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第 0 章 前言
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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第 1 章 初识算法
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第 1 章 初识算法
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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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1.2 算法是什么
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1.3 小结
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<span class="md-ellipsis">
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第 2 章 复杂度分析
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第 2 章 复杂度分析
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2.1 算法效率评估
|
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</span>
|
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|
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|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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2.2 迭代与递归
|
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|
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|
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|
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|
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|
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<span class="md-status md-status--new" title="最近添加">
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2.3 时间复杂度
|
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|
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|
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|
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</a>
|
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</li>
|
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2.4 空间复杂度
|
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|
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|
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|
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</a>
|
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</li>
|
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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2.5 小结
|
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</span>
|
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|
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|
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</a>
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<span class="md-ellipsis">
|
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第 3 章 数据结构
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3.1 数据结构分类
|
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</a>
|
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</li>
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3.3 数字编码 *
|
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|
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|
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|
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</a>
|
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</li>
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3.4 字符编码 *
|
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|
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|
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|
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</a>
|
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</li>
|
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3.5 小结
|
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|
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|
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第 4 章 数组与链表
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4.1 数组
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</a>
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4.2 链表
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|
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</a>
|
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</li>
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4.3 列表
|
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|
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|
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|
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</a>
|
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</li>
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4.4 小结
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|
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|
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第 5 章 栈与队列
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5.1 栈
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</a>
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</li>
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5.2 队列
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</a>
|
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</li>
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5.3 双向队列
|
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</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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5.4 小结
|
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|
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|
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|
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<span class="md-ellipsis">
|
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第 6 章 哈希表
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</a>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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第 6 章 哈希表
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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6.1 哈希表
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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|
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6.2 哈希冲突
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.3 哈希算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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<span class="md-ellipsis">
|
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6.4 小结
|
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</span>
|
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|
||
|
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</a>
|
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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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第 7 章 树
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|
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|
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7.1 二叉树
|
||
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|
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|
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|
||
</a>
|
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</li>
|
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|
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|
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7.2 二叉树遍历
|
||
</span>
|
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|
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|
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</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
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7.3 二叉树数组表示
|
||
</span>
|
||
|
||
|
||
</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">
|
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
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|
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|
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7.4 二叉搜索树
|
||
</span>
|
||
|
||
|
||
</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_tree/avl_tree/" class="md-nav__link">
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7.5 AVL 树 *
|
||
</span>
|
||
|
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|
||
</a>
|
||
</li>
|
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|
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|
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7.6 小结
|
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</span>
|
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|
||
|
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</a>
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|
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|
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|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 8 章 堆
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " 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>
|
||
第 8 章 堆
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.1 堆
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.2 建堆操作
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.3 Top-K 问题
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
8.4 小结
|
||
</span>
|
||
|
||
|
||
</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__container">
|
||
<a href="../../chapter_graph/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 9 章 图
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " 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>
|
||
第 9 章 图
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.1 图
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.2 图基础操作
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.3 图的遍历
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
9.4 小结
|
||
</span>
|
||
|
||
|
||
</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__container">
|
||
<a href="../../chapter_searching/" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 10 章 搜索
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " 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>
|
||
第 10 章 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">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.1 二分查找
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.2 二分查找插入点
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.3 二分查找边界
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.4 哈希优化策略
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.5 重识搜索算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
10.6 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--active md-nav__item--nested">
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
|
||
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../" class="md-nav__link ">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
第 11 章 排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
|
||
<label class="md-nav__link " 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="true">
|
||
<label class="md-nav__title" for="__nav_12">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
第 11 章 排序
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../sorting_algorithm/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.1 排序算法
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../selection_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.2 选择排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../bubble_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.3 冒泡排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../insertion_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.4 插入排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../quick_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.5 快速排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../merge_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.6 归并排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../heap_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.7 堆排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../bucket_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.8 桶排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../counting_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.9 计数排序
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../radix_sort/" class="md-nav__link">
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
11.10 基数排序
|
||
</span>
|
||
|
||
|
||
</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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<label class="md-nav__link md-nav__link--active" for="__toc">
|
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|
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|
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<span class="md-ellipsis">
|
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11.11 小结
|
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</span>
|
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|
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|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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<a href="./" class="md-nav__link md-nav__link--active">
|
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|
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|
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<span class="md-ellipsis">
|
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11.11 小结
|
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</span>
|
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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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目录
|
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|
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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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1. 重点回顾
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#2-q-a" class="md-nav__link">
|
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2. Q & A
|
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|
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|
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|
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</ul>
|
||
|
||
</nav>
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|
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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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|
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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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<span class="md-ellipsis">
|
||
第 12 章 分治
|
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|
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<span class="md-nav__icon md-icon"></span>
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第 12 章 分治
|
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12.1 分治算法
|
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</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
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|
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12.2 分治搜索策略
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
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<li class="md-nav__item">
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12.3 构建树问题
|
||
</span>
|
||
|
||
|
||
</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_divide_and_conquer/hanota_problem/" class="md-nav__link">
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|
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12.4 汉诺塔问题
|
||
</span>
|
||
|
||
|
||
</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_divide_and_conquer/summary/" class="md-nav__link">
|
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|
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|
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|
||
12.5 小结
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
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|
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</ul>
|
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</nav>
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||
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|
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||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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<span class="md-ellipsis">
|
||
第 13 章 回溯
|
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</span>
|
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|
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</a>
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|
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<label class="md-nav__link " for="__nav_14">
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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第 13 章 回溯
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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13.1 回溯算法
|
||
</span>
|
||
|
||
|
||
</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">
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.2 全排列问题
|
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</span>
|
||
|
||
|
||
</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_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3 子集和问题
|
||
</span>
|
||
|
||
|
||
</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">
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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|
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<span class="md-ellipsis">
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13.4 N 皇后问题
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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<a href="../../chapter_backtracking/summary/" class="md-nav__link">
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|
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<span class="md-ellipsis">
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13.5 小结
|
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</span>
|
||
|
||
|
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</a>
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</li>
|
||
|
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|
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</ul>
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</li>
|
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|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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<span class="md-ellipsis">
|
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第 14 章 动态规划
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第 14 章 动态规划
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14.1 初探动态规划
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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14.2 DP 问题特性
|
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|
||
|
||
|
||
</a>
|
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</li>
|
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14.3 DP 解题思路
|
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</span>
|
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|
||
|
||
</a>
|
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</li>
|
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|
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14.4 0-1 背包问题
|
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</span>
|
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|
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|
||
</a>
|
||
</li>
|
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14.5 完全背包问题
|
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</span>
|
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|
||
|
||
</a>
|
||
</li>
|
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14.6 编辑距离问题
|
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</span>
|
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|
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|
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</a>
|
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</li>
|
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14.7 小结
|
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</span>
|
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|
||
|
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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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<!-- Page content -->
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<h1 id="1111">11.11 小结<a class="headerlink" href="#1111" title="Permanent link">¶</a></h1>
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<h3 id="1">1. 重点回顾<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
|
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<ul>
|
||
<li>冒泡排序通过交换相邻元素来实现排序。通过添加一个标志位来实现提前返回，我们可以将冒泡排序的最佳时间复杂度优化到 <span class="arithmatex">\(O(n)\)</span> 。</li>
|
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<li>插入排序每轮将未排序区间内的元素插入到已排序区间的正确位置，从而完成排序。虽然插入排序的时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，但由于单元操作相对较少，它在小数据量的排序任务中非常受欢迎。</li>
|
||
<li>快速排序基于哨兵划分操作实现排序。在哨兵划分中，有可能每次都选取到最差的基准数，导致时间复杂度劣化至 <span class="arithmatex">\(O(n^2)\)</span> 。引入中位数基准数或随机基准数可以降低这种劣化的概率。尾递归方法可以有效地减少递归深度，将空间复杂度优化到 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
|
||
<li>归并排序包括划分和合并两个阶段，典型地体现了分治策略。在归并排序中，排序数组需要创建辅助数组，空间复杂度为 <span class="arithmatex">\(O(n)\)</span> ；然而排序链表的空间复杂度可以优化至 <span class="arithmatex">\(O(1)\)</span> 。</li>
|
||
<li>桶排序包含三个步骤：数据分桶、桶内排序和合并结果。它同样体现了分治策略，适用于数据体量很大的情况。桶排序的关键在于对数据进行平均分配。</li>
|
||
<li>计数排序是桶排序的一个特例，它通过统计数据出现的次数来实现排序。计数排序适用于数据量大但数据范围有限的情况，并且要求数据能够转换为正整数。</li>
|
||
<li>基数排序通过逐位排序来实现数据排序，要求数据能够表示为固定位数的数字。</li>
|
||
<li>总的来说，我们希望找到一种排序算法，具有高效率、稳定、原地以及正向自适应性等优点。然而，正如其他数据结构和算法一样，没有一种排序算法能够同时满足所有这些条件。在实际应用中，我们需要根据数据的特性来选择合适的排序算法。</li>
|
||
<li>图 11-19 对比了主流排序算法的效率、稳定性、就地性和自适应性等。</li>
|
||
</ul>
|
||
<p><a class="glightbox" href="../summary.assets/sorting_algorithms_comparison.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="排序算法对比" src="../summary.assets/sorting_algorithms_comparison.png" /></a></p>
|
||
<p align="center"> 图 11-19 排序算法对比 </p>
|
||
|
||
<h3 id="2-q-a">2. Q & A<a class="headerlink" href="#2-q-a" title="Permanent link">¶</a></h3>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">排序算法稳定性在什么情况下是必须的？</p>
|
||
<p>在现实中，我们有可能是在对象的某个属性上进行排序。例如，学生有姓名和身高两个属性，我们希望实现一个多级排序/</p>
|
||
<p>先按照姓名进行排序，得到 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> ；接下来对身高进行排序。由于排序算法不稳定，我们可能得到 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> 。</p>
|
||
<p>可以发现，学生 D 和 C 的位置发生了交换，姓名的有序性被破坏了，而这是我们不希望看到的。</p>
|
||
</div>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">哨兵划分中“从右往左查找”与“从左往右查找”的顺序可以交换吗？</p>
|
||
<p>不行，当我们以最左端元素为基准数时，必须先“从右往左查找”再“从左往右查找”。这个结论有些反直觉，我们来剖析一下原因。</p>
|
||
<p>哨兵划分 <code>partition()</code> 的最后一步是交换 <code>nums[left]</code> 和 <code>nums[i]</code> 。完成交换后，基准数左边的元素都 <code><=</code> 基准数，<strong>这就要求最后一步交换前 <code>nums[left] >= nums[i]</code> 必须成立</strong>。假设我们先“从左往右查找”，那么如果找不到比基准数更小的元素，<strong>则会在 <code>i == j</code> 时跳出循环，此时可能 <code>nums[j] == nums[i] > nums[left]</code></strong>。也就是说，此时最后一步交换操作会把一个比基准数更大的元素交换至数组最左端，导致哨兵划分失败。</p>
|
||
<p>举个例子，给定数组 <code>[0, 0, 0, 0, 1]</code> ，如果先“从左向右查找”，哨兵划分后数组为 <code>[1, 0, 0, 0, 0]</code> ，这个结果是不正确的。</p>
|
||
<p>再深入思考一下，如果我们选择 <code>nums[right]</code> 为基准数，那么正好反过来，必须先“从左往右查找”。</p>
|
||
</div>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">关于尾递归优化，为什么选短的数组能保证递归深度不超过 <span class="arithmatex">\(\log n\)</span> ？</p>
|
||
<p>递归深度就是当前未返回的递归方法的数量。每轮哨兵划分我们将原数组划分为两个子数组。在尾递归优化后，向下递归的子数组长度最大为原数组的一半长度。假设最差情况，一直为一半长度，那么最终的递归深度就是 <span class="arithmatex">\(\log n\)</span> 。</p>
|
||
<p>回顾原始的快速排序，我们有可能会连续地递归长度较大的数组，最差情况下为 <span class="arithmatex">\(n\)</span>、<span class="arithmatex">\(n - 1\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</span> ，递归深度为 <span class="arithmatex">\(n\)</span> 。尾递归优化可以避免这种情况的出现。</p>
|
||
</div>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">当数组中所有元素都相等时，快速排序的时间复杂度是 <span class="arithmatex">\(O(n^2)\)</span> 吗？该如何处理这种退化情况？</p>
|
||
<p>是的。这种情况可以考虑通过哨兵划分将数组划分为三个部分：小于、等于、大于基准数。仅向下递归小于和大于的两部分。在该方法下，输入元素全部相等的数组，仅一轮哨兵划分即可完成排序。</p>
|
||
</div>
|
||
<div class="admonition question">
|
||
<p class="admonition-title">桶排序的最差时间复杂度为什么是 <span class="arithmatex">\(O(n^2)\)</span> ？</p>
|
||
<p>最差情况下，所有元素被分至同一个桶中。如果我们采用一个 <span class="arithmatex">\(O(n^2)\)</span> 算法来排序这些元素，则时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||
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