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0.1. 关于本书
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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|
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|
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|
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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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|
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|
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|
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|
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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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|
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|
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</ul>
|
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1.1. 算法无处不在
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1.2. 算法是什么
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1.3. 小结
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2. 复杂度分析
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2. 复杂度分析
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|
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2.1. 算法效率评估
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|
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|
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|
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|
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|
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|
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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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|
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|
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|
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|
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|
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|
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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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||
|
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|
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|
||
|
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|
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|
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|
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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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|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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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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|
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|
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|
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|
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</ul>
|
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</nav>
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|
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|
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|
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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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|
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|
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|
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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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<span class="md-nav__icon md-icon"></span>
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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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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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|
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|
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|
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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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|
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|
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|
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|
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|
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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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|
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|
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|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<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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|
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|
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|
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|
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</ul>
|
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</nav>
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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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|
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|
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|
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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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|
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|
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|
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|
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|
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|
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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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|
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|
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|
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|
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|
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|
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|
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|
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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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|
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|
||
|
||
|
||
|
||
|
||
|
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|
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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
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
||
<li class="md-nav__item">
|
||
<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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|
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|
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|
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|
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</ul>
|
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</nav>
|
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</li>
|
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|
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|
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|
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|
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|
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|
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|
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|
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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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|
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|
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|
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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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</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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|
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|
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|
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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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|
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|
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|
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|
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|
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|
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|
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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>
|
||
</li>
|
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|
||
|
||
|
||
|
||
|
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|
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|
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|
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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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|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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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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|
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|
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|
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|
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</ul>
|
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</nav>
|
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</li>
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|
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|
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|
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|
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|
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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_7" checked>
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
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<span class="md-nav__icon md-icon"></span>
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6. 散列表
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../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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|
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|
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|
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|
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|
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<li class="md-nav__item md-nav__item--active">
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6.2. 哈希冲突处理
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<a href="./" class="md-nav__link md-nav__link--active">
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6.2. 哈希冲突处理
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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目录
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<li class="md-nav__item">
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<a href="#621" class="md-nav__link">
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6.2.1. 哈希表扩容
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</li>
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|
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<li class="md-nav__item">
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<a href="#622" class="md-nav__link">
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6.2.2. 链式地址
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|
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<li class="md-nav__item">
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<a href="#623" class="md-nav__link">
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6.2.3. 开放寻址
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<nav class="md-nav" aria-label="6.2.3. 开放寻址">
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<a href="#_1" class="md-nav__link">
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线性探测
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</a>
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#_2" class="md-nav__link">
|
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多次哈希
|
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</li>
|
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|
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</ul>
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</nav>
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</ul>
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</nav>
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</li>
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|
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|
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|
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|
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<li class="md-nav__item">
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<a href="../summary/" class="md-nav__link">
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6.3. 小结
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|
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|
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|
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|
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</ul>
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</nav>
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</li>
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|
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|
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|
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|
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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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7. 树
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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
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
||
7.2. 二叉树遍历
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
||
7.3. 二叉搜索树
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
7.4. AVL 树 *
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
||
7.5. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
||
8.1. 堆
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
||
8.2. 建堆操作 *
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
||
8.3. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
||
9.1. 图
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
||
9.2. 图基础操作
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
||
9.3. 图的遍历（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
||
9.4. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
||
10.1. 线性查找
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
||
10.2. 二分查找
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
||
10.3. 哈希查找
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
10.4. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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="false">
|
||
<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="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
|
||
11.1. 排序简介
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
||
11.2. 冒泡排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
||
11.3. 插入排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
||
11.4. 快速排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
11.5. 归并排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
||
11.6. 桶排序（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
11.7. 计数排序（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
11.8. 基数排序（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
||
11.9. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
|
||
12. 附录
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_13">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
12. 附录
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_appendix/installation/" class="md-nav__link">
|
||
12.1. 编程环境安装
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
|
||
12.2. 一起参与创作
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_reference/">参考文献</a>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_14">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
参考文献
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
||
<div class="md-sidebar__scrollwrap">
|
||
<div class="md-sidebar__inner">
|
||
|
||
|
||
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__title" for="__toc">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
目录
|
||
</label>
|
||
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#621" class="md-nav__link">
|
||
6.2.1. 哈希表扩容
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#622" class="md-nav__link">
|
||
6.2.2. 链式地址
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#623" class="md-nav__link">
|
||
6.2.3. 开放寻址
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="6.2.3. 开放寻址">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_1" class="md-nav__link">
|
||
线性探测
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_2" class="md-nav__link">
|
||
多次哈希
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="md-content" data-md-component="content">
|
||
<article class="md-content__inner md-typeset">
|
||
|
||
|
||
|
||
|
||
<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_hashing/hash_collision.md" title="编辑此页" class="md-content__button md-icon">
|
||
|
||
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<h1 id="62">6.2. 哈希冲突<a class="headerlink" href="#62" title="Permanent link">¶</a></h1>
|
||
<p>在理想情况下，哈希函数应为每个输入生成唯一的输出，实现 key 和 value 的一一对应。然而实际上，向哈希函数输入不同的 key 却产生相同输出的情况是存在的，这种现象被称为「哈希冲突 Hash Collision」。哈希冲突可能导致查询结果错误，从而严重影响哈希表的可用性。</p>
|
||
<p>那么，为何会出现哈希冲突呢？从本质上看，由于哈希函数的输入空间通常远大于输出空间，因此多个输入产生相同输出的情况是不可避免的。例如，若输入空间为全体整数，而输出空间为固定大小的数组，则必然有多个整数映射至同一数组索引。</p>
|
||
<p>为了减轻哈希冲突，一方面，<strong>可以通过扩大哈希表容量来降低冲突概率</strong>。极端情况下，当输入空间和输出空间大小相等时，哈希表等同于数组，每个 key 都对应唯一的数组索引，可谓“大力出奇迹”。</p>
|
||
<p>另一方面，<strong>可以考虑优化哈希表的表示以缓解哈希冲突</strong>，常用方法包括「链式地址 Separate Chaining」和「开放寻址 Open Addressing」。</p>
|
||
<h2 id="621">6.2.1. 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">¶</a></h2>
|
||
<p>哈希函数的最后一步通常是对桶数量 <span class="arithmatex">\(n\)</span> 取余，作用是将哈希值映射到桶索引范围，从而将 key 放入对应的桶中。当哈希表容量越大（即 <span class="arithmatex">\(n\)</span> 越大）时，多个 key 被分配到同一个桶中的概率就越低，冲突就越少。</p>
|
||
<p>因此，<strong>当哈希表内的冲突总体较为严重时，编程语言通常通过扩容哈希表来缓解冲突</strong>。类似于数组扩容，哈希表扩容需将所有键值对从原哈希表迁移至新哈希表，开销较大。</p>
|
||
<p>编程语言通常使用「负载因子 Load Factor」来衡量哈希冲突的严重程度，<strong>定义为哈希表中元素数量除以桶数量</strong>，常作为哈希表扩容的触发条件。在 Java 中，当负载因子 <span class="arithmatex">\(> 0.75\)</span> 时，系统会将 HashMap 容量扩展为原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
|
||
<h2 id="622">6.2.2. 链式地址<a class="headerlink" href="#622" title="Permanent link">¶</a></h2>
|
||
<p>在原始哈希表中，每个桶仅能存储一个键值对。<strong>链式地址将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中</strong>。</p>
|
||
<p><img alt="链式地址" src="../hash_collision.assets/hash_collision_chaining.png" /></p>
|
||
<p align="center"> Fig. 链式地址 </p>
|
||
|
||
<p>链式地址下，哈希表的操作方法包括：</p>
|
||
<ul>
|
||
<li><strong>查询元素</strong>：输入 key ，经过哈希函数得到数组索引，即可访问链表头节点，然后遍历链表并对比 key 以查找目标键值对。</li>
|
||
<li><strong>添加元素</strong>：先通过哈希函数访问链表头节点，然后将节点（即键值对）添加到链表中。</li>
|
||
<li><strong>删除元素</strong>：根据哈希函数的结果访问链表头部，接着遍历链表以查找目标节点，并将其删除。</li>
|
||
</ul>
|
||
<p>尽管链式地址法解决了哈希冲突问题，但仍存在一些局限性，包括：</p>
|
||
<ul>
|
||
<li><strong>占用空间增大</strong>，由于链表或二叉树包含节点指针，相比数组更加耗费内存空间；</li>
|
||
<li><strong>查询效率降低</strong>，因为需要线性遍历链表来查找对应元素；</li>
|
||
</ul>
|
||
<p>为了提高操作效率，<strong>可以将链表转换为「AVL 树」或「红黑树」</strong>，将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
|
||
<h2 id="623">6.2.3. 开放寻址<a class="headerlink" href="#623" title="Permanent link">¶</a></h2>
|
||
<p>「开放寻址」方法不引入额外的数据结构，而是通过“多次探测”来解决哈希冲突，<strong>探测方主要包括线性探测、平方探测、多次哈希</strong>。</p>
|
||
<h3 id="_1">线性探测<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||
<p>「线性探测」采用固定步长的线性查找来解决哈希冲突。</p>
|
||
<p><strong>插入元素</strong>：若出现哈希冲突，则从冲突位置向后线性遍历（步长通常为 <span class="arithmatex">\(1\)</span> ），直至找到空位，将元素插入其中。</p>
|
||
<p><strong>查找元素</strong>：在出现哈希冲突时，使用相同步长进行线性查找，可能遇到以下两种情况。</p>
|
||
<ol>
|
||
<li>找到对应元素，返回 value 即可；</li>
|
||
<li>若遇到空位，说明目标键值对不在哈希表中；</li>
|
||
</ol>
|
||
<p><img alt="线性探测" src="../hash_collision.assets/hash_collision_linear_probing.png" /></p>
|
||
<p align="center"> Fig. 线性探测 </p>
|
||
|
||
<p>线性探测存在以下缺陷：</p>
|
||
<ul>
|
||
<li><strong>不能直接删除元素</strong>。删除元素会在数组内产生一个空位，查找其他元素时，该空位可能导致程序误判元素不存在（即上述第 <code>2.</code> 种情况）。因此，需要借助一个标志位来标记已删除元素。</li>
|
||
<li><strong>容易产生聚集</strong>。数组内连续被占用位置越长，这些连续位置发生哈希冲突的可能性越大，进一步促使这一位置的“聚堆生长”，最终导致增删查改操作效率降低。</li>
|
||
</ul>
|
||
<h3 id="_2">多次哈希<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||
<p>顾名思义，「多次哈希」方法是使用多个哈希函数 <span class="arithmatex">\(f_1(x)\)</span> , <span class="arithmatex">\(f_2(x)\)</span> , <span class="arithmatex">\(f_3(x)\)</span> , <span class="arithmatex">\(\cdots\)</span> 进行探测。</p>
|
||
<p><strong>插入元素</strong>：若哈希函数 <span class="arithmatex">\(f_1(x)\)</span> 出现冲突，则尝试 <span class="arithmatex">\(f_2(x)\)</span> ，以此类推，直到找到空位后插入元素。</p>
|
||
<p><strong>查找元素</strong>：在相同的哈希函数顺序下进行查找，存在以下两种情况：</p>
|
||
<ol>
|
||
<li>如果找到目标元素，则返回之；</li>
|
||
<li>若遇到空位或已尝试所有哈希函数，则说明哈希表中不存在该元素；</li>
|
||
</ol>
|
||
<p>与线性探测相比，多次哈希方法不易产生聚集，但多个哈希函数会增加额外的计算量。</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">哈希表设计方案</p>
|
||
<p>Java 采用「链式地址」。自 JDK 1.8 以来，当 HashMap 内数组长度大于 64 且链表长度大于 8 时，链表会被转换为「红黑树」以提升查找性能。</p>
|
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<p>Python 采用「开放寻址」。字典 dict 使用伪随机数进行探测。</p>
|
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<p>Golang 采用「链式地址」。Go 规定每个桶最多存储 8 个键值对，超出容量则连接一个溢出桶；当溢出桶过多时，会执行一次特殊的等量扩容操作，以确保性能。</p>
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