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||
|
||
|
||
|
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|
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|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
|
||
0.1. 关于本书
|
||
</a>
|
||
</li>
|
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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
|
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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">
|
||
<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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|
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|
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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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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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|
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|
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|
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|
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|
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_introduction/">1. 初识算法</a>
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<label for="__nav_2">
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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|
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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. 初识算法
|
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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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|
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|
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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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|
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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_introduction/summary/" class="md-nav__link">
|
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1.3. 小结
|
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</a>
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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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<li class="md-nav__item 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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|
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_computational_complexity/">2. 复杂度</a>
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<label for="__nav_3">
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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|
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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
|
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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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|
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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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|
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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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|
||
|
||
<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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|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
|
||
2.4. 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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|
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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--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" checked>
|
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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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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
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<a href="../">3. 数据结构</a>
|
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|
||
<label for="__nav_4">
|
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<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
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|
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</div>
|
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|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
|
||
<label class="md-nav__title" for="__nav_4">
|
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<span class="md-nav__icon md-icon"></span>
|
||
3. 数据结构
|
||
</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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|
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|
||
<li class="md-nav__item">
|
||
<a href="../classification_of_data_structure/" class="md-nav__link">
|
||
3.1. 数据结构分类
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../basic_data_types/" class="md-nav__link">
|
||
3.2. 基本数据类型
|
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</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../number_encoding/" class="md-nav__link">
|
||
3.3. 数字编码 *
|
||
</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 md-nav__item--active">
|
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|
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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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|
||
|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
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3.4. 字符编码 *
|
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<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<a href="./" class="md-nav__link md-nav__link--active">
|
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3.4. 字符编码 *
|
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</a>
|
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|
||
|
||
|
||
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
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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__title" for="__toc">
|
||
<span class="md-nav__icon md-icon"></span>
|
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目录
|
||
</label>
|
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#341-ascii" class="md-nav__link">
|
||
3.4.1. ASCII 字符集
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#342-gbk" class="md-nav__link">
|
||
3.4.2. GBK 字符集
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#343-unicode" class="md-nav__link">
|
||
3.4.3. Unicode 字符集
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#344-utf-8" class="md-nav__link">
|
||
3.4.4. UTF-8 编码
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#345" class="md-nav__link">
|
||
3.4.5. 编程语言的字符编码
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
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|
||
</nav>
|
||
|
||
</li>
|
||
|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../summary/" class="md-nav__link">
|
||
3.5. 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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|
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<li class="md-nav__item md-nav__item--nested">
|
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|
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|
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|
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|
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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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|
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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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|
||
<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_array_and_linkedlist/">4. 数组与链表</a>
|
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|
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<label for="__nav_5">
|
||
<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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|
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</div>
|
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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>
|
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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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|
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|
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<li class="md-nav__item">
|
||
<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">
|
||
<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">
|
||
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
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4.3. 列表
|
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</a>
|
||
</li>
|
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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/summary/" class="md-nav__link">
|
||
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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</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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<li class="md-nav__item 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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|
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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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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_stack_and_queue/">5. 栈与队列</a>
|
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|
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<label for="__nav_6">
|
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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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</div>
|
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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
|
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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">
|
||
<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">
|
||
<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>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item">
|
||
<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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|
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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--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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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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|
||
<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_hashing/">6. 散列表</a>
|
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|
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<label for="__nav_7">
|
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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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</div>
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|
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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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|
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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_hashing/hash_map/" class="md-nav__link">
|
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6.1. 哈希表（New）
|
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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">
|
||
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
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6.2. 哈希冲突（New）
|
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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">
|
||
<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
||
6.3. 哈希算法（New）
|
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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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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
||
6.4. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_tree/">7. 树</a>
|
||
|
||
<label for="__nav_8">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_8">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
7. 树
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
||
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/array_representation_of_tree/" class="md-nav__link">
|
||
7.3. 二叉树数组表示
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
||
7.4. 二叉搜索树
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
7.5. AVL 树 *
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
||
7.6. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_heap/">8. 堆</a>
|
||
|
||
<label for="__nav_9">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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/top_k/" class="md-nav__link">
|
||
8.3. Top-K 问题（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
||
8.4. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_graph/">9. 图</a>
|
||
|
||
<label for="__nav_10">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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
|
||
</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--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_searching/">10. 搜索</a>
|
||
|
||
<label for="__nav_11">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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. 搜索
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
||
10.1. 二分查找
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
||
10.2. 二分查找边界
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
||
10.3. 哈希优化策略
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
||
10.4. 重识搜索算法
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
10.5. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__link--index ">
|
||
<a href="../../chapter_sorting/">11. 排序</a>
|
||
|
||
<label for="__nav_12">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<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
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
||
11.1. 排序算法
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
||
11.2. 选择排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
||
11.3. 冒泡排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
||
11.4. 插入排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
||
11.5. 快速排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
11.6. 归并排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
||
11.7. 堆排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
||
11.8. 桶排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
11.9. 计数排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
11.10. 基数排序
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
||
11.11. 小结
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item 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_divide_and_conquer/">12. 分治</a>
|
||
|
||
<label for="__nav_13">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</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>
|
||
12. 分治
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
|
||
12.1. 分治算法（New）
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_divide_and_conquer/build_binary_tree/" class="md-nav__link">
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12.2. 构建树问题（New）
|
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_backtracking/">13. 回溯</a>
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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_14_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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13. 回溯
|
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13.1. 回溯算法
|
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13.2. 全排列问题
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13.3. 子集和问题
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/summary/" class="md-nav__link">
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13.5. 小结
|
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|
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|
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14.3. DP 解题思路（New）
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|
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<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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14.5. 完全背包问题（New）
|
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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_dynamic_programming/edit_distance_problem/" class="md-nav__link">
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|
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|
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|
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<a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
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14.7. 小结（New）
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15.1. 编程环境安装
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15.2. 一起参与创作
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<a href="#341-ascii" class="md-nav__link">
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3.4.1. ASCII 字符集
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<a href="#342-gbk" class="md-nav__link">
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3.4.2. GBK 字符集
|
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<a href="#343-unicode" class="md-nav__link">
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3.4.3. Unicode 字符集
|
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<a href="#344-utf-8" class="md-nav__link">
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3.4.4. UTF-8 编码
|
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|
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|
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<li class="md-nav__item">
|
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<a href="#345" class="md-nav__link">
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3.4.5. 编程语言的字符编码
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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_data_structure/character_encoding.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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|
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<h1 id="34">3.4. 字符编码 *<a class="headerlink" href="#34" title="Permanent link">¶</a></h1>
|
||
<p>在计算机中，所有数据都是以二进制数的形式存储的，字符 <code>char</code> 也不例外。为了表示字符，我们需要建立一套「字符集」，规定每个字符和二进制数之间的一一对应关系。有了字符集之后，计算机就可以通过查表完成二进制数到字符的转换。</p>
|
||
<h2 id="341-ascii">3.4.1. ASCII 字符集<a class="headerlink" href="#341-ascii" title="Permanent link">¶</a></h2>
|
||
<p>「ASCII 码」是最早出现的字符集，全称为“美国标准信息交换代码”。它使用 7 位二进制数（即一个字节的低 7 位）表示一个字符，最多能够表示 128 个不同的字符。这包括英文字母的大小写、数字 0-9 、一些标点符号，以及一些控制字符（如换行符和制表符）。</p>
|
||
<p><img alt="ASCII 码" src="../character_encoding.assets/ascii_table.png" /></p>
|
||
<p align="center"> Fig. ASCII 码 </p>
|
||
|
||
<p>然而，<strong>ASCII 码仅能够表示英文</strong>。随着计算机的全球化，诞生了一种能够表示更多语言的字符集「EASCII」。它在 ASCII 的 7 位基础上扩展到 8 位，能够表示 256 个不同的字符。在世界范围内，陆续出现了一批适用于不同地区的 EASCII 字符集。这些字符集的前 128 个字符统一为 ASCII 码，后 128 个字符定义不同，以适应不同语言的需求。</p>
|
||
<h2 id="342-gbk">3.4.2. GBK 字符集<a class="headerlink" href="#342-gbk" title="Permanent link">¶</a></h2>
|
||
<p>后来人们发现，<strong>EASCII 码仍然无法满足许多语言的字符数量要求</strong>。例如，汉字大约有近十万个，光日常使用的就有几千个。中国国家标准总局于 1980 年发布了「GB2312」字符集，其收录了 6763 个汉字，基本满足了汉字的计算机处理需要。</p>
|
||
<p>然而，GB2312 无法处理部分的罕见字和繁体字。之后在 GB2312 的基础上，扩展得到了「GBK」字符集，它共收录了 21886 个汉字。在 GBK 编码方案中，ASCII 字符使用一个字节表示，汉字使用两个字节表示。</p>
|
||
<h2 id="343-unicode">3.4.3. Unicode 字符集<a class="headerlink" href="#343-unicode" title="Permanent link">¶</a></h2>
|
||
<p>随着计算机的蓬勃发展，字符集与编码标准百花齐放，而这带来了许多问题。一方面，这些字符集一般只定义了特定语言的字符，无法在多语言环境下正常工作；另一方面，同一种语言也存在多种字符集标准，如果两台电脑安装的是不同的编码标准，则在信息传递时就会出现乱码。</p>
|
||
<p>那个时代的人们就在想：<strong>如果推出一个足够完整的字符集，将世界范围内的所有语言和符号都收录其中，不就可以解决跨语言环境和乱码问题了吗</strong>？在这种想法的驱动下，一个大而全的字符集 Unicode 应运而生。</p>
|
||
<p>「Unicode」的全称为“统一字符编码”，理论上能容纳一百多万个字符。它致力于将全球范围内的字符纳入到统一的字符集之中，提供一种通用的字符集来处理和显示各种语言文字，减少因为编码标准不同而产生的乱码问题。</p>
|
||
<p>自 1991 年发布以来，Unicode 不断扩充新的语言与字符。截止 2022 年 9 月，Unicode 已经包含 149186 个字符，包括各种语言的字符、符号、甚至是表情符号等。在庞大的 Unicode 字符集中，常用的字符占用 2 字节，有些生僻的字符占 3 字节甚至 4 字节。</p>
|
||
<p>Unicode 是一种字符集标准，本质上是给每个字符分配一个编号（称为“码点”），<strong>但它并没有规定在计算机中如何存储这些字符码点</strong>。我们不禁会问：当多种长度的 Unicode 码点同时出现在同一个文本中时，系统如何解析字符？例如，给定一个长度为 2 字节的编码，系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符？</p>
|
||
<p>对于以上问题，<strong>一种直接的解决方案是将所有字符存储为等长的编码</strong>。如下图所示，“Hello”中的每个字符占用 1 字节，“算法”中的每个字符占用 2 字节。我们可以通过高位填 0 ，将“Hello 算法”中的所有字符都编码为 2 字节长度。这样系统就可以每隔 2 字节解析一个字符，恢复出这个短语的内容了。</p>
|
||
<p><img alt="Unicode 编码示例" src="../character_encoding.assets/unicode_hello_algo.png" /></p>
|
||
<p align="center"> Fig. Unicode 编码示例 </p>
|
||
|
||
<p>然而，ASCII 码已经向我们证明，编码英文只需要 1 字节。若采用上述方案，英文文本占用空间的大小将会是 ASCII 编码下大小的 2 倍，非常浪费内存空间。因此，我们需要一种更加高效的 Unicode 编码方法。</p>
|
||
<h2 id="344-utf-8">3.4.4. UTF-8 编码<a class="headerlink" href="#344-utf-8" title="Permanent link">¶</a></h2>
|
||
<p>目前，UTF-8 已成为国际上使用最广泛的 Unicode 编码方法。<strong>它是一种可变长的编码</strong>，使用 1 到 4 个字节来表示一个字符，根据字符的复杂性而变。ASCII 字符只需要 1 个字节，拉丁字母和希腊字母需要 2 个字节，常用的中文字符需要 3 个字节，其他的一些生僻字符需要 4 个字节。</p>
|
||
<p>UTF-8 的编码规则并不复杂，分为两种情况：</p>
|
||
<ul>
|
||
<li>对于长度为 1 字节的字符，将最高位设置为 <span class="arithmatex">\(0\)</span> 、其余 7 位设置为 Unicode 码点。值得注意的是，ASCII 字符在 Unicode 字符集中占据了前 128 个码点。也就是说，<strong>UTF-8 编码可以向下兼容 ASCII 码</strong>。这意味着我们可以使用 UTF-8 来解析年代久远的 ASCII 码文本。</li>
|
||
<li>对于长度为 <span class="arithmatex">\(n\)</span> 字节的字符（其中 <span class="arithmatex">\(n > 1\)</span>），将首个字节的高 <span class="arithmatex">\(n\)</span> 位都设置为 <span class="arithmatex">\(1\)</span> 、第 <span class="arithmatex">\(n + 1\)</span> 位设置为 <span class="arithmatex">\(0\)</span> ；从第二个字节开始，将每个字节的高 2 位都设置为 <span class="arithmatex">\(10\)</span> ；其余所有位用于填充字符的 Unicode 码点。</li>
|
||
</ul>
|
||
<p>下图展示了“Hello算法”对应的 UTF-8 编码。将最高 <span class="arithmatex">\(n\)</span> 位设置为 <span class="arithmatex">\(1\)</span> 比较容易理解，可以向系统指出字符的长度为 <span class="arithmatex">\(n\)</span> 。那么，为什么要将其余所有字节的高 2 位都设置为 <span class="arithmatex">\(10\)</span> 呢？实际上，这个 <span class="arithmatex">\(10\)</span> 能够起到校验符的作用，因为在 UTF-8 编码规则下，不可能有字符的最高两位是 <span class="arithmatex">\(10\)</span> 。这是因为长度为 1 字节的字符的最高一位是 <span class="arithmatex">\(0\)</span> 。假设系统从一个错误的字节开始解析文本，字节头部的 <span class="arithmatex">\(10\)</span> 能够帮助系统快速的判断出异常。</p>
|
||
<p><img alt="UTF-8 编码示例" src="../character_encoding.assets/utf-8_hello_algo.png" /></p>
|
||
<p align="center"> Fig. UTF-8 编码示例 </p>
|
||
|
||
<p>除了 UTF-8 之外，常见的编码方式还包括 UTF-16 和 UTF-32 。它们为 Unicode 字符集提供了不同的编码方法。</p>
|
||
<ul>
|
||
<li><strong>UTF-16 编码</strong>：使用 2 或 4 个字节来表示一个字符。所有的 ASCII 字符和很多常用的非英文字符，都用 2 个字节表示；少数字符需要用到 4 个字节表示。对于 2 字节的字符，UTF-16 编码与 Unicode 码点相等。</li>
|
||
<li><strong>UTF-32 编码</strong>：每个字符都使用 4 个字节。这意味着 UTF-32 会比 UTF-8 和 UTF-16 更占用空间，特别是对于主要使用 ASCII 字符的文本。</li>
|
||
</ul>
|
||
<p>从存储空间的角度看，使用 UTF-8 表示英文字符非常高效，因为它仅需 1 个字节；使用 UTF-16 编码某些非英文字符（例如中文）会更加高效，因为它只需要 2 个字节，而 UTF-8 可能需要 3 个字节。从兼容性的角度看，UTF-8 的通用性最佳，许多工具和库都优先支持 UTF-8 。</p>
|
||
<h2 id="345">3.4.5. 编程语言的字符编码<a class="headerlink" href="#345" title="Permanent link">¶</a></h2>
|
||
<p>对于以往的大多数编程语言，程序运行中的字符串都采用 UTF-16 或 UTF-32 这类等长的编码。这是因为在等长编码下，我们可以将字符串看作数组来处理，具体来说：</p>
|
||
<ul>
|
||
<li><strong>随机访问</strong>: UTF-16 编码的字符串可以很容易地进行随机访问。UTF-8 是一种变长编码，要找到第 <span class="arithmatex">\(i\)</span> 个字符，我们需要从字符串的开始处遍历到第 <span class="arithmatex">\(i\)</span> 个字符，这需要 <span class="arithmatex">\(O(n)\)</span> 的时间。</li>
|
||
<li><strong>字符计数</strong>: 与随机访问类似，计算 UTF-16 字符串的长度也是 <span class="arithmatex">\(O(1)\)</span> 的操作。但是，计算 UTF-8 编码的字符串的长度需要遍历整个字符串。</li>
|
||
<li><strong>字符串操作</strong>: 在 UTF-16 编码的字符串中，很多字符串操作（如分割、连接、插入、删除等）都更容易进行。在 UTF-8 编码的字符串上进行这些操作通常需要额外的计算，以确保不会产生无效的 UTF-8 编码。</li>
|
||
</ul>
|
||
<p>编程语言的字符编码方案设计是一个很有趣的话题，涉及到许多因素：</p>
|
||
<ul>
|
||
<li>Java 的 <code>String</code> 类型使用 UTF-16 编码，每个字符占用 2 字节。这是因为 Java 语言设计之初，人们认为 16 位足以表示所有可能的字符。然而，这是一个不正确的判断。后来 Unicode 规范扩展到了超过 16 位，所以 Java 中的字符现在可能由一对 16 位的值（称为“代理对”）表示。</li>
|
||
<li>JavaScript 和 TypeScript 的字符串使用 UTF-16 编码的原因与 Java 类似。当 JavaScript 语言在 1995 年被 Netscape 公司首次引入时，Unicode 还处于相对早期的阶段，那时候使用 16 位的编码就足够表示所有的 Unicode 字符了。</li>
|
||
<li>C# 使用 UTF-16 编码，主要因为 .NET 平台是由 Microsoft 设计的，而 Microsoft 的很多技术，包括 Windows 操作系统，都广泛地使用 UTF-16 编码。</li>
|
||
</ul>
|
||
<p>由于以上编程语言对字符数量的低估，它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面，包含代理对的字符串中，一个字符可能占用 2 字节或 4 字节，因此丧失了等长编码的优势。另一方面，处理代理对需要增加额外代码，这增加了编程的复杂性和 Debug 难度。</p>
|
||
<p>出于以上原因，部分编程语言提出了不同的编码方案：</p>
|
||
<ul>
|
||
<li>Python 3 使用一种灵活的字符串表示，存储的字符长度取决于字符串中最大的 Unicode 码点。对于全部是 ASCII 字符的字符串，每个字符占用 1 个字节；如果字符串中包含的字符超出了 ASCII 范围，但全部在基本多语言平面（BMP）内，每个字符占用 2 个字节；如果字符串中有超出 BMP 的字符，那么每个字符占用 4 个字节。</li>
|
||
<li>Go 语言的 <code>string</code> 类型在内部使用 UTF-8 编码。Go 语言还提供了 <code>rune</code> 类型，它用于表示单个 Unicode 码点。</li>
|
||
<li>Rust 语言的 str 和 String 类型在内部使用 UTF-8 编码。Rust 也提供了 char 类型，用于表示单个 Unicode 码点。</li>
|
||
</ul>
|
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<p>需要注意的是，以上讨论的都是字符串在编程语言中的存储方式，<strong>这和字符串如何在文件中存储或在网络中传输是两个不同的问题</strong>。在文件存储或网络传输中，我们一般会将字符串编码为 UTF-8 格式，以达到最优的兼容性和空间效率。</p>
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