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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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<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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<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">
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<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">
|
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6. 散列表
|
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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_7_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_7">
|
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<span class="md-nav__icon md-icon"></span>
|
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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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<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
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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_8_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
|
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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">
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
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7.1. 二叉树
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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/binary_tree_traversal/" class="md-nav__link">
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7.2. 二叉树遍历
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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/binary_search_tree/" class="md-nav__link">
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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">
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7.4. AVL 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/summary/" class="md-nav__link">
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7.5. 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--active md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
|
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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>
|
|
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</label>
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
|
|
|
<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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8. 堆
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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="../heap/" class="md-nav__link">
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8.1. 堆
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|
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
8.2. 建堆操作 *
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
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|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
8.2. 建堆操作 *
|
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|
</a>
|
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<label class="md-nav__title" for="__toc">
|
|
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<span class="md-nav__icon md-icon"></span>
|
|
|
目录
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
|
|
<a href="#821" class="md-nav__link">
|
|
|
8.2.1. 两种建堆方法
|
|
|
</a>
|
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|
|
<nav class="md-nav" aria-label="8.2.1. 两种建堆方法">
|
|
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<ul class="md-nav__list">
|
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<li class="md-nav__item">
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
借助入堆方法实现
|
|
|
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
基于堆化操作实现
|
|
|
</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">
|
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<a href="#822" class="md-nav__link">
|
|
|
8.2.2. 复杂度分析
|
|
|
</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">
|
|
|
<a href="../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>
|
|
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|
|
<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>
|
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9. 图
|
|
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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_graph/graph/" class="md-nav__link">
|
|
|
9.1. 图
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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9.2. 图基础操作
|
|
|
</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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|
|
9.3. 图的遍历（New）
|
|
|
</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|
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9.4. 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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<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">
|
|
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10. 查找算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
|
<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
|
|
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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_searching/linear_search/" class="md-nav__link">
|
|
|
10.1. 线性查找
|
|
|
</a>
|
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</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>
|
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</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>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|
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10.4. 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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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
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11. 排序算法
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11. 排序算法
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11.1. 排序简介
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11.2. 冒泡排序
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11.3. 插入排序
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11.4. 快速排序
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11.5. 归并排序
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11.6. 桶排序（New）
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11.7. 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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12.1. 编程环境安装
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12.2. 一起参与创作
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8.2.1. 两种建堆方法
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借助入堆方法实现
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基于堆化操作实现
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8.2.2. 复杂度分析
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<article class="md-content__inner md-typeset">
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_heap/build_heap.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="82">8.2. 建堆操作 *<a class="headerlink" href="#82" title="Permanent link">¶</a></h1>
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<p>如果我们想要根据输入列表来生成一个堆，这样的操作被称为「建堆」。</p>
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<h2 id="821">8.2.1. 两种建堆方法<a class="headerlink" href="#821" title="Permanent link">¶</a></h2>
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<h3 id="_1">借助入堆方法实现<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
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<p>最直接地，考虑借助「元素入堆」方法，先建立一个空堆，<strong>再将列表元素依次入堆即可</strong>。</p>
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<p>设元素数量为 <span class="arithmatex">\(n\)</span> ，则最后一个元素入堆的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> ，在依次入堆时，堆的平均长度为 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此该方法的总体时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
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<h3 id="_2">基于堆化操作实现<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
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<p>有趣的是，存在一种更加高效的建堆方法，时间复杂度可以达到 <span class="arithmatex">\(O(n)\)</span> 。我们先将列表所有元素原封不动添加进堆，<strong>然后迭代地对各个结点执行「从顶至底堆化」</strong>。当然，<strong>无需对叶结点执行堆化</strong>，因为其没有子结点。</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="tabbed-block">
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<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 构造方法，根据输入列表建堆 */</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">List</span><span class="o"><</span><span class="n">Integer</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">// 将列表元素原封不动添加进堆</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="n">maxHeap</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><span class="n">nums</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="c1">// 堆化除叶结点以外的其他所有结点</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="n">parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</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="mi">0</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">siftDown</span><span class="p">(</span><span class="n">i</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="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">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 构造方法，根据输入列表建堆 */</span>
|
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</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-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// 将列表元素原封不动添加进堆</span>
|
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|
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
|
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></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="n">parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</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="mi">0</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-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
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<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></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">my_heap.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">""" 构造方法 """</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="c1"># 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span> <span class="o">=</span> <span class="n">nums</span>
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<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="c1"># 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
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<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="bp">self</span><span class="o">.</span><span class="n">sift_down</span><span class="p">(</span><span class="n">i</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">my_heap.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 构造方法，根据切片建堆 */</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">newMaxHeap</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</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">// 将列表元素原封不动添加进堆</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">h</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&</span><span class="nx">maxHeap</span><span class="p">{</span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nx">nums</span><span class="p">}</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="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="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</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="o">--</span><span class="w"> </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="c1">// 堆化除叶结点以外的其他所有结点</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">h</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</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="k">return</span><span class="w"> </span><span class="nx">h</span>
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<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></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">my_heap.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 构造方法，建立空堆或根据输入列表建堆 */</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kr">constructor</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="c1">// 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">[...</span><span class="nx">nums</span><span class="p">];</span>
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<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">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-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
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<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></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">my_heap.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 构造方法，建立空堆或根据输入列表建堆 */</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kr">constructor</span><span class="p">(</span><span class="nx">nums?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w"> </span><span class="c1">// 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">[...</span><span class="nx">nums</span><span class="p">];</span>
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<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w"> </span><span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">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-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
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<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></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">my_heap.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="n">maxHeap</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">newMaxHeap</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">my_heap.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 构造函数，根据输入列表建堆 */</span>
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<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">IEnumerable</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">nums</span><span class="p">)</span>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="p">{</span>
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<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="c1">// 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="n">maxHeap</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">List</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
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<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w"> </span><span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w"> </span><span class="kt">var</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
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<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></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="n">size</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="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span>
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<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
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<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></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">my_heap.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 构造方法，根据输入列表建堆 */</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">init</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</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">// 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="n">maxHeap</span> <span class="p">=</span> <span class="n">nums</span>
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<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span> <span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">through</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="n">by</span><span class="p">:</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
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<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="n">i</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="p">}</span>
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<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></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">my_heap.zig</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// 构造方法，根据输入列表建堆</span>
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<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">init</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">,</span><span class="w"> </span><span class="n">allocator</span><span class="o">:</span><span class="w"> </span><span class="n">std</span><span class="p">.</span><span class="n">mem</span><span class="p">.</span><span class="n">Allocator</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">:</span><span class="w"> </span><span class="p">[]</span><span class="kr">const</span><span class="w"> </span><span class="n">T</span><span class="p">)</span><span class="w"> </span><span class="o">!</span><span class="kt">void</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">self</span><span class="p">.</span><span class="n">max_heap</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">max_heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">.</span><span class="n">ArrayList</span><span class="p">(</span><span class="n">T</span><span class="p">).</span><span class="n">init</span><span class="p">(</span><span class="n">allocator</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="w"> </span><span class="c1">// 将列表元素原封不动添加进堆</span>
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<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="k">try</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">appendSlice</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
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<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="c1">// 堆化除叶结点以外的其他所有结点</span>
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<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
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<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</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="o">:</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w"> </span><span class="k">try</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
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<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><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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<h2 id="822">8.2.2. 复杂度分析<a class="headerlink" href="#822" title="Permanent link">¶</a></h2>
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<p>第二种建堆方法的时间复杂度为什么是 <span class="arithmatex">\(O(n)\)</span> 呢？我们来展开推算一下。</p>
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<ul>
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<li>完全二叉树中，设结点总数为 <span class="arithmatex">\(n\)</span> ，则叶结点数量为 <span class="arithmatex">\((n + 1) / 2\)</span> ，其中 <span class="arithmatex">\(/\)</span> 为向下整除。因此在排除叶结点后，需要堆化结点数量为 <span class="arithmatex">\((n - 1)/2\)</span> ，即为 <span class="arithmatex">\(O(n)\)</span> ；</li>
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<li>从顶至底堆化中，每个结点最多堆化至叶结点，因此最大迭代次数为二叉树高度 <span class="arithmatex">\(O(\log n)\)</span> ；</li>
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</ul>
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<p>将上述两者相乘，可得时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。这个估算结果不够准确，因为我们没有考虑到 <strong>二叉树底层结点远多于顶层结点</strong> 的性质。</p>
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<p>下面我们来展开计算。为了减小计算难度，我们假设树是一个「完美二叉树」，该假设不会影响计算结果的正确性。设二叉树（即堆）结点数量为 <span class="arithmatex">\(n\)</span> ，树高度为 <span class="arithmatex">\(h\)</span> 。上文提到，<strong>结点堆化最大迭代次数等于该结点到叶结点的距离，而这正是“结点高度”</strong>。</p>
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<p><img alt="完美二叉树的各层结点数量" src="../build_heap.assets/heapify_operations_count.png" /></p>
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<p align="center"> Fig. 完美二叉树的各层结点数量 </p>
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<p>因此，我们将各层的“结点数量 <span class="arithmatex">\(\times\)</span> 结点高度”求和，即可得到 <strong>所有结点的堆化的迭代次数总和</strong>。</p>
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<div class="arithmatex">\[
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T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1
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\]</div>
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<p>化简上式需要借助中学的数列知识，先对 <span class="arithmatex">\(T(h)\)</span> 乘以 <span class="arithmatex">\(2\)</span> ，易得</p>
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<div class="arithmatex">\[
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\begin{aligned}
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T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline
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2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline
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\end{aligned}
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\]</div>
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<p><strong>使用错位相减法</strong>，令下式 <span class="arithmatex">\(2 T(h)\)</span> 减去上式 <span class="arithmatex">\(T(h)\)</span> ，可得</p>
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<div class="arithmatex">\[
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2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h
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\]</div>
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<p>观察上式，<span class="arithmatex">\(T(h)\)</span> 是一个等比数列，可直接使用求和公式，得到时间复杂度为</p>
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<div class="arithmatex">\[
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\begin{aligned}
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T(h) & = 2 \frac{1 - 2^h}{1 - 2} - h \newline
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& = 2^{h+1} - h \newline
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& = O(2^h)
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\end{aligned}
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\]</div>
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<p>进一步地，高度为 <span class="arithmatex">\(h\)</span> 的完美二叉树的结点数量为 <span class="arithmatex">\(n = 2^{h+1} - 1\)</span> ，易得复杂度为 <span class="arithmatex">\(O(2^h) = O(n)\)</span>。以上推算表明，<strong>输入列表并建堆的时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，非常高效</strong>。</p>
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