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Hello 算法
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12.1 分治算法
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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 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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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 内存与缓存 *
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4.5 小结
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< a href = "../../chapter_stack_and_queue/" class = "md-nav__link " >
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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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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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< 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 >
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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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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
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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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< div class = "md-nav__link md-nav__container" >
< a href = "../" class = "md-nav__link " >
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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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12.1 分治算法
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目录
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12.1.1 如何判断分治问题
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12.1.2 通过分治提升效率
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1. 操作数量优化
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2. 并行计算优化
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12.1.3 分治常见应用
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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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15.1 贪心算法
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15.5 小结
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12.1.1 如何判断分治问题
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2. 并行计算优化
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< h1 id = "121" > 12.1 分治算法< a class = "headerlink" href = "#121" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 「分治 divide and conquer」， ， ， < / p >
< ol >
< 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 >
< 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 >
< / ol >
< p > 如图 12-1 所示，“归并排序”是分治策略的典型应用之一。< / p >
< ol >
< 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 >
< li > < strong > 治< / strong > ：从底至顶地将有序的子数组（子问题的解）进行合并，从而得到有序的原数组（原问题的解）。< / li >
< / ol >
< p > < a class = "glightbox" href = "../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "归并排序的分治策略" class = "animation-figure" src = "../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" / > < / a > < / p >
< p align = "center" > 图 12-1 归并排序的分治策略 < / p >
< h2 id = "1211" > 12.1.1 如何判断分治问题< a class = "headerlink" href = "#1211" title = "Permanent link" > ¶ < / a > < / h2 >
< 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 >
< ol >
< li > < strong > 问题可以分解< / strong > ：原问题可以分解成规模更小、类似的子问题，以及能够以相同方式递归地进行划分。< / li >
< li > < strong > 子问题是独立的< / strong > ：子问题之间没有重叠，互不依赖，可以独立解决。< / li >
< li > < strong > 子问题的解可以合并< / strong > ：原问题的解通过合并子问题的解得来。< / li >
< / ol >
< p > 显然，归并排序满足以上三条判断依据。< / p >
< ol >
< li > < strong > 问题可以分解< / strong > ：递归地将数组（原问题）划分为两个子数组（子问题）。< / li >
< li > < strong > 子问题是独立的< / strong > ：每个子数组都可以独立地进行排序（子问题可以独立进行求解）。< / li >
< li > < strong > 子问题的解可以合并< / strong > ：两个有序子数组（子问题的解）可以合并为一个有序数组（原问题的解）。< / li >
< / ol >
< h2 id = "1212" > 12.1.2 通过分治提升效率< a class = "headerlink" href = "#1212" title = "Permanent link" > ¶ < / a > < / h2 >
< p > < strong > 分治不仅可以有效地解决算法问题，往往还可以提升算法效率< / strong > 。在排序算法中，快速排序、归并排序、堆排序相较于选择、冒泡、插入排序更快，就是因为它们应用了分治策略。< / p >
< 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，这几步的效率为什么比直接解决原问题的效率更高？这个问题可以从操作数量和并行计算两方面来讨论。< / p >
< h3 id = "1" > 1. 操作数量优化< a class = "headerlink" href = "#1" title = "Permanent link" > ¶ < / a > < / h3 >
< 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。假设我们按照图 12-2 所示的方式，将数组从中点处分为两个子数组，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 >
< div class = "arithmatex" > \[
O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
\]< / div >
< p > < a class = "glightbox" href = "../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "划分数组前后的冒泡排序" class = "animation-figure" src = "../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" / > < / a > < / p >
< p align = "center" > 图 12-2 划分数组前后的冒泡排序 < / p >
< 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 >
< div class = "arithmatex" > \[
\begin{aligned}
n^2 & > \frac{n^2}{2} + 2n \newline
n^2 - \frac{n^2}{2} - 2n & > 0 \newline
n(n - 4) & > 0
\end{aligned}
\]< / div >
< p > < strong > 这意味着当 < span class = "arithmatex" > \(n > 4\)< / span > 时，划分后的操作数量更少，排序效率应该更高< / strong > 。请注意，划分后的时间复杂度仍然是平方阶 < span class = "arithmatex" > \(O(n^2)\)< / span > ，只是复杂度中的常数项变小了。< / p >
< p > 进一步想，< strong > 如果我们把子数组不断地再从中点处划分为两个子数组< / strong > ，直至子数组只剩一个元素时停止划分呢？这种思路实际上就是“归并排序”，时间复杂度为 < span class = "arithmatex" > \(O(n \log n)\)< / span > 。< / p >
< 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？这种情况与“桶排序”非常类似，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 >
< h3 id = "2" > 2. 并行计算优化< a class = "headerlink" href = "#2" title = "Permanent link" > ¶ < / a > < / h3 >
< 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 >
< 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 >
< p > 比如在图 12-3 所示的“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，完成后再合并结果。< / p >
< p > < a class = "glightbox" href = "../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "桶排序的并行计算" class = "animation-figure" src = "../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" / > < / a > < / p >
< p align = "center" > 图 12-3 桶排序的并行计算 < / p >
< h2 id = "1213" > 12.1.3 分治常见应用< a class = "headerlink" href = "#1213" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 一方面，分治可以用来解决许多经典算法问题。< / p >
< ul >
< 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，最后找出跨越两部分的最近点对。< / li >
< li > < strong > 大整数乘法< / strong > ：例如 Karatsuba 算法，它将大整数乘法分解为几个较小的整数的乘法和加法。< / li >
< li > < strong > 矩阵乘法< / strong > ：例如 Strassen 算法，它将大矩阵乘法分解为多个小矩阵的乘法和加法。< / li >
< li > < strong > 汉诺塔问题< / strong > ：汉诺塔问题可以通过递归解决，这是典型的分治策略应用。< / li >
< 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利用分治的思想，借助归并排序进行求解。< / li >
< / ul >
< p > 另一方面，分治在算法和数据结构的设计中应用非常广泛。< / p >
< ul >
< li > < strong > 二分查找< / strong > ：二分查找是将有序数组从中点索引处分为两部分，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，并在剩余区间执行相同的二分操作。< / li >
< li > < strong > 归并排序< / strong > ：本节开头已介绍，不再赘述。< / li >
< 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，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 >
< 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 >
< li > < strong > 树< / strong > ： < / li >
< 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 >
< 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方案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 >
< / ul >
< 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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