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Hello 算法
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12.5 小结
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第 0 章 前言
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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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1.2 算法是什么
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1.3 小结
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< a href = "../../chapter_computational_complexity/" class = "md-nav__link " >
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第 2 章 时空复杂度
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第 2 章 时空复杂度
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2.1 算法效率评估
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2.2 迭代与递归
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2.3 时间复杂度
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2.4 空间复杂度
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2.5 小结
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< a href = "../../chapter_data_structure/" class = "md-nav__link " >
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第 3 章 数据结构
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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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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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< a href = "../../chapter_array_and_linkedlist/" class = "md-nav__link " >
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第 4 章 数组与链表
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第 4 章 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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4.4 小结
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< a href = "../../chapter_stack_and_queue/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z" / > < / svg >
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第 5 章 栈与队列
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第 5 章 栈与队列
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5.1 栈
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5.2 队列
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5.3 双向队列
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5.4 小结
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< a href = "../../chapter_hashing/" class = "md-nav__link " >
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第 6 章 哈希表
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第 6 章 哈希表
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6.1 哈希表
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6.2 哈希冲突
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6.3 哈希算法
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6.4 小结
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< a href = "../../chapter_tree/" class = "md-nav__link " >
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第 7 章 树
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第 7 章 树
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7.1 二叉树
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7.2 二叉树遍历
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7.3 二叉树数组表示
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7.4 二叉搜索树
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7.5 AVL 树 *
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7.6 小结
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< a href = "../../chapter_heap/" class = "md-nav__link " >
< 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 >
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第 8 章 堆
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第 8 章 堆
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8.1 堆
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8.2 建堆操作
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8.3 Top-K 问题
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8.4 小结
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< 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 >
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第 9 章 图
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第 9 章 图
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9.1 图
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9.2 图基础操作
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9.3 图的遍历
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9.4 小结
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< 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 >
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第 10 章 搜索
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第 10 章 搜索
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10.1 二分查找
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10.2 二分查找插入点
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10.3 二分查找边界
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10.4 哈希优化策略
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10.5 重识搜索算法
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10.6 小结
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< a href = "../../chapter_sorting/" class = "md-nav__link " >
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第 11 章 排序
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第 11 章 排序
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11.1 排序算法
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11.2 选择排序
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11.3 冒泡排序
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11.4 插入排序
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11.5 快速排序
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11.6 归并排序
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11.7 堆排序
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11.8 桶排序
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11.9 计数排序
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11.10 基数排序
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11.11 小结
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< a href = "../" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z" / > < / svg >
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第 12 章 分治
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第 12 章 分治
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12.1 分治算法
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12.2 分治搜索策略
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
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< a href = "../../chapter_backtracking/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z" / > < / svg >
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第 13 章 回溯
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第 13 章 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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13.4 N 皇后问题
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13.5 小结
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< a href = "../../chapter_dynamic_programming/" class = "md-nav__link " >
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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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14.2 DP 问题特性
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14.3 DP 解题思路
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14.4 0-1 背包问题
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14.5 完全背包问题
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14.6 编辑距离问题
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14.7 小结
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< a href = "../../chapter_greedy/" class = "md-nav__link " >
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第 15 章 贪心
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第 15 章 贪心
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< h1 id = "125" > 12.5 小结< a class = "headerlink" href = "#125" title = "Permanent link" > ¶ < / a > < / h1 >
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< li > 分治算法是一种常见的算法设计策略，包括分（划分）和治（合并）两个阶段，通常基于递归实现。< / li >
< li > 判断是否是分治算法问题的依据包括：问题能否被分解、子问题是否独立、子问题是否可以被合并。< / li >
< li > 归并排序是分治策略的典型应用，其递归地将数组划分为等长的两个子数组，直到只剩一个元素时开始逐层合并，从而完成排序。< / li >
< li > 引入分治策略往往可以带来算法效率的提升。一方面，分治策略减少了操作数量；另一方面，分治后有利于系统的并行优化。< / li >
< li > 分治既可以解决许多算法问题，也广泛应用于数据结构与算法设计中，处处可见其身影。< / li >
< li > 相较于暴力搜索，自适应搜索效率更高。时间复杂度为 < span class = "arithmatex" > \(O(\log n)\)< / span > 的搜索算法通常都是基于分治策略实现的。< / li >
< li > 二分查找是分治策略的另一个典型应用，它不包含将子问题的解进行合并的步骤。我们可以通过递归分治实现二分查找。< / li >
< li > 在构建二叉树问题中，构建树（原问题）可以被划分为构建左子树和右子树（子问题），其可以通过划分前序遍历和中序遍历的索引区间来实现。< / li >
< li > 在汉诺塔问题中，一个规模为 < span class = "arithmatex" > \(n\)< / span > 的问题可以被划分为两个规模为 < span class = "arithmatex" > \(n-1\)< / span > 的子问题和一个规模为 < span class = "arithmatex" > \(1\)< / span > 的子问题。按顺序解决这三个子问题后，原问题随之得到解决。< / li >
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