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桶排序
11.7. 桶排序（ ）
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11. 排序算法
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11.7. 桶排序（ ）
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11.7. 桶排序（ ）
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11.7.1. 算法流程
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11.7.2. 算法特性
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11.7.3. 如何实现平均分配
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11.7. 小结
11.8 . 小结
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11.7.1. 算法流程
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11.7.2. 算法特性
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11.7.3. 如何实现平均分配
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< h1 id = "_1" > 桶排序< a class = "headerlink" href = "#_1" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 「桶排序 Bucket Sort」考虑设置 < span class = "arithmatex" > \(k\)< / span > 个桶，并将 < span class = "arithmatex" > \(n\)< / span > 个元素根据大小分配到 < span class = "arithmatex" > \(k\)< / span > 个桶中，< strong > 并在每个桶内部分别执行排序< / strong > ，由于桶之间的大小关系的确定的，因此最后按照桶之间的顺序将元素依次展开即可。< / p >
< p > 假设元素平均分布在各个桶内，则每个桶内元素数量为 < span class = "arithmatex" > \(\frac{n}{k}\)< / span > ；如果使用「快速排序」来实现桶内排序，则排序单个桶使用 < span class = "arithmatex" > \(O(\frac{n}{k} \log\frac{n}{k})\)< / span > 时间，排序所有桶使用 < span class = "arithmatex" > \(O(n \log\frac{n}{k})\)< / span > 时间。< strong > 当桶数量 < span class = "arithmatex" > \(k\)< / span > 比较大时，时间复杂度则趋向于 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > 。< / p >
< div class = "admonition note 计数排序与桶排序的关系" >
< p class = "admonition-title" > Note< / p >
< p > < strong > 计数排序可以看作是桶排序的一种特例< / strong > 。我们可以把计数排序中 < code > counter< / code > 的每个索引想象成一个桶，将统计数量的过程想象成把 < span class = "arithmatex" > \(n\)< / span > 个元素分配到对应的桶中，再根据桶之间的有序性输出结果，从而实现排序。< / p >
< h1 id = "117" > 11.7. 桶排序< a class = "headerlink" href = "#117" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 「桶排序 Bucket Sort」是分治思想的典型体现， ， ， ， ， < / p >
< h2 id = "1171" > 11.7.1. 算法流程< a class = "headerlink" href = "#1171" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 输入一个长度为 < span class = "arithmatex" > \(n\)< / span > 的数组，元素是范围 < span class = "arithmatex" > \([0, 1)\)< / span > 的浮点数，桶排序流程为：< / p >
< ol >
< li > 初始化 < span class = "arithmatex" > \(k\)< / span > 个桶，将 < span class = "arithmatex" > \(n\)< / span > 个元素分配至 < span class = "arithmatex" > \(k\)< / span > 个桶中；< / li >
< li > 对每个桶分别执行排序（本文采用编程语言的内置排序函数）；< / li >
< li > 按照桶的从小到大的顺序，合并结果；< / li >
< / ol >
< p > < img alt = "桶排序算法流程" src = "../bucket_sort.assets/bucket_sort_overview.png" / > < / p >
< p align = "center" > Fig. 桶排序算法流程 < / 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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< div class = "highlight" > < span class = "filename" > bucket_sort.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 桶排序 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kt" > void< / span > < span class = "w" > < / span > < span class = "nf" > bucketSort< / span > < span class = "p" > (< / span > < span class = "kt" > float< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > nums< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 k = n/3 个桶，预期向每个桶分配 3 个元素< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > k< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > nums< / span > < span class = "p" > .< / span > < span class = "na" > length< / span > < span class = "w" > < / span > < span class = "o" > /< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "w" > < / span > < span class = "n" > List< / span > < span class = "o" > < < / span > < span class = "n" > List< / span > < span class = "o" > < < / span > < span class = "n" > Float< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "n" > buckets< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > ArrayList< / span > < span class = "o" > < > < / span > < span class = "p" > ();< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > k< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "w" > < / span > < span class = "n" > buckets< / span > < span class = "p" > .< / span > < span class = "na" > add< / span > < span class = "p" > (< / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "n" > ArrayList< / span > < span class = "o" > < > < / span > < span class = "p" > ());< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "w" > < / span > < span class = "c1" > // 1. 将数组元素分配到各个桶中< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > float< / span > < span class = "w" > < / span > < span class = "n" > num< / span > < span class = "w" > < / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > nums< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "n" > num< / span > < span class = "w" > < / span > < span class = "o" > *< / span > < span class = "w" > < / span > < span class = "n" > k< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 将 num 添加进桶 i< / span >
< a id = "__codelineno-0-14" name = "__codelineno-0-14" href = "#__codelineno-0-14" > < / a > < span class = "w" > < / span > < span class = "n" > buckets< / span > < span class = "p" > .< / span > < span class = "na" > get< / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "p" > ).< / span > < span class = "na" > add< / span > < span class = "p" > (< / span > < span class = "n" > num< / span > < span class = "p" > );< / span >
< a id = "__codelineno-0-15" name = "__codelineno-0-15" href = "#__codelineno-0-15" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-16" name = "__codelineno-0-16" href = "#__codelineno-0-16" > < / a > < span class = "w" > < / span > < span class = "c1" > // 2. 对各个桶执行排序< / span >
< a id = "__codelineno-0-17" name = "__codelineno-0-17" href = "#__codelineno-0-17" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > List< / span > < span class = "o" > < < / span > < span class = "n" > Float< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > bucket< / span > < span class = "w" > < / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > buckets< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-18" name = "__codelineno-0-18" href = "#__codelineno-0-18" > < / a > < span class = "w" > < / span > < span class = "c1" > // 使用内置排序函数，也可以替换成其它排序算法< / span >
< a id = "__codelineno-0-19" name = "__codelineno-0-19" href = "#__codelineno-0-19" > < / a > < span class = "w" > < / span > < span class = "n" > Collections< / span > < span class = "p" > .< / span > < span class = "na" > sort< / span > < span class = "p" > (< / span > < span class = "n" > bucket< / span > < span class = "p" > );< / span >
< a id = "__codelineno-0-20" name = "__codelineno-0-20" href = "#__codelineno-0-20" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-21" name = "__codelineno-0-21" href = "#__codelineno-0-21" > < / a > < span class = "w" > < / span > < span class = "c1" > // 3. 遍历桶合并结果< / span >
< a id = "__codelineno-0-22" name = "__codelineno-0-22" href = "#__codelineno-0-22" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-23" name = "__codelineno-0-23" href = "#__codelineno-0-23" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > List< / span > < span class = "o" > < < / span > < span class = "n" > Float< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > bucket< / span > < span class = "w" > < / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > buckets< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-24" name = "__codelineno-0-24" href = "#__codelineno-0-24" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > float< / span > < span class = "w" > < / span > < span class = "n" > num< / span > < span class = "w" > < / span > < span class = "p" > :< / span > < span class = "w" > < / span > < span class = "n" > bucket< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-25" name = "__codelineno-0-25" href = "#__codelineno-0-25" > < / a > < span class = "w" > < / span > < span class = "n" > nums< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ++]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > num< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-26" name = "__codelineno-0-26" href = "#__codelineno-0-26" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-27" name = "__codelineno-0-27" href = "#__codelineno-0-27" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-28" name = "__codelineno-0-28" href = "#__codelineno-0-28" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< p > （图）< / p >
< p > 理论上桶排序的时间复杂度是 < span class = "arithmatex" > \(O(n)\)< / span > ， < strong > 但前提是需要将元素均匀分配到各个桶中< / strong > ，而这并不容易做到。假设想要把淘宝的 < span class = "arithmatex" > \(100\)< / span > 万件商品根据价格范围平均分配到 < span class = "arithmatex" > \(100\)< / span > 个桶中，而商品价格不是均匀分布的，例如 < span class = "arithmatex" > \(100\)< / span > 元以下的商品非常多、< span class = "arithmatex" > \(1000\)< / span > 元以上的商品非常少等。如果我们将价格区间平均划分为 < span class = "arithmatex" > \(100\)< / span > 份，那么各个桶内的商品数量差距会非常大。为了实现平均分配，我们一般这样做：< / p >
< ul >
< li > 先粗略设置分界线，将元素分配完后，< strong > 把元素较多的桶继续划分为多个桶< / strong > ，直至所有桶内元素数量合理为止；该做法本质上是一个递归树；< / li >
< li > 如果我们提前知道商品价格的概率分布，< strong > 则可以根据已知分布来设置每个桶的价格分界线< / strong > ；值得说明的是，数据分布不一定需要特意统计，也可以根据数据特点采用某种常见概率模型来近似，例如自然界的正态分布等；< / li >
< / ul >
< p > （图）< / p >
< p > 另外，排序桶内元素需要选择一种合适的排序算法，比如快速排序。< / p >
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< p class = "admonition-title" > 桶排序是计数排序的一种推广< / p >
< p > 从桶排序的角度，我们可以把计数排序中计数数组 < code > counter< / code > 的每个索引想象成一个桶，将统计数量的过程想象成把各个元素分配到对应的桶中，再根据桶之间的有序性输出结果，从而实现排序。< / p >
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< h2 id = "1172" > 11.7.2. 算法特性< a class = "headerlink" href = "#1172" title = "Permanent link" > ¶ < / a > < / h2 >
< p > < strong > 时间复杂度 < span class = "arithmatex" > \(O(n + k)\)< / span > < / strong > ：假设元素平均分布在各个桶内，则每个桶内元素数量为 < span class = "arithmatex" > \(\frac{n}{k}\)< / span > 。假设排序单个桶使用 < span class = "arithmatex" > \(O(\frac{n}{k} \log\frac{n}{k})\)< / span > 时间，则排序所有桶使用 < span class = "arithmatex" > \(O(n \log\frac{n}{k})\)< / span > 时间，< strong > 当桶数量 < span class = "arithmatex" > \(k\)< / span > 比较大时，时间复杂度则趋向于 < span class = "arithmatex" > \(O(n)\)< / span > < / strong > 。最后合并结果需要遍历 < span class = "arithmatex" > \(n\)< / span > 个桶，使用 < span class = "arithmatex" > \(O(k)\)< / span > 时间。< / p >
< p > 最差情况下，所有数据被分配到一个桶中，且排序算法退化至 < span class = "arithmatex" > \(O(n^2)\)< / span > ，此时使用 < span class = "arithmatex" > \(O(n^2)\)< / span > 时间，因此是“自适应排序”。< / p >
< p > < strong > 空间复杂度 < span class = "arithmatex" > \(O(n + k)\)< / span > < / strong > ：需要借助 < span class = "arithmatex" > \(k\)< / span > 个桶和共 < span class = "arithmatex" > \(n\)< / span > 个元素的额外空间，是“非原地排序”。< / p >
< p > 桶排序是否稳定取决于排序桶内元素的算法是否稳定。< / p >
< h2 id = "1173" > 11.7.3. 如何实现平均分配< a class = "headerlink" href = "#1173" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 桶排序的时间复杂度理论上可以达到 < span class = "arithmatex" > \(O(n)\)< / span > ， < strong > 难点是需要将元素均匀分配到各个桶中< / strong > ，因为现实中的数据往往都不是均匀分布的。举个例子，假设我们想要把淘宝的所有商品根据价格范围平均分配到 10 个桶中， ， < / p >
< p > 为了实现平均分配，我们可以先大致设置一个分界线，将数据粗略分到 3 个桶，分配完后，< strong > 再把商品较多的桶继续划分为 3 个桶，直至所有桶内元素数量大致平均为止< / strong > 。此方法本质上是生成一个递归树，让叶结点的值尽量平均。当然，不一定非要划分为 3 个桶，可以根据数据特点灵活选取。< / p >
< p > < img alt = "递归划分桶" src = "../bucket_sort.assets/scatter_in_buckets_recursively.png" / > < / p >
< p align = "center" > Fig. 递归划分桶 < / p >
< p > 如果我们提前知道商品价格的概率分布，< strong > 那么也可以根据数据概率分布来设置每个桶的价格分界线< / strong > 。注意，数据分布不一定需要特意去统计，也可以根据数据特点采用某种概率模型来近似。如下图所示，我们假设商品价格服从正态分布，就可以合理设置价格区间，将商品平均分配到各个桶中。< / p >
< p > < img alt = "根据概率分布划分桶" src = "../bucket_sort.assets/scatter_in_buckets_distribution.png" / > < / p >
< p align = "center" > Fig. 根据概率分布划分桶 < / p >
< h2 id = "__comments" > 评论< / h2 >
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krahets
2 years ago
