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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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4.4. 小结
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5. 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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6. 散列表
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7. 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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8. 堆
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8.1. 堆
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8.2. 建堆操作
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9.2. 图基础操作
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10.3. 哈希优化策略
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10.4. 重识搜索算法
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10.5. 小结
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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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12.3. 子集和问题（ ）
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12.4. N 皇后问题
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12.5. 小结
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13. 动态规划
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13.1. 初探动态规划（ ）
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13.2. DP 问题特性（ ）
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目录
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14. 附录
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14.1. 编程环境安装
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14.2. 一起参与创作
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目录
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13.2.1. 最优子结构
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13.2.2. 无后效性
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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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< h1 id = "132" > 13.2. 动态规划问题特性< a class = "headerlink" href = "#132" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 在上节中，我们学习了动态规划问题的暴力解法，从递归树中观察到海量的重叠子问题，以及了解到动态规划是如何通过记录解来优化时间复杂度的。< / p >
< p > 实际上，动态规划最常用来求解最优方案问题，例如寻找最短路径、最大利润、最少时间等。< strong > 这类问题不仅包含重叠子问题，往往还具有另外两大特性：最优子结构、无后效性< / strong > 。< / p >
< h2 id = "1321" > 13.2.1. 最优子结构< a class = "headerlink" href = "#1321" title = "Permanent link" > ¶ < / a > < / h2 >
< 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 >
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< 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 >
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< p > 如下图所示，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 > < img alt = "爬到第 3 阶的最小代价" src = "../dp_problem_features.assets/min_cost_cs_example.png" / > < / p >
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< p align = "center" > Fig. 爬到第 3 阶的最小代价 < / p >
< 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，即：< / p >
< div class = "arithmatex" > \[
dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
\]< / div >
< 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 > 。对于本题，我们从两个子问题最优解 < span class = "arithmatex" > \(dp[i-1]\)< / span > , < 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 > 相较于分治问题，动态规划问题的解也是由其子问题的解构成的。不同的是，< strong > 动态规划中子问题的解不仅揭示了问题的局部最优解，而且还通过特定的递推关系链接起来，共同构建出原问题的全局最优解< / strong > 。< / 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？它要求解的是方案数量，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，最优子结构的是一个比较宽泛的概念，在不同问题中会有不同的含义。< / p >
< p > 根据以上状态转移方程，以及初始状态 < span class = "arithmatex" > \(dp[1] = cost[1]\)< / span > , < span class = "arithmatex" > \(dp[2] = cost[2]\)< / span > ，我们可以得出动态规划解题代码。< / p >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "na" > length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
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< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
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< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > []< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "na" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-14" name = "__codelineno-0-14" href = "#__codelineno-0-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-15" name = "__codelineno-0-15" href = "#__codelineno-0-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-16" name = "__codelineno-0-16" href = "#__codelineno-0-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > size< / span > < span class = "p" > ()< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
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< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
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< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-1-8" name = "__codelineno-1-8" href = "#__codelineno-1-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-1-9" name = "__codelineno-1-9" href = "#__codelineno-1-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-10" name = "__codelineno-1-10" href = "#__codelineno-1-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-11" name = "__codelineno-1-11" href = "#__codelineno-1-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-1-12" name = "__codelineno-1-12" href = "#__codelineno-1-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-13" name = "__codelineno-1-13" href = "#__codelineno-1-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-14" name = "__codelineno-1-14" href = "#__codelineno-1-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-15" name = "__codelineno-1-15" href = "#__codelineno-1-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-16" name = "__codelineno-1-16" href = "#__codelineno-1-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_cost_climbing_stairs_dp< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ])< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 爬楼梯最小代价：动态规划" " " < / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "n" > n< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 1< / span > < span class = "ow" > or< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "k" > return< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "c1" > # 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-8" name = "__codelineno-2-8" href = "#__codelineno-2-8" > < / a > < span class = "c1" > # 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-2-9" name = "__codelineno-2-9" href = "#__codelineno-2-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-2-10" name = "__codelineno-2-10" href = "#__codelineno-2-10" > < / a > < span class = "c1" > # 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-2-11" name = "__codelineno-2-11" href = "#__codelineno-2-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-2-12" name = "__codelineno-2-12" href = "#__codelineno-2-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ])< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-2-13" name = "__codelineno-2-13" href = "#__codelineno-2-13" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-07-03 16:55:46 +08:00
< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：动态规划 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-7-6" name = "__codelineno-7-6" href = "#__codelineno-7-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-03 16:55:46 +08:00
< a id = "__codelineno-7-7" name = "__codelineno-7-7" href = "#__codelineno-7-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-8" name = "__codelineno-7-8" href = "#__codelineno-7-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-7-9" name = "__codelineno-7-9" href = "#__codelineno-7-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-10" name = "__codelineno-7-10" href = "#__codelineno-7-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-11" name = "__codelineno-7-11" href = "#__codelineno-7-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-7-12" name = "__codelineno-7-12" href = "#__codelineno-7-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-13" name = "__codelineno-7-13" href = "#__codelineno-7-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-14" name = "__codelineno-7-14" href = "#__codelineno-7-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
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< a id = "__codelineno-7-16" name = "__codelineno-7-16" href = "#__codelineno-7-16" > < / a > < span class = "p" > }< / span >
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< p > < 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" / > < / p >
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< p align = "center" > Fig. 爬楼梯最小代价的动态规划过程 < / p >
< p > 这道题同样也可以进行状态压缩，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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< a id = "__codelineno-13-2" name = "__codelineno-13-2" href = "#__codelineno-13-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 爬楼梯最小代价：状态压缩后的动态规划" " " < / span >
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< a id = "__codelineno-13-8" name = "__codelineno-13-8" href = "#__codelineno-13-8" > < / a > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "o" > =< / span > < span class = "n" > b< / span > < span class = "p" > ,< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-13-9" name = "__codelineno-13-9" href = "#__codelineno-13-9" > < / a > < span class = "k" > return< / span > < span class = "n" > b< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "cm" > /* 爬楼梯最小代价：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-18-2" name = "__codelineno-18-2" href = "#__codelineno-18-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minCostClimbingStairsDPComp< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-3" name = "__codelineno-18-3" href = "#__codelineno-18-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-18-4" name = "__codelineno-18-4" href = "#__codelineno-18-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span >
< a id = "__codelineno-18-5" name = "__codelineno-18-5" href = "#__codelineno-18-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-6" name = "__codelineno-18-6" href = "#__codelineno-18-6" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-7" name = "__codelineno-18-7" href = "#__codelineno-18-7" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-8" name = "__codelineno-18-8" href = "#__codelineno-18-8" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-18-9" name = "__codelineno-18-9" href = "#__codelineno-18-9" > < / a > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > cost< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-10" name = "__codelineno-18-10" href = "#__codelineno-18-10" > < / a > < span class = "w" > < / span > < span class = "n" > a< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > tmp< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-18-11" name = "__codelineno-18-11" href = "#__codelineno-18-11" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-18-12" name = "__codelineno-18-12" href = "#__codelineno-18-12" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > b< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-18-13" name = "__codelineno-18-13" href = "#__codelineno-18-13" > < / a > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-19-1" name = "__codelineno-19-1" href = "#__codelineno-19-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-20-1" name = "__codelineno-20-1" href = "#__codelineno-20-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_cost_climbing_stairs_dp.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-21-1" name = "__codelineno-21-1" href = "#__codelineno-21-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minCostClimbingStairsDPComp< / span > < span class = "p" > }< / span >
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< h2 id = "1322" > 13.2.2. 无后效性< a class = "headerlink" href = "#1322" title = "Permanent link" > ¶ < / a > < / h2 >
< 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 >
< 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，我们无需考虑状态 < span class = "arithmatex" > \(i\)< / span > 之前的状态，即它们对状态 < span class = "arithmatex" > \(i\)< / span > 的未来没有影响。< / p >
< 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 >
< p > 例如，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 > < 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" / > < / p >
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< p align = "center" > Fig. 带约束爬到第 3 阶的方案数量 < / p >
< p > 在该问题中，< 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 > 。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。< / 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，我们不能将 < span class = "arithmatex" > \(dp[i-1]\)< / span > 直接计入 < span class = "arithmatex" > \(dp[i]\)< / span > 中。< / p >
< p > 为了解决该问题，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，我们可以据此来决定下一步该怎么跳：< / p >
< ul >
< li > 当 < span class = "arithmatex" > \(j\)< / span > 等于 < span class = "arithmatex" > \(1\)< / span > ，即上一轮跳了 < span class = "arithmatex" > \(1\)< / span > 阶时，这一轮只能选择跳 < span class = "arithmatex" > \(2\)< / span > 阶；< / li >
< li > 当 < span class = "arithmatex" > \(j\)< / span > 等于 < span class = "arithmatex" > \(2\)< / span > ，即上一轮跳了 < span class = "arithmatex" > \(2\)< / span > 阶时，这一轮可选择跳 < span class = "arithmatex" > \(1\)< / span > 阶或跳 < span class = "arithmatex" > \(2\)< / span > 阶；< / li >
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< p > 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。由此，我们便能推导出以下的状态转移方程：< / p >
< 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 > < img alt = "考虑约束下的递推关系" src = "../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" / > < / p >
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< p align = "center" > Fig. 考虑约束下的递推关系 < / p >
< 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 >
< div class = "tabbed-set tabbed-alternate" data-tabs = "3:11" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_11" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Java< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Python< / label > < label for = "__tabbed_3_4" > Go< / label > < label for = "__tabbed_3_5" > JavaScript< / label > < label for = "__tabbed_3_6" > TypeScript< / label > < label for = "__tabbed_3_7" > C< / label > < label for = "__tabbed_3_8" > C#< / label > < label for = "__tabbed_3_9" > Swift< / label > < label for = "__tabbed_3_10" > Zig< / label > < label for = "__tabbed_3_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-22-2" name = "__codelineno-22-2" href = "#__codelineno-22-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-22-3" name = "__codelineno-22-3" href = "#__codelineno-22-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-22-4" name = "__codelineno-22-4" href = "#__codelineno-22-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-5" name = "__codelineno-22-5" href = "#__codelineno-22-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-22-6" name = "__codelineno-22-6" href = "#__codelineno-22-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-22-7" name = "__codelineno-22-7" href = "#__codelineno-22-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [][]< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 3< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-8" name = "__codelineno-22-8" href = "#__codelineno-22-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-22-9" name = "__codelineno-22-9" href = "#__codelineno-22-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-10" name = "__codelineno-22-10" href = "#__codelineno-22-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-11" name = "__codelineno-22-11" href = "#__codelineno-22-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-12" name = "__codelineno-22-12" href = "#__codelineno-22-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-13" name = "__codelineno-22-13" href = "#__codelineno-22-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-22-14" name = "__codelineno-22-14" href = "#__codelineno-22-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-22-15" name = "__codelineno-22-15" href = "#__codelineno-22-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-16" name = "__codelineno-22-16" href = "#__codelineno-22-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-17" name = "__codelineno-22-17" href = "#__codelineno-22-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-22-18" name = "__codelineno-22-18" href = "#__codelineno-22-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ][< / span > < span class = "mi" > 1< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "o" > [< / span > < span class = "n" > n< / span > < span class = "o" > ][< / span > < span class = "mi" > 2< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-22-19" name = "__codelineno-22-19" href = "#__codelineno-22-19" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-23-1" name = "__codelineno-23-1" href = "#__codelineno-23-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-23-2" name = "__codelineno-23-2" href = "#__codelineno-23-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-3" name = "__codelineno-23-3" href = "#__codelineno-23-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-4" name = "__codelineno-23-4" href = "#__codelineno-23-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-5" name = "__codelineno-23-5" href = "#__codelineno-23-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-23-6" name = "__codelineno-23-6" href = "#__codelineno-23-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-23-7" name = "__codelineno-23-7" href = "#__codelineno-23-7" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ));< / span >
< a id = "__codelineno-23-8" name = "__codelineno-23-8" href = "#__codelineno-23-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-23-9" name = "__codelineno-23-9" href = "#__codelineno-23-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-10" name = "__codelineno-23-10" href = "#__codelineno-23-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-11" name = "__codelineno-23-11" href = "#__codelineno-23-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-12" name = "__codelineno-23-12" href = "#__codelineno-23-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-23-13" name = "__codelineno-23-13" href = "#__codelineno-23-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-23-14" name = "__codelineno-23-14" href = "#__codelineno-23-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-15" name = "__codelineno-23-15" href = "#__codelineno-23-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-16" name = "__codelineno-23-16" href = "#__codelineno-23-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-17" name = "__codelineno-23-17" href = "#__codelineno-23-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-18" name = "__codelineno-23-18" href = "#__codelineno-23-18" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-19" name = "__codelineno-23-19" href = "#__codelineno-23-19" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > climbing_stairs_constraint_dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 带约束爬楼梯：动态规划" " " < / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / a > < span class = "k" > if< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 1< / span > < span class = "ow" > or< / span > < span class = "n" > n< / span > < span class = "o" > ==< / span > < span class = "mi" > 2< / span > < span class = "p" > :< / span >
< a id = "__codelineno-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "k" > return< / span > < span class = "n" > n< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / a > < span class = "c1" > # 初始化 dp 表，用于存储子问题的解< / span >
2023-07-01 22:39:28 +08:00
< a id = "__codelineno-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [[< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "mi" > 3< / span > < span class = "k" > for< / span > < span class = "n" > _< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > )]< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / a > < span class = "c1" > # 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-24-8" name = "__codelineno-24-8" href = "#__codelineno-24-8" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span >
< a id = "__codelineno-24-9" name = "__codelineno-24-9" href = "#__codelineno-24-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 1< / span >
< a id = "__codelineno-24-10" name = "__codelineno-24-10" href = "#__codelineno-24-10" > < / a > < span class = "c1" > # 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-24-11" name = "__codelineno-24-11" href = "#__codelineno-24-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "o" > +< / span > < span class = "mi" > 1< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-12" name = "__codelineno-24-12" href = "#__codelineno-24-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-13" name = "__codelineno-24-13" href = "#__codelineno-24-13" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 2< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-14" name = "__codelineno-24-14" href = "#__codelineno-24-14" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ][< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-27-1" name = "__codelineno-27-1" href = "#__codelineno-27-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > climbingStairsConstraintDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
2023-07-03 16:55:46 +08:00
< div class = "highlight" > < span class = "filename" > climbing_stairs_constraint_dp.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a > < span class = "cm" > /* 带约束爬楼梯：动态规划 */< / span >
< a id = "__codelineno-29-2" name = "__codelineno-29-2" href = "#__codelineno-29-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > climbingStairsConstraintDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-3" name = "__codelineno-29-3" href = "#__codelineno-29-3" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-4" name = "__codelineno-29-4" href = "#__codelineno-29-4" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-5" name = "__codelineno-29-5" href = "#__codelineno-29-5" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
2023-07-09 03:02:15 +08:00
< a id = "__codelineno-29-6" name = "__codelineno-29-6" href = "#__codelineno-29-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表，用于存储子问题的解< / span >
2023-07-03 16:55:46 +08:00
< a id = "__codelineno-29-7" name = "__codelineno-29-7" href = "#__codelineno-29-7" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [,]< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-8" name = "__codelineno-29-8" href = "#__codelineno-29-8" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始状态：预设最小子问题的解< / span >
< a id = "__codelineno-29-9" name = "__codelineno-29-9" href = "#__codelineno-29-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-10" name = "__codelineno-29-10" href = "#__codelineno-29-10" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-11" name = "__codelineno-29-11" href = "#__codelineno-29-11" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-12" name = "__codelineno-29-12" href = "#__codelineno-29-12" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-13" name = "__codelineno-29-13" href = "#__codelineno-29-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：从较小子问题逐步求解较大子问题< / span >
< a id = "__codelineno-29-14" name = "__codelineno-29-14" href = "#__codelineno-29-14" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 3< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < =< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-15" name = "__codelineno-29-15" href = "#__codelineno-29-15" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-16" name = "__codelineno-29-16" href = "#__codelineno-29-16" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 2< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-17" name = "__codelineno-29-17" href = "#__codelineno-29-17" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
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< a id = "__codelineno-29-19" name = "__codelineno-29-19" href = "#__codelineno-29-19" > < / a > < span class = "p" > }< / span >
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< / code > < / pre > < / div >
< / div >
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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，使得问题恢复无后效性。然而，许多问题具有非常严重的“有后效性”，例如：< / p >
< div class = "admonition question" >
< p class = "admonition-title" > 爬楼梯与障碍生成< / 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 > 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" > \(2, 3\)< / span > 阶上，则之后就不能跳到第 < span class = "arithmatex" > \(4, 6\)< / span > 阶上。请问有多少种方案可以爬到楼顶。< / p >
< / div >
< 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，或是因为计算复杂度过高而难以应用。< / p >
< p > 实际上，许多组合优化问题（例如著名的旅行商问题）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，从而降低时间复杂度，在有限时间内得到能够接受的局部最优解。< / p >
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