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0. 前言
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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3. 数据结构
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3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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4.3. 列表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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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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<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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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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5. 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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5.2. 队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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5.3. 双向队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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5.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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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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<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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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
|
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<span class="md-nav__icon md-icon"></span>
|
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6. 散列表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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6.1. 哈希表（New）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
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6.2. 哈希冲突（New）
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
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6.3. 哈希算法（New）
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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6.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_tree/">7. 树</a>
|
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<label for="__nav_8">
|
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
|
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7. 树
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
7.1. 二叉树
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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7.2. 二叉树遍历
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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7.3. 二叉树数组表示
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
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7.4. 二叉搜索树
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
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7.5. AVL 树 *
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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7.6. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_heap/">8. 堆</a>
|
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<label for="__nav_9">
|
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
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<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
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8. 堆
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
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8.1. 堆
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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8.2. 建堆操作
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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8.3. Top-K 问题（New）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/summary/" class="md-nav__link">
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8.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_graph/">9. 图</a>
|
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|
<label for="__nav_10">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_10">
|
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<span class="md-nav__icon md-icon"></span>
|
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9. 图
|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph/" class="md-nav__link">
|
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9.1. 图
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
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9.2. 图基础操作
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
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9.3. 图的遍历
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/summary/" class="md-nav__link">
|
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9.4. 小结
|
|
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_searching/">10. 搜索</a>
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<label for="__nav_11">
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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10. 搜索
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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_searching/binary_search/" class="md-nav__link">
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10.1. 二分查找
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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10.2. 二分查找边界
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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10.3. 哈希优化策略
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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10.4. 重识搜索算法
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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10.5. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_sorting/">11. 排序</a>
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<label for="__nav_12">
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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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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_12">
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|
<span class="md-nav__icon md-icon"></span>
|
|
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11. 排序
|
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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_sorting/sorting_algorithm/" class="md-nav__link">
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11.1. 排序算法
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
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11.2. 选择排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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11.3. 冒泡排序
|
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
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11.4. 插入排序
|
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|
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11.5. 快速排序
|
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
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|
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11.6. 归并排序
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
|
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11.7. 堆排序
|
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
|
|
11.8. 桶排序
|
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
|
|
11.9. 计数排序
|
|
|
</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
|
|
11.10. 基数排序
|
|
|
</a>
|
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|
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11.11. 小结
|
|
|
</a>
|
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</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
|
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<div class="md-nav__link md-nav__link--index ">
|
|
|
<a href="../">12. 分治</a>
|
|
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|
|
<label for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
|
</div>
|
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
|
|
|
<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
12. 分治
|
|
|
</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item md-nav__item--active">
|
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
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<label class="md-nav__link md-nav__link--active" for="__toc">
|
|
|
12.1. 分治算法（New）
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
12.1. 分治算法（New）
|
|
|
</a>
|
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<label class="md-nav__title" for="__toc">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
目录
|
|
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</label>
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="#1211" class="md-nav__link">
|
|
|
12.1.1. 通过分治提升效率
|
|
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</a>
|
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|
|
<nav class="md-nav" aria-label="12.1.1. 通过分治提升效率">
|
|
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<ul class="md-nav__list">
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<li class="md-nav__item">
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
操作数量优化
|
|
|
</a>
|
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</li>
|
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|
<li class="md-nav__item">
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
并行计算优化
|
|
|
</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
|
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<a href="#1212" class="md-nav__link">
|
|
|
12.1.2. 分治常见应用
|
|
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
|
|
|
<a href="../build_binary_tree_problem/" class="md-nav__link">
|
|
|
12.2. 构建树问题（New）
|
|
|
</a>
|
|
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</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../hanota_problem/" class="md-nav__link">
|
|
|
12.3. 汉诺塔问题（New）
|
|
|
</a>
|
|
|
</li>
|
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
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<div class="md-nav__link md-nav__link--index ">
|
|
|
<a href="../../chapter_backtracking/">13. 回溯</a>
|
|
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|
|
<label for="__nav_14">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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|
</div>
|
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_14">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
13. 回溯
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
|
|
|
13.1. 回溯算法
|
|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
|
|
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13.2. 全排列问题
|
|
|
</a>
|
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</li>
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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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13.3. 子集和问题
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</a>
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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13.4. N 皇后问题
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</a>
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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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</a>
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14.7. 小结（New）
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15. 附录
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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="#1211" class="md-nav__link">
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12.1.1. 通过分治提升效率
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</a>
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<nav class="md-nav" aria-label="12.1.1. 通过分治提升效率">
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操作数量优化
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并行计算优化
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12.1.2. 分治常见应用
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</a>
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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_divide_and_conquer/divide_and_conquer.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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</a>
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<h1 id="121">12.1. 分治算法<a class="headerlink" href="#121" title="Permanent link">¶</a></h1>
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<p>「分治 Divide and Conquer」，全称分而治之，是一种非常重要的算法策略。分治通常基于递归实现，包括“分”和“治”两部分，主要步骤如下：</p>
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<ol>
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<li><strong>分（划分阶段）</strong>：递归地将原问题分解为两个或多个子问题，直至到达最小子问题时终止；</li>
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<li><strong>治（合并阶段）</strong>：从已知解的最小子问题开始，从底至顶地将子问题的解进行合并，从而构建出原问题的解；</li>
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</ol>
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<p>之前学过的「归并排序」是分治策略的典型应用之一，对于该算法：</p>
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<ol>
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<li><strong>分</strong>：递归地将原数组（原问题）划分为两个子数组（子问题），直到子数组只剩一个元素（最小子问题）。</li>
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<li><strong>治</strong>：从底至顶地将有序的子数组进行合并，从而得到有序的原数组。</li>
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</ol>
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<p><img alt="归并排序的分治策略" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></p>
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<p align="center"> Fig. 归并排序的分治策略 </p>
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<p>一个问题是否适合使用分治解决，通常可以参考以下几个判断依据：</p>
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<ol>
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<li><strong>问题可以被分解</strong>：原问题可以被分解成规模更小、类似的子问题，以及能够以相同方式递归地进行划分。</li>
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<li><strong>子问题是独立的</strong>：子问题之间是没有重叠的，互相没有依赖，可以被独立解决。</li>
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<li><strong>子问题的解可以被合并</strong>：原问题的解可以通过合并子问题的解得来。</li>
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</ol>
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<p>归并排序显然是满足以上三条判断依据的：</p>
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<ol>
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<li>递归地将数组（原问题）划分为两个子数组（子问题），当子数组只有一个元素时天然有序；</li>
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<li>每个子数组都可以独立地进行排序，因此子问题是独立的；</li>
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<li>两个有序子数组（子问题的解）可以被合并为一个有序数组（原问题的解）；</li>
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</ol>
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<h2 id="1211">12.1.1. 通过分治提升效率<a class="headerlink" href="#1211" title="Permanent link">¶</a></h2>
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<p>分治不仅可以有效地解决算法问题，<strong>往往还可以提升算法效率</strong>。在排序算法中，归并排序相较于选择、冒泡、插入排序更快，就是因为其应用了分治策略。</p>
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<p>那么，我们不禁发问：<strong>为什么分治可以提升算法效率，其底层逻辑是什么</strong>？换句话说，将大问题分解为多个子问题、解决子问题、将子问题的解合并为原问题的解，这三步为什么比直接解决原问题效率更高？这个问题可以从操作数量和并行计算两方面来讨论。</p>
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<h3 id="_1">操作数量优化<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
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<p>以「冒泡排序」为例，其处理一个长度为 <span class="arithmatex">\(n\)</span> 的数组需要 <span class="arithmatex">\(O(n^2)\)</span> 时间。假设我们把数组从中点分为两个子数组，则划分需要 <span class="arithmatex">\(O(n)\)</span> 时间，排序每个子数组需要 <span class="arithmatex">\(O((n/2)^2)\)</span> 时间，合并两个子数组需要 <span class="arithmatex">\(O(n)\)</span> 时间，总体时间复杂度为：</p>
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<div class="arithmatex">\[
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O(n + (n/2)^2 \times 2 + n) = O(n^2 / 2 + 2n)
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\]</div>
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<p><img alt="划分数组前后的冒泡排序" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></p>
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<p align="center"> Fig. 划分数组前后的冒泡排序 </p>
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<p>接下来，我们计算以下不等式，其左边和右边分别为划分前和划分后的操作总数：</p>
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<div class="arithmatex">\[
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\begin{aligned}
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n^2 & > \frac{n^2}{2} + 2n \newline
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|
n^2 - \frac{n^2}{2} - 2n & > 0 \newline
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|
|
n(n - 4) & > 0
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\end{aligned}
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\]</div>
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<p><strong>这意味着当 <span class="arithmatex">\(n > 4\)</span> 时，划分后的操作数量更少，排序效率可能更高</strong>。当然，划分后的时间复杂度仍然是平方阶 <span class="arithmatex">\(O(n^2)\)</span> ，即复杂度并没有降低，只是其中的常数项变小了。</p>
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<p>那么，<strong>如果我们把子数组不断地再从中点划分为两个子数组</strong>，直至子数组只剩一个元素时停止划分呢？这就达到了「归并排序」的情况，时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
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<p>再思考，<strong>如果我们多设置几个划分点</strong>，将原数组平均划分为 <span class="arithmatex">\(k\)</span> 个子数组呢？这种情况就与「桶排序」非常类似了，它非常适合排序海量数据，理论上时间复杂度可以达到 <span class="arithmatex">\(O(n + k)\)</span> 。</p>
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<h3 id="_2">并行计算优化<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
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<p>我们知道，分治生成的子问题是相互独立的，<strong>因此通常可以并行解决</strong>。也就是说，分治不仅可以降低算法的时间复杂度，<strong>还有利于操作系统的并行优化</strong>。</p>
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<p>并行优化在多核或多处理器的环境中尤其有效，因为系统可以同时处理多个子问题，更加充分地利用计算资源，从而显著减少总体的运行时间。</p>
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<p>例如在桶排序中，我们将海量的数据平均分配到各个桶中，则可所有桶的排序任务分散到各个计算单元，完成后再进行结果合并。</p>
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<p><img alt="桶排序的并行计算" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></p>
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<p align="center"> Fig. 桶排序的并行计算 </p>
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<h2 id="1212">12.1.2. 分治常见应用<a class="headerlink" href="#1212" title="Permanent link">¶</a></h2>
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<p>一方面，分治可以用来解决许多经典算法问题：</p>
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<ul>
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<li><strong>寻找最近点对</strong>：该算法首先将点集分成两部分，然后分别找出两部分中的最近点对，最后再找出跨越两部分的最近点对。</li>
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<li><strong>大整数乘法</strong>：例如 Karatsuba 算法，它是将大整数乘法分解为几个较小的整数的乘法和加法。</li>
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<li><strong>矩阵乘法</strong>：例如 Strassen 算法，它是将大矩阵乘法分解为多个小矩阵的乘法和加法。</li>
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<li><strong>汉诺塔问题</strong>：汉诺塔问题可以视为典型的分治策略，通过递归解决。</li>
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<li><strong>求解逆序对</strong>：在一个序列中，如果前面的数字大于后面的数字，那么这两个数字构成一个逆序对。求解逆序对问题可以通过分治的思想，借助归并排序进行求解。</li>
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</ul>
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<p>另一方面，<strong>分治在算法和数据结构的设计中应用非常广泛</strong>，举几个已经学过的例子：</p>
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<ul>
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<li><strong>二分查找</strong>：二分查找是将有序数组从中点索引分为两部分，然后根据目标值与中间元素值比较结果，决定排除哪一半区间，然后在剩余区间执行相同的二分操作。</li>
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<li><strong>归并排序</strong>：文章开头已介绍，不再赘述。</li>
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<li><strong>快速排序</strong>：快速排序是选取一个基准值，然后把数组分为两个子数组，一个子数组的元素比基准值小，另一子数组的元素比基准值大，然后再对这两部分进行相同的划分操作，直至子数组只剩下一个元素。</li>
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<li><strong>桶排序</strong>：桶排序的基本思想是将数据分散到多个桶，然后对每个桶内的元素进行排序，最后将各个桶的元素依次取出，从而得到一个有序数组。</li>
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<li><strong>树</strong>：例如二叉搜索树、AVL 树、红黑树、B 树、B+ 树等，它们的查找、插入和删除等操作都可以视为分治的应用。</li>
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<li><strong>堆</strong>：堆是一种特殊的完全二叉树，其各种操作，如插入、删除和堆化，实际上都隐含了分治的思想。</li>
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<li><strong>哈希表</strong>：虽然哈希表来并不直接应用分治，但某些哈希冲突解决策略间接应用了分治策略，例如，链式地址中的长链表会被转化为红黑树，以提升查询效率。</li>
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</ul>
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<p>可以看出，<strong>分治是一种“润物细无声”的算法思想</strong>，隐含在各种算法与数据结构之中。</p>
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