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0.1. 关于本书
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
|
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0.2. 如何使用本书
|
||
</a>
|
||
</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_preface/installation/" class="md-nav__link">
|
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0.3. 编程环境安装
|
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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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|
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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/contribution/" class="md-nav__link">
|
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0.4. 一起参与创作
|
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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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</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_2" >
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1. 引言
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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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|
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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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|
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<li class="md-nav__item">
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1.2. 算法是什么
|
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</li>
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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_3" >
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|
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|
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|
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
|
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2. 计算复杂度
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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. 计算复杂度
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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_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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|
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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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</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">
|
||
<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>
|
||
</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_computational_complexity/space_time_tradeoff/" class="md-nav__link">
|
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2.4. 权衡时间与空间
|
||
</a>
|
||
</li>
|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<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>
|
||
</nav>
|
||
</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 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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</label>
|
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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
|
||
<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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|
||
|
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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/data_and_memory/" class="md-nav__link">
|
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3.1. 数据与内存
|
||
</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_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>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_data_structure/summary/" class="md-nav__link">
|
||
3.3. 小结
|
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</a>
|
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</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</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 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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|
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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>
|
||
<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="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
|
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4.1. 数组（Array）
|
||
</a>
|
||
</li>
|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
||
4.2. 链表（LinkedList）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
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4.3. 列表（List）
|
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</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
||
4.4. 小结
|
||
</a>
|
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</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</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--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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<span class="md-nav__icon md-icon"></span>
|
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</label>
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|
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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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|
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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（Stack）
|
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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">
|
||
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
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5.2. 队列（Queue）
|
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</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
||
5.3. 双向队列（Deque）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
||
5.4. 小结
|
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</a>
|
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</li>
|
||
|
||
|
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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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|
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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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|
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|
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|
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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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|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
|
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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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|
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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（HashMap）
|
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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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|
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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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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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|
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|
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|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
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6.2. 哈希冲突处理
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<span class="md-nav__icon md-icon"></span>
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</label>
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|
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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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|
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|
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|
||
<nav class="md-nav md-nav--secondary" aria-label="目录">
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|
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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目录
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<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">
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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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</a>
|
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|
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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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</a>
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|
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</li>
|
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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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</a>
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|
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<nav class="md-nav" aria-label="6.2.3. 开放寻址">
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<ul class="md-nav__list">
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|
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<li class="md-nav__item">
|
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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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|
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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">
|
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多次哈希
|
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</a>
|
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|
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</li>
|
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|
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</ul>
|
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</nav>
|
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|
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</li>
|
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|
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</ul>
|
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|
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</nav>
|
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|
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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">
|
||
<a href="../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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|
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|
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|
||
|
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</ul>
|
||
</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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<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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|
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|
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|
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|
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|
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
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7. 二叉树
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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. 二叉树
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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">
|
||
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
||
7.1. 二叉树（Binary Tree）
|
||
</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（Heap）
|
||
</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（Graph）
|
||
</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
|
||
</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/summary/" class="md-nav__link">
|
||
11.6. 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" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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_13_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_13">
|
||
<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">
|
||
|
||
<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>
|
||
</a>
|
||
|
||
|
||
|
||
|
||
|
||
<h1 id="62">6.2. 哈希冲突<a class="headerlink" href="#62" title="Permanent link">¶</a></h1>
|
||
<p>理想情况下，哈希函数应该为每个输入产生唯一的输出，使得 key 和 value 一一对应。而实际上，往往存在向哈希函数输入不同的 key 而产生相同输出的情况，这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误，从而严重影响哈希表的可用性。</p>
|
||
<p>那么，为什么会出现哈希冲突呢？本质上看，<strong>由于哈希函数的输入空间往往远大于输出空间</strong>，因此不可避免地会出现多个输入产生相同输出的情况，即为哈希冲突。比如，输入空间是全体整数，输出空间是一个固定大小的桶（数组）的索引范围，那么必定会有多个整数同时映射到一个桶索引。</p>
|
||
<p>为了缓解哈希冲突，一方面，我们可以通过「哈希表扩容」来减小冲突概率。极端情况下，当输入空间和输出空间大小相等时，哈希表就等价于数组了，可谓“大力出奇迹”。</p>
|
||
<p>另一方面，<strong>考虑通过优化数据结构以缓解哈希冲突</strong>，常见的方法有「链式地址」和「开放寻址」。</p>
|
||
<h2 id="621">6.2.1. 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">¶</a></h2>
|
||
<p>「负载因子 Load Factor」定义为 <strong>哈希表中元素数量除以桶槽数量（即数组大小）</strong>，代表哈希冲突的严重程度。</p>
|
||
<p><strong>负载因子常用作哈希表扩容的触发条件</strong>。比如在 Java 中，当负载因子 <span class="arithmatex">\(> 0.75\)</span> 时则触发扩容，将 HashMap 大小扩充至原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
|
||
<p>与数组扩容类似，<strong>哈希表扩容操作的开销很大</strong>，因为需要将所有键值对从原哈希表依次移动至新哈希表。</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，桶内的每个地址只能存储一个元素（即键值对）。<strong>考虑将单个元素转化成一个链表，将所有冲突元素都存储在一个链表中</strong>。</p>
|
||
<p><img alt="hash_collision_chaining" src="../hash_collision.assets/hash_collision_chaining.png" /></p>
|
||
<p>链式地址下，哈希表操作方法为：</p>
|
||
<ul>
|
||
<li><strong>查询元素</strong>：先将 key 输入到哈希函数得到桶内索引，即可访问链表头结点，再通过遍历链表查找对应 value 。</li>
|
||
<li><strong>添加元素</strong>：先通过哈希函数访问链表头部，再将结点（即键值对）添加到链表头部即可。</li>
|
||
<li><strong>删除元素</strong>：同样先根据哈希函数结果访问链表头部，再遍历链表查找对应结点，删除之即可。</li>
|
||
</ul>
|
||
<p>链式地址虽然解决了哈希冲突问题，但仍存在局限性，包括：</p>
|
||
<ul>
|
||
<li><strong>占用空间变大</strong>，因为链表或二叉树包含结点指针，相比于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>为了缓解时间效率问题，<strong>可以把「链表」转化为「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」不引入额外数据结构，而是通过“多次探测”来解决哈希冲突。根据探测方法的不同，主要分为 <strong>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」使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>：如果出现哈希冲突，则从冲突位置向后线性遍历（步长一般取 1 ），直到找到一个空位，则将元素插入到该空位中。</p>
|
||
<p><strong>查找元素</strong>：若出现哈希冲突，则使用相同步长执行线性查找，会遇到两种情况：</p>
|
||
<ol>
|
||
<li>找到对应元素，返回 value 即可；</li>
|
||
<li>若遇到空位，则说明查找键值对不在哈希表中；</li>
|
||
</ol>
|
||
<p><img alt="hash_collision_linear_probing" src="../hash_collision.assets/hash_collision_linear_probing.png" /></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>。删除元素会导致桶内出现一个空位，在查找其他元素时，该空位有可能导致程序认为元素不存在（即上述第 <code>2.</code> 种情况）。因此需要借助一个标志位来标记删除元素。</li>
|
||
<li><strong>容易产生聚集</strong>。桶内被占用的连续位置越长，这些连续位置发生哈希冲突的可能性越大，从而进一步促进这一位置的“聚堆生长”，最终导致增删查改操作效率的劣化。</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」的思路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>：以相同的哈希函数顺序查找，存在两种情况：</p>
|
||
<ol>
|
||
<li>找到目标元素，则返回之；</li>
|
||
<li>到空位或已尝试所有哈希函数，说明哈希表中无此元素；</li>
|
||
</ol>
|
||
<p>相比于「线性探测」，「多次哈希」方法更不容易产生聚集，代价是多个哈希函数增加了额外计算量。</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">工业界方案</p>
|
||
<p>Java 采用「链式地址」。在 JDK 1.8 之后，HashMap 内数组长度大于 64 时，长度大于 8 的链表会被转化为「红黑树」，以提升查找性能。</p>
|
||
<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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