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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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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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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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第 4 章 数组与链表
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第 4 章 数组与链表
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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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第 7 章 树
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第 7 章 树
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7.5 AVL 树 *
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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 >
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第 8 章 堆
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第 8 章 堆
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< / label >
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8.1 堆
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8.2 建堆操作
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< / span >
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8.3 Top-K 问题
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< / span >
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8.4 小结
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< / span >
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< 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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< / label >
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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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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
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< div class = "md-nav__link md-nav__container" >
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< 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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< / span >
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第 10 章 搜索
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< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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10.1 二分查找
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< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search_insertion/" class = "md-nav__link" >
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10.2 二分查找插入点
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< / span >
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10.3 二分查找边界
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< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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10.4 哈希优化策略
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< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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10.5 重识搜索算法
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< / span >
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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10.6 小结
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< / span >
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< / a >
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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__container" >
< a href = "../../chapter_sorting/" 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 >
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第 11 章 排序
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< / span >
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第 11 章 排序
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< / label >
< ul class = "md-nav__list" data-md-scrollfix >
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< li class = "md-nav__item" >
< a href = "../../chapter_sorting/sorting_algorithm/" class = "md-nav__link" >
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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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11.5 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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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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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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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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14.2 DP 问题特性
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14.2.1 最优子结构
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14.2.2 无后效性
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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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第 15 章 贪心
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15.1 贪心算法
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15.2 分数背包问题
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15.3 最大容量问题
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15.4 最大切分乘积问题
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15.5 小结
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第 16 章 附录
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第 16 章 附录
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16.1 编程环境安装
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16.2 一起参与创作
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14.2.1 最优子结构
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14.2.2 无后效性
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< h1 id = "142" > 14.2 动态规划问题特性< a class = "headerlink" href = "#142" title = "Permanent link" > ¶ < / a > < / h1 >
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< p > 在上节中，我们学习了动态规划是如何通过子问题分解来求解问题的。实际上，子问题分解是一种通用的算法思路，在分治、动态规划、回溯中的侧重点不同。< / p >
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< ul >
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< li > 分治算法递归地将原问题划分为多个相互独立的子问题，直至最小子问题，并在回溯中合并子问题的解，最终得到原问题的解。< / li >
< li > 动态规划也对问题进行递归分解，但与分治算法的主要区别是，动态规划中的子问题是相互依赖的，在分解过程中会出现许多重叠子问题。< / li >
< li > 回溯算法在尝试和回退中穷举所有可能的解，并通过剪枝避免不必要的搜索分支。原问题的解由一系列决策步骤构成，我们可以将每个决策步骤之前的子序列看作为一个子问题。< / li >
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< / ul >
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< p > 实际上，动态规划常用来求解最优化问题，它们不仅包含重叠子问题，还具有另外两大特性：最优子结构、无后效性。< / p >
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< h2 id = "1421" > 14.2.1 最优子结构< a class = "headerlink" href = "#1421" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 我们对爬楼梯问题稍作改动，使之更加适合展示最优子结构概念。< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 爬楼梯最小代价< / p >
< p > 给定一个楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶，每一阶楼梯上都贴有一个非负整数，表示你在该台阶所需要付出的代价。给定一个非负整数数组 < span class = "arithmatex" > \(cost\)< / span > ，其中 < span class = "arithmatex" > \(cost[i]\)< / span > 表示在第 < span class = "arithmatex" > \(i\)< / span > 个台阶需要付出的代价，< span class = "arithmatex" > \(cost[0]\)< / span > 为地面起始点。请计算最少需要付出多少代价才能到达顶部？< / p >
< / div >
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< p > 如图 14-6 所示，若第 < span class = "arithmatex" > \(1\)< / span > 、< span class = "arithmatex" > \(2\)< / span > 、< span class = "arithmatex" > \(3\)< / span > 阶的代价分别为 < span class = "arithmatex" > \(1\)< / span > 、< span class = "arithmatex" > \(10\)< / span > 、< span class = "arithmatex" > \(1\)< / span > ，则从地面爬到第 < span class = "arithmatex" > \(3\)< / span > 阶的最小代价为 < span class = "arithmatex" > \(2\)< / span > 。< / p >
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< p > < a class = "glightbox" href = "../dp_problem_features.assets/min_cost_cs_example.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "爬到第 3 阶的最小代价" src = "../dp_problem_features.assets/min_cost_cs_example.png" / > < / a > < / p >
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< p align = "center" > 图 14-6 爬到第 3 阶的最小代价 < / p >
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< p > 设 < span class = "arithmatex" > \(dp[i]\)< / span > 为爬到第 < span class = "arithmatex" > \(i\)< / span > 阶累计付出的代价，由于第 < span class = "arithmatex" > \(i\)< / span > 阶只可能从 < span class = "arithmatex" > \(i - 1\)< / span > 阶或 < span class = "arithmatex" > \(i - 2\)< / span > 阶走来，因此 < span class = "arithmatex" > \(dp[i]\)< / span > 只可能等于 < span class = "arithmatex" > \(dp[i - 1] + cost[i]\)< / span > 或 < span class = "arithmatex" > \(dp[i - 2] + cost[i]\)< / span > 。为了尽可能减少代价，我们应该选择两者中较小的那一个：< / p >
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< div class = "arithmatex" > \[
dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
\]< / div >
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< p > 这便可以引出最优子结构的含义：< strong > 原问题的最优解是从子问题的最优解构建得来的< / strong > 。< / p >
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< p > 本题显然具有最优子结构：我们从两个子问题最优解 < span class = "arithmatex" > \(dp[i-1]\)< / span > 和 < span class = "arithmatex" > \(dp[i-2]\)< / span > 中挑选出较优的那一个，并用它构建出原问题 < span class = "arithmatex" > \(dp[i]\)< / span > 的最优解。< / p >
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< p > 那么，上节的爬楼梯题目有没有最优子结构呢？它的目标是求解方案数量，看似是一个计数问题，但如果换一种问法：“求解最大方案数量”。我们意外地发现，< strong > 虽然题目修改前后是等价的，但最优子结构浮现出来了< / strong > ：第 < span class = "arithmatex" > \(n\)< / span > 阶最大方案数量等于第 < span class = "arithmatex" > \(n-1\)< / span > 阶和第 < span class = "arithmatex" > \(n-2\)< / span > 阶最大方案数量之和。所以说，最优子结构的解释方式比较灵活，在不同问题中会有不同的含义。< / p >
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< p > 根据状态转移方程，以及初始状态 < span class = "arithmatex" > \(dp[1] = cost[1]\)< / span > 和 < span class = "arithmatex" > \(dp[2] = cost[2]\)< / span > ，我们就可以得到动态规划代码。< / p >
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< p > 图 14-7 展示了以上代码的动态规划过程。< / p >
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< p > < a class = "glightbox" href = "../dp_problem_features.assets/min_cost_cs_dp.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "爬楼梯最小代价的动态规划过程" src = "../dp_problem_features.assets/min_cost_cs_dp.png" / > < / a > < / p >
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< p align = "center" > 图 14-7 爬楼梯最小代价的动态规划过程 < / p >
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< p > 本题也可以进行空间优化，将一维压缩至零维，使得空间复杂度从 < span class = "arithmatex" > \(O(n)\)< / span > 降低至 < span class = "arithmatex" > \(O(1)\)< / span > 。< / p >
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< h2 id = "1422" > 14.2.2 无后效性< a class = "headerlink" href = "#1422" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 无后效性是动态规划能够有效解决问题的重要特性之一，定义为：< strong > 给定一个确定的状态，它的未来发展只与当前状态有关，而与当前状态过去所经历过的所有状态无关< / strong > 。< / p >
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< p > 以爬楼梯问题为例，给定状态 < span class = "arithmatex" > \(i\)< / span > ，它会发展出状态 < span class = "arithmatex" > \(i+1\)< / span > 和状态 < span class = "arithmatex" > \(i+2\)< / span > ，分别对应跳 < span class = "arithmatex" > \(1\)< / span > 步和跳 < span class = "arithmatex" > \(2\)< / span > 步。在做出这两种选择时，我们无须考虑状态 < span class = "arithmatex" > \(i\)< / span > 之前的状态，它们对状态 < span class = "arithmatex" > \(i\)< / span > 的未来没有影响。< / p >
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< p > 然而，如果我们向爬楼梯问题添加一个约束，情况就不一样了。< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 带约束爬楼梯< / p >
< p > 给定一个共有 < span class = "arithmatex" > \(n\)< / span > 阶的楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶，< strong > 但不能连续两轮跳 < span class = "arithmatex" > \(1\)< / span > 阶< / strong > ，请问有多少种方案可以爬到楼顶。< / p >
< / div >
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< p > 例如图 14-8 ，爬上第 < span class = "arithmatex" > \(3\)< / span > 阶仅剩 < span class = "arithmatex" > \(2\)< / span > 种可行方案，其中连续三次跳 < span class = "arithmatex" > \(1\)< / span > 阶的方案不满足约束条件，因此被舍弃。< / p >
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< p > < a class = "glightbox" href = "../dp_problem_features.assets/climbing_stairs_constraint_example.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "带约束爬到第 3 阶的方案数量" src = "../dp_problem_features.assets/climbing_stairs_constraint_example.png" / > < / a > < / p >
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< p align = "center" > 图 14-8 带约束爬到第 3 阶的方案数量 < / p >
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< p > 在该问题中，如果上一轮是跳 < span class = "arithmatex" > \(1\)< / span > 阶上来的，那么下一轮就必须跳 < span class = "arithmatex" > \(2\)< / span > 阶。这意味着，< strong > 下一步选择不能由当前状态（当前楼梯阶数）独立决定，还和前一个状态（上轮楼梯阶数）有关< / strong > 。< / p >
< p > 不难发现，此问题已不满足无后效性，状态转移方程 < span class = "arithmatex" > \(dp[i] = dp[i-1] + dp[i-2]\)< / span > 也失效了，因为 < span class = "arithmatex" > \(dp[i-1]\)< / span > 代表本轮跳 < span class = "arithmatex" > \(1\)< / span > 阶，但其中包含了许多“上一轮跳 < span class = "arithmatex" > \(1\)< / span > 阶上来的”方案，而为了满足约束，我们就不能将 < span class = "arithmatex" > \(dp[i-1]\)< / span > 直接计入 < span class = "arithmatex" > \(dp[i]\)< / span > 中。< / p >
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< p > 为此，我们需要扩展状态定义：< strong > 状态 < span class = "arithmatex" > \([i, j]\)< / span > 表示处在第 < span class = "arithmatex" > \(i\)< / span > 阶、并且上一轮跳了 < span class = "arithmatex" > \(j\)< / span > 阶< / strong > ，其中 < span class = "arithmatex" > \(j \in \{1, 2\}\)< / span > 。此状态定义有效地区分了上一轮跳了 < span class = "arithmatex" > \(1\)< / span > 阶还是 < span class = "arithmatex" > \(2\)< / span > 阶，我们可以据此来判断当前状态是从何而来的。< / p >
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< ul >
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< li > 当上一轮跳了 < span class = "arithmatex" > \(1\)< / span > 阶时，上上一轮只能选择跳 < span class = "arithmatex" > \(2\)< / span > 阶，即 < span class = "arithmatex" > \(dp[i, 1]\)< / span > 只能从 < span class = "arithmatex" > \(dp[i-1, 2]\)< / span > 转移过来。< / li >
< li > 当上一轮跳了 < span class = "arithmatex" > \(2\)< / span > 阶时，上上一轮可选择跳 < span class = "arithmatex" > \(1\)< / span > 阶或跳 < span class = "arithmatex" > \(2\)< / span > 阶，即 < span class = "arithmatex" > \(dp[i, 2]\)< / span > 可以从 < span class = "arithmatex" > \(dp[i-2, 1]\)< / span > 或 < span class = "arithmatex" > \(dp[i-2, 2]\)< / span > 转移过来。< / li >
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< p > 如图 14-9 所示，在该定义下，< span class = "arithmatex" > \(dp[i, j]\)< / span > 表示状态 < span class = "arithmatex" > \([i, j]\)< / span > 对应的方案数。此时状态转移方程为：< / p >
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< div class = "arithmatex" > \[
\begin{cases}
dp[i, 1] = dp[i-1, 2] \\
dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
\end{cases}
\]< / div >
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< p > < a class = "glightbox" href = "../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" data-type = "image" data-width = "100%" data-height = "auto" data-desc-position = "bottom" > < img alt = "考虑约束下的递推关系" src = "../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" / > < / a > < / p >
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< p align = "center" > 图 14-9 考虑约束下的递推关系 < / p >
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< p > 最终，返回 < span class = "arithmatex" > \(dp[n, 1] + dp[n, 2]\)< / span > 即可，两者之和代表爬到第 < span class = "arithmatex" > \(n\)< / span > 阶的方案总数。< / p >
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< p > 在上面的案例中，由于仅需多考虑前面一个状态，我们仍然可以通过扩展状态定义，使得问题重新满足无后效性。然而，某些问题具有非常严重的“有后效性”。< / p >
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< div class = "admonition question" >
< p class = "admonition-title" > 爬楼梯与障碍生成< / p >
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< p > 给定一个共有 < span class = "arithmatex" > \(n\)< / span > 阶的楼梯，你每步可以上 < span class = "arithmatex" > \(1\)< / span > 阶或者 < span class = "arithmatex" > \(2\)< / span > 阶。< strong > 规定当爬到第 < span class = "arithmatex" > \(i\)< / span > 阶时，系统自动会给第 < span class = "arithmatex" > \(2i\)< / span > 阶上放上障碍物，之后所有轮都不允许跳到第 < span class = "arithmatex" > \(2i\)< / span > 阶上< / strong > 。例如，前两轮分别跳到了第 < span class = "arithmatex" > \(2\)< / span > 、< span class = "arithmatex" > \(3\)< / span > 阶上，则之后就不能跳到第 < span class = "arithmatex" > \(4\)< / span > 、< span class = "arithmatex" > \(6\)< / span > 阶上。请问有多少种方案可以爬到楼顶。< / p >
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< p > 在这个问题中，下次跳跃依赖于过去所有的状态，因为每一次跳跃都会在更高的阶梯上设置障碍，并影响未来的跳跃。对于这类问题，动态规划往往难以解决。< / p >
< p > 实际上，许多复杂的组合优化问题（例如旅行商问题）都不满足无后效性。对于这类问题，我们通常会选择使用其他方法，例如启发式搜索、遗传算法、强化学习等，从而在有限时间内得到可用的局部最优解。< / p >
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