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krahets/hello-algo
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</div>
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</a>
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</div>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
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0. 写在前面
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_1">
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<span class="md-nav__icon md-icon"></span>
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0. 写在前面
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
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0.1. 关于本书
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_preface/suggestions/" class="md-nav__link">
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0.2. 如何使用本书
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_preface/summary/" class="md-nav__link">
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0.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
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1. 引言
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_2">
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<span class="md-nav__icon md-icon"></span>
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1. 引言
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
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1.1. 算法无处不在
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
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1.2. 算法是什么
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_introduction/summary/" class="md-nav__link">
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1.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 复杂度分析
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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2. 复杂度分析
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
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2.1. 算法效率评估
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
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2.2. 时间复杂度
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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2.3. 空间复杂度
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
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2.4. 权衡时间与空间
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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2.5. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
|
|
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3. 数据结构简介
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|
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<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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|
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
|
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3. 数据结构简介
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/data_and_memory/" class="md-nav__link">
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3.1. 数据与内存
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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3.2. 数据结构分类
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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3.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
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4. 数组与链表
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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4.1. 数组
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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4.2. 链表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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4.3. 列表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
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5. 栈与队列
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|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
|
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5. 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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5.2. 队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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5.3. 双向队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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5.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</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" >
|
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
|
|
|
6. 散列表
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
|
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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">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
6. 散列表
|
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</label>
|
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|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
|
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6.1. 哈希表
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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6.2. 哈希冲突处理
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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6.3. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
|
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7. 树
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
|
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|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_8">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
7. 树
|
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|
</label>
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<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
7.1. 二叉树
|
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|
</a>
|
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</li>
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
|
|
7.2. 二叉树遍历
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
|
|
7.3. 二叉搜索树
|
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|
</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
|
|
7.4. AVL 树 *
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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7.5. 小结
|
|
|
</a>
|
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</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
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|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
|
|
8. 堆
|
|
|
<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. 堆
|
|
|
</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. 堆
|
|
|
</a>
|
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
8.2. 建堆操作 *
|
|
|
</a>
|
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|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
|
8.3. 小结
|
|
|
</a>
|
|
|
</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
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|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
|
|
|
9. 图
|
|
|
<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. 图
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
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|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
|
|
9.1. 图
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
|
|
9.2. 图基础操作
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|
|
9.3. 图的遍历（New）
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|
|
9.4. 小结
|
|
|
</a>
|
|
|
</li>
|
|
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|
</ul>
|
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|
</nav>
|
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</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_11" >
|
|
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|
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
|
|
|
10. 查找算法
|
|
|
<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. 查找算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
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|
|
|
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|
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|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|
|
10.1. 线性查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|
|
10.2. 二分查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
|
|
10.3. 哈希查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|
|
10.4. 小结
|
|
|
</a>
|
|
|
</li>
|
|
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|
|
|
</ul>
|
|
|
</nav>
|
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</li>
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|
|
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
|
|
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|
|
|
|
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|
|
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
|
|
|
11. 排序算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
|
|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
11. 排序算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="../intro_to_sort/" class="md-nav__link">
|
|
|
11.1. 排序简介
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../bubble_sort/" class="md-nav__link">
|
|
|
11.2. 冒泡排序
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../insertion_sort/" class="md-nav__link">
|
|
|
11.3. 插入排序
|
|
|
</a>
|
|
|
</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="../quick_sort/" class="md-nav__link">
|
|
|
11.4. 快速排序
|
|
|
</a>
|
|
|
</li>
|
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<li class="md-nav__item">
|
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<a href="../merge_sort/" class="md-nav__link">
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11.5. 归并排序
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11.6. 桶排序（New）
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11.6. 桶排序（New）
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<a href="#1161" class="md-nav__link">
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11.6.1. 算法流程
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#1162" class="md-nav__link">
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11.6.2. 算法特性
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#1163" class="md-nav__link">
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11.6.3. 如何实现平均分配
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</a>
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</ul>
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11.7. 计数排序（New）
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11.8. 基数排序（New）
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11.9. 小结
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12. 附录
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12.1. 编程环境安装
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12.2. 一起参与创作
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11.6.1. 算法流程
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</a>
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<a href="#1162" class="md-nav__link">
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11.6.2. 算法特性
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<li class="md-nav__item">
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<a href="#1163" class="md-nav__link">
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11.6.3. 如何实现平均分配
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</a>
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_sorting/bucket_sort.md" title="编辑此页" class="md-content__button md-icon">
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<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>
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<h1 id="116">11.6. 桶排序<a class="headerlink" href="#116" title="Permanent link">¶</a></h1>
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<p>前面介绍的几种排序算法都属于 <strong>基于比较的排序算法</strong>，即通过比较元素之间的大小来实现排序，此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将学习几种 <strong>非比较排序算法</strong> ，其时间复杂度可以达到线性级别。</p>
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<p>「桶排序 Bucket Sort」是分治思想的典型体现，其通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中，并在每个桶内部分别执行排序，最终按照桶的顺序将所有数据合并即可。</p>
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<h2 id="1161">11.6.1. 算法流程<a class="headerlink" href="#1161" title="Permanent link">¶</a></h2>
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<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>
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<ol>
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<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>
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<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>
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<li>按照桶的从小到大的顺序，合并结果；</li>
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</ol>
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<p><img alt="桶排序算法流程" src="../bucket_sort.assets/bucket_sort_overview.png" /></p>
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<p align="center"> Fig. 桶排序算法流程 </p>
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<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="tabbed-content">
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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>
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<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>
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<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/2 个桶，预期向每个桶分配 2 个元素</span>
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<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>
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<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>
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<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>
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<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>
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<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="p">}</span>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="p">}</span>
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<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>
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<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>
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<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>
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<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>
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<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="p">}</span>
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<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>
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<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>
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<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>
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<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>
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<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>
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<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.py</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="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">bucket_sort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 桶排序 */</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">bucketSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">float64</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="c1">// 初始化 k = n/2 个桶，预期向每个桶分配 2 个元素</span>
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<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
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<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w"> </span><span class="nx">buckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">float64</span><span class="p">,</span><span class="w"> </span><span class="nx">k</span><span class="p">)</span>
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<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">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="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w"> </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">float64</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
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<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="c1">// 1. 将数组元素分配到各个桶中</span>
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<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></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>
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<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">k</span>
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<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="c1">// 将 num 添加进桶 i</span>
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<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w"> </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span>
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<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="c1">// 2. 对各个桶执行排序</span>
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<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">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="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w"> </span><span class="c1">// 使用内置切片排序函数，也可以替换成其它排序算法</span>
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<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w"> </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Float64s</span><span class="p">(</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
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<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w"> </span><span class="c1">// 3. 遍历桶合并结果</span>
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<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
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<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">buckets</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">num</span>
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<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w"> </span><span class="nx">i</span><span class="o">++</span>
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<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.js</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="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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.ts</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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.c</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="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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">bucket_sort</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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 桶排序 */</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">func</span> <span class="nf">bucketSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Double</span><span class="p">])</span> <span class="p">{</span>
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="c1">// 初始化 k = n/2 个桶，预期向每个桶分配 2 个元素</span>
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<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">let</span> <span class="nv">k</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span> <span class="o">/</span> <span class="mi">2</span>
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<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">buckets</span> <span class="p">=</span> <span class="p">(</span><span class="mi">0</span> <span class="p">..</span><span class="o"><</span> <span class="n">k</span><span class="p">).</span><span class="bp">map</span> <span class="p">{</span> <span class="kc">_</span> <span class="k">in</span> <span class="p">[</span><span class="nb">Double</span><span class="p">]()</span> <span class="p">}</span>
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<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="c1">// 1. 将数组元素分配到各个桶中</span>
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<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="k">for</span> <span class="n">num</span> <span class="k">in</span> <span class="n">nums</span> <span class="p">{</span>
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<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <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>
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<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a> <span class="kd">let</span> <span class="nv">i</span> <span class="p">=</span> <span class="nb">Int</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">*</span> <span class="n">k</span>
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<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="c1">// 将 num 添加进桶 i</span>
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<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">append</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
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<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a> <span class="p">}</span>
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<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <span class="c1">// 2. 对各个桶执行排序</span>
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<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a> <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="n">buckets</span><span class="p">.</span><span class="bp">indices</span> <span class="p">{</span>
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<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a> <span class="c1">// 使用内置排序函数，也可以替换成其它排序算法</span>
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<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a> <span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="bp">sort</span><span class="p">()</span>
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<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a> <span class="p">}</span>
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<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a> <span class="c1">// 3. 遍历桶合并结果</span>
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<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a> <span class="kd">var</span> <span class="nv">i</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
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<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a> <span class="k">for</span> <span class="n">bucket</span> <span class="k">in</span> <span class="n">buckets</span> <span class="p">{</span>
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<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a> <span class="k">for</span> <span class="n">num</span> <span class="k">in</span> <span class="n">bucket</span> <span class="p">{</span>
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<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">=</span> <span class="n">num</span>
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<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a> <span class="n">nums</span><span class="p">.</span><span class="n">formIndex</span><span class="p">(</span><span class="n">after</span><span class="p">:</span> <span class="p">&</span><span class="n">i</span><span class="p">)</span>
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<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a> <span class="p">}</span>
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<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a> <span class="p">}</span>
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<a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">bucket_sort.zig</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="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">bucketSort</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<div class="admonition question">
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<p class="admonition-title">桶排序的应用场景是什么？</p>
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<p>桶排序一般用于排序超大体量的数据。例如输入数据包含 100 万个元素，由于空间有限，系统无法一次性将所有数据加载进内存，那么可以先将数据划分到 1000 个桶里，再依次排序每个桶，最终合并结果即可。</p>
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</div>
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<h2 id="1162">11.6.2. 算法特性<a class="headerlink" href="#1162" title="Permanent link">¶</a></h2>
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<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>
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<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>
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<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>
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<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>
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<h2 id="1163">11.6.3. 如何实现平均分配<a class="headerlink" href="#1163" title="Permanent link">¶</a></h2>
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<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，然而商品价格不是均匀分布的，100 元以下非常多、1000 元以上非常少；如果我们将价格区间平均划为 10 份，那么各个桶内的商品数量差距会非常大。</p>
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<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>
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<p><img alt="递归划分桶" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></p>
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<p align="center"> Fig. 递归划分桶 </p>
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<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>
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<p><img alt="根据概率分布划分桶" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
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<p align="center"> Fig. 根据概率分布划分桶 </p>
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