = "__codelineno-3-22" href = "#__codelineno-3-22" > < / a > < span class = "nb" > hash< / span > < span class = "o" > ^=< / span > < span class = "nb" > ord< / span > < span class = "p" > (< / span > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-3-23" name = "__codelineno-3-23" href = "#__codelineno-3-23" > < / a > < span class = "k" > return< / span > < span class = "nb" > hash< / span > < span class = "o" > %< / span > < span class = "n" > modulus< / span >
< a id = "__codelineno-3-24" name = "__codelineno-3-24" href = "#__codelineno-3-24" > < / a >
< a id = "__codelineno-3-25" name = "__codelineno-3-25" href = "#__codelineno-3-25" > < / a > < span class = "k" > def< / span > < span class = "nf" > rot_hash< / span > < span class = "p" > (< / span > < span class = "n" > key< / span > < span class = "p" > :< / span > < span class = "nb" > str< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-26" name = "__codelineno-3-26" href = "#__codelineno-3-26" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 旋转哈希" " " < / span >
< a id = "__codelineno-3-27" name = "__codelineno-3-27" href = "#__codelineno-3-27" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-3-28" name = "__codelineno-3-28" href = "#__codelineno-3-28" > < / a > < span class = "n" > modulus< / span > < span class = "o" > =< / span > < span class = "mi" > 1000000007< / span >
< a id = "__codelineno-3-29" name = "__codelineno-3-29" href = "#__codelineno-3-29" > < / a > < span class = "k" > for< / span > < span class = "n" > c< / span > < span class = "ow" > in< / span > < span class = "n" > key< / span > < span class = "p" > :< / span >
< a id = "__codelineno-3-30" name = "__codelineno-3-30" href = "#__codelineno-3-30" > < / a > < span class = "nb" > hash< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "nb" > hash< / span > < span class = "o" > < < < / span > < span class = "mi" > 4< / span > < span class = "p" > )< / span > < span class = "o" > ^< / span > < span class = "p" > (< / span > < span class = "nb" > hash< / span > < span class = "o" > > > < / span > < span class = "mi" > 28< / span > < span class = "p" > )< / span > < span class = "o" > ^< / span > < span class = "nb" > ord< / span > < span class = "p" > (< / span > < span class = "n" > c< / span > < span class = "p" > )< / span >
< a id = "__codelineno-3-31" name = "__codelineno-3-31" href = "#__codelineno-3-31" > < / a > < span class = "k" > return< / span > < span class = "nb" > hash< / span > < span class = "o" > %< / span > < span class = "n" > modulus< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-4-2" name = "__codelineno-4-2" href = "#__codelineno-4-2" > < / a >
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< a id = "__codelineno-4-4" name = "__codelineno-4-4" href = "#__codelineno-4-4" > < / a >
< a id = "__codelineno-4-5" name = "__codelineno-4-5" href = "#__codelineno-4-5" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > xorHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-4-6" name = "__codelineno-4-6" href = "#__codelineno-4-6" > < / a >
< a id = "__codelineno-4-7" name = "__codelineno-4-7" href = "#__codelineno-4-7" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-5-2" name = "__codelineno-5-2" href = "#__codelineno-5-2" > < / a >
< a id = "__codelineno-5-3" name = "__codelineno-5-3" href = "#__codelineno-5-3" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > mulHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-5-4" name = "__codelineno-5-4" href = "#__codelineno-5-4" > < / a >
< a id = "__codelineno-5-5" name = "__codelineno-5-5" href = "#__codelineno-5-5" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > xorHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-5-6" name = "__codelineno-5-6" href = "#__codelineno-5-6" > < / a >
< a id = "__codelineno-5-7" name = "__codelineno-5-7" href = "#__codelineno-5-7" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-6-2" name = "__codelineno-6-2" href = "#__codelineno-6-2" > < / a >
< a id = "__codelineno-6-3" name = "__codelineno-6-3" href = "#__codelineno-6-3" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > mulHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-6-4" name = "__codelineno-6-4" href = "#__codelineno-6-4" > < / a >
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< a id = "__codelineno-6-6" name = "__codelineno-6-6" href = "#__codelineno-6-6" > < / a >
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< div class = "highlight" > < span class = "filename" > simple_hash.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a >
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< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > xorHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-8-2" name = "__codelineno-8-2" href = "#__codelineno-8-2" > < / a >
< a id = "__codelineno-8-3" name = "__codelineno-8-3" href = "#__codelineno-8-3" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > mulHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-8-4" name = "__codelineno-8-4" href = "#__codelineno-8-4" > < / a >
< a id = "__codelineno-8-5" name = "__codelineno-8-5" href = "#__codelineno-8-5" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > xorHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-8-6" name = "__codelineno-8-6" href = "#__codelineno-8-6" > < / a >
< a id = "__codelineno-8-7" name = "__codelineno-8-7" href = "#__codelineno-8-7" > < / a > < span class = "na" > [class]< / span > < span class = "p" > {< / span > < span class = "n" > simple_hash< / span > < span class = "p" > }< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-9-2" name = "__codelineno-9-2" href = "#__codelineno-9-2" > < / a >
< a id = "__codelineno-9-3" name = "__codelineno-9-3" href = "#__codelineno-9-3" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > mulHash< / span > < span class = "p" > }< / span >
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< a id = "__codelineno-9-5" name = "__codelineno-9-5" href = "#__codelineno-9-5" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > xorHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > addHash< / span > < span class = "p" > }< / span >
< a id = "__codelineno-10-2" name = "__codelineno-10-2" href = "#__codelineno-10-2" > < / a >
< a id = "__codelineno-10-3" name = "__codelineno-10-3" href = "#__codelineno-10-3" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > mulHash< / span > < span class = "p" > }< / span >
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< a id = "__codelineno-10-7" name = "__codelineno-10-7" href = "#__codelineno-10-7" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > rotHash< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > simple_hash.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-11-1" name = "__codelineno-11-1" href = "#__codelineno-11-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > add_hash< / span > < span class = "p" > }< / span >
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< p > 观察发现，每种哈希算法的最后一步都是对大质数 < span class = "arithmatex" > \(1000000007\)< / span > 取模，以确保哈希值在合适的范围内。值得思考的是，为什么要强调对质数取模，对合数取模的弊端是什么？这是一个有趣的问题。< / p >
< p > 先抛出结论：< strong > 当我们使用大质数作为模数时，可以最大化地保证哈希值的均匀分布< / strong > 。因为质数不会与其他数字存在公约数，可以减少因取模操作而产生的周期性模式，从而避免哈希冲突。< / p >
< p > 举个例子，假设我们选择合数 < span class = "arithmatex" > \(9\)< / span > 作为模数，它可以被 < span class = "arithmatex" > \(3\)< / span > 整除。那么所有可以被 < span class = "arithmatex" > \(3\)< / span > 整除的 < code > key< / code > 都会被映射到 < span class = "arithmatex" > \(0\)< / span > , < span class = "arithmatex" > \(3\)< / span > , < span class = "arithmatex" > \(6\)< / span > 这三个哈希值。< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\text{modulus} & = 9 \newline
\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \cdots \} \newline
\text{hash} & = \{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\cdots \}
\end{aligned}
\]< / div >
< p > 如果输入 < code > key< / code > 恰好满足这种等差数列的数据分布，那么哈希值就会出现聚堆，从而加重哈希冲突。现在，假设将 < code > modulus< / code > 替换为质数 < span class = "arithmatex" > \(13\)< / span > ，由于 < code > key< / code > 和 < code > modulus< / code > 之间不存在公约数，输出的哈希值的均匀性会明显提升。< / p >
< div class = "arithmatex" > \[
\begin{aligned}
\text{modulus} & = 13 \newline
\text{key} & = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \cdots \} \newline
\text{hash} & = \{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \cdots \}
\end{aligned}
\]< / div >
< p > 值得强调的是，如果能够保证 < code > key< / code > 是随机均匀分布的，那么选择质数或者合数作为模数都是可以的，它们都能输出均匀分布的哈希值。而当 < code > key< / code > 的分布存在某种周期性时，对合数取模更容易出现聚集现象。< / p >
< p > 总而言之，我们通常选取质数作为模数，并且这个质数最好大一些，以提升哈希算法的稳健性。< / p >
< h2 id = "633" > 6.3.3. 常见哈希算法< a class = "headerlink" href = "#633" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 不难发现，以上介绍的简单哈希算法都比较“脆弱”，远远没有达到哈希算法的设计目标。例如，由于加法和异或满足交换律，因此加法哈希和异或哈希无法区分内容相同但顺序不同的字符串，这可能会加剧哈希冲突，并引起一些安全问题。< / p >
< p > 在实际中，我们通常会用一些标准哈希算法，例如 MD5, SHA-1, SHA-2, SHA3 等。它们可以将任意长度的输入数据映射到恒定长度的哈希值。近一个世纪以来，哈希算法处在不断升级与优化的过程中。一部分研究人员努力提升哈希算法的性能，另一部分研究人员和黑客则致力于寻找哈希算法的安全性问题。< / p >
< p > 直至目前， ， ， ， < / p >
< table >
< thead >
< tr >
< th > < / th >
< th > MD5< / th >
< th > SHA-1< / th >
< th > SHA-2< / th >
< th > SHA-3< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > 推出时间< / td >
< td > 1992< / td >
< td > 1995< / td >
< td > 2002< / td >
< td > 2008< / td >
< / tr >
< tr >
< td > 输出长度< / td >
< td > 128 bits< / td >
< td > 160 bits< / td >
< td > 256 / 512 bits< / td >
< td > 224/256/384/512 bits< / td >
< / tr >
< tr >
< td > 哈希冲突< / td >
< td > 较多< / td >
< td > 较多< / td >
< td > 很少< / td >
< td > 很少< / td >
< / tr >
< tr >
< td > 安全等级< / td >
< td > 低，已被成功攻击< / td >
< td > 低，已被成功攻击< / td >
< td > 高< / td >
< td > 高< / td >
< / tr >
< tr >
< td > 应用< / td >
< td > 已被弃用，仍用于数据完整性检查< / td >
< td > 已被弃用< / td >
< td > 加密货币交易验证、数字签名等< / td >
< td > 可用于替代 SHA-2< / td >
< / tr >
< / tbody >
< / table >
< h2 id = "634" > 6.3.4. 数据结构的哈希值< a class = "headerlink" href = "#634" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 我们知道，哈希表的 < code > key< / code > 可以是整数、小数或字符串等数据类型。编程语言通常会为这些数据类型提供内置的哈希算法，用于计算哈希表中的桶索引。以 Python 为例，我们可以调用 < code > hash()< / code > 函数来计算各种数据类型的哈希值，包括：< / p >
< ul >
< li > 整数和布尔量的哈希值就是其本身。< / li >
< li > 浮点数和字符串的哈希值计算较为复杂，有兴趣的同学请自行学习。< / li >
< li > 元组的哈希值是对其中每一个元素进行哈希，然后将这些哈希值组合起来，得到单一的哈希值。< / li >
< li > 对象的哈希值基于其内存地址生成。通过重写对象的哈希方法，可实现基于内容生成哈希值。< / li >
< / ul >
< div class = "tabbed-set tabbed-alternate" data-tabs = "2:11" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_11" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JavaScript< / label > < label for = "__tabbed_2_6" > TypeScript< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < label for = "__tabbed_2_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > built_in_hash.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > built_in_hash.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > built_in_hash.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "n" > num< / span > < span class = "o" > =< / span > < span class = "mi" > 3< / span >
< a id = "__codelineno-14-2" name = "__codelineno-14-2" href = "#__codelineno-14-2" > < / a > < span class = "n" > hash_num< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "n" > num< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-3" name = "__codelineno-14-3" href = "#__codelineno-14-3" > < / a > < span class = "c1" > # 整数 3 的哈希值为 3< / span >
< a id = "__codelineno-14-4" name = "__codelineno-14-4" href = "#__codelineno-14-4" > < / a >
< a id = "__codelineno-14-5" name = "__codelineno-14-5" href = "#__codelineno-14-5" > < / a > < span class = "n" > bol< / span > < span class = "o" > =< / span > < span class = "kc" > True< / span >
< a id = "__codelineno-14-6" name = "__codelineno-14-6" href = "#__codelineno-14-6" > < / a > < span class = "n" > hash_bol< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "n" > bol< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-7" name = "__codelineno-14-7" href = "#__codelineno-14-7" > < / a > < span class = "c1" > # 布尔量 True 的哈希值为 1< / span >
< a id = "__codelineno-14-8" name = "__codelineno-14-8" href = "#__codelineno-14-8" > < / a >
< a id = "__codelineno-14-9" name = "__codelineno-14-9" href = "#__codelineno-14-9" > < / a > < span class = "n" > dec< / span > < span class = "o" > =< / span > < span class = "mf" > 3.14159< / span >
< a id = "__codelineno-14-10" name = "__codelineno-14-10" href = "#__codelineno-14-10" > < / a > < span class = "n" > hash_dec< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "n" > dec< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-11" name = "__codelineno-14-11" href = "#__codelineno-14-11" > < / a > < span class = "c1" > # 小数 3.14159 的哈希值为 326484311674566659< / span >
< a id = "__codelineno-14-12" name = "__codelineno-14-12" href = "#__codelineno-14-12" > < / a >
< a id = "__codelineno-14-13" name = "__codelineno-14-13" href = "#__codelineno-14-13" > < / a > < span class = "nb" > str< / span > < span class = "o" > =< / span > < span class = "s2" > " Hello 算法" < / span >
< a id = "__codelineno-14-14" name = "__codelineno-14-14" href = "#__codelineno-14-14" > < / a > < span class = "n" > hash_str< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "nb" > str< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-15" name = "__codelineno-14-15" href = "#__codelineno-14-15" > < / a > < span class = "c1" > # 字符串 Hello 算法 的哈希值为 4617003410720528961< / span >
< a id = "__codelineno-14-16" name = "__codelineno-14-16" href = "#__codelineno-14-16" > < / a >
< a id = "__codelineno-14-17" name = "__codelineno-14-17" href = "#__codelineno-14-17" > < / a > < span class = "n" > tup< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "mi" > 12836< / span > < span class = "p" > ,< / span > < span class = "s2" > " 小哈" < / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-18" name = "__codelineno-14-18" href = "#__codelineno-14-18" > < / a > < span class = "n" > hash_tup< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "n" > tup< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-19" name = "__codelineno-14-19" href = "#__codelineno-14-19" > < / a > < span class = "c1" > # 元组 (12836, ' 小哈' ) 的哈希值为 1029005403108185979< / span >
< a id = "__codelineno-14-20" name = "__codelineno-14-20" href = "#__codelineno-14-20" > < / a >
< a id = "__codelineno-14-21" name = "__codelineno-14-21" href = "#__codelineno-14-21" > < / a > < span class = "n" > obj< / span > < span class = "o" > =< / span > < span class = "n" > ListNode< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-22" name = "__codelineno-14-22" href = "#__codelineno-14-22" > < / a > < span class = "n" > hash_obj< / span > < span class = "o" > =< / span > < span class = "nb" > hash< / span > < span class = "p" > (< / span > < span class = "n" > obj< / span > < span class = "p" > )< / span >
< a id = "__codelineno-14-23" name = "__codelineno-14-23" href = "#__codelineno-14-23" > < / a > < span class = "c1" > # 节点对象 < ListNode object at 0x1058fd810> 的哈希值为 274267521< / span >
< / code > < / pre > < / div >
< / div >
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< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > 在大多数编程语言中，< strong > 只有不可变对象才可作为哈希表的 < code > key< / code > < / strong > 。假如我们将列表（动态数组）作为 < code > key< / code > ，当列表的内容发生变化时，它的哈希值也随之改变，我们就无法在哈希表中查询到原先的 < code > value< / code > 了。< / p >
< p > 虽然自定义对象（例如链表节点）的成员变量是可变的，但它是可哈希的，这是因为对象的哈希值默认基于内存地址生成。即使对象的内容发生了变化，但它的内存地址不变，哈希值仍然是不变的。< / p >
< div class = "admonition tip" >
< p class = "admonition-title" > 向哈希函数加盐< / p >
< p > Python 解释器在每次启动时， （ ） < / p >
< / div >
< h2 id = "__comments" > 评论< / h2 >
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