From 920512d87221b68876c682a169f801e120fca329 Mon Sep 17 00:00:00 2001 From: krahets Date: Sat, 1 Jul 2023 03:02:50 +0800 Subject: [PATCH] Update the section of intro to DP. --- .../climbing_stairs_constraint_dp.cpp | 37 +++++ .../climbing_stairs_constraint_dp.java | 36 +++++ .../climbing_stairs_constraint_dp.py | 29 ++++ .../climbing_stairs_constraint_example.png | Bin 0 -> 72034 bytes ...mbing_stairs_constraint_state_transfer.png | Bin 0 -> 64797 bytes .../climbing_stairs_state_transfer.png | Bin 62905 -> 62431 bytes .../intro_to_dynamic_programming.md | 147 ++++++++++++++++-- 7 files changed, 235 insertions(+), 14 deletions(-) create mode 100644 codes/cpp/chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp create mode 100644 codes/java/chapter_dynamic_programming/climbing_stairs_constraint_dp.java create mode 100644 codes/python/chapter_dynamic_programming/climbing_stairs_constraint_dp.py create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_constraint_example.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_constraint_state_transfer.png diff --git a/codes/cpp/chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp b/codes/cpp/chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp new file mode 100644 index 00000000..3b47667d --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/climbing_stairs_constraint_dp.cpp @@ -0,0 +1,37 @@ +/** + * File: climbing_stairs_constraint_dp.cpp + * Created Time: 2023-07-01 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 带约束爬楼梯:动态规划 */ +int climbingStairsConstraintDP(int n) { + if (n == 1 || n == 2) { + return n; + } + // 初始化 dp 列表,用于存储子问题的解 + vector> dp(n + 1, vector(3, 0)); + // 初始状态:预设最小子问题的解 + dp[1][1] = 1; + dp[1][2] = 0; + dp[2][1] = 0; + dp[2][2] = 1; + // 状态转移:从较小子问题逐步求解较大子问题 + for (int i = 3; i <= n; i++) { + dp[i][1] = dp[i - 1][2]; + dp[i][2] = dp[i - 2][1] + dp[i - 2][2]; + } + return dp[n][1] + dp[n][2]; +} + +/* Driver Code */ +int main() { + int n = 9; + + int res = climbingStairsConstraintDP(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + return 0; +} diff --git a/codes/java/chapter_dynamic_programming/climbing_stairs_constraint_dp.java b/codes/java/chapter_dynamic_programming/climbing_stairs_constraint_dp.java new file mode 100644 index 00000000..dafd51a0 --- /dev/null +++ b/codes/java/chapter_dynamic_programming/climbing_stairs_constraint_dp.java @@ -0,0 +1,36 @@ +/** + * File: climbing_stairs_constraint_dp.java + * Created Time: 2023-07-01 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +public class climbing_stairs_constraint_dp { + /* 带约束爬楼梯:动态规划 */ + static int climbingStairsConstraintDP(int n) { + if (n == 1 || n == 2) { + return n; + } + // 初始化 dp 列表,用于存储子问题的解 + int[][] dp = new int[n + 1][3]; + // 初始状态:预设最小子问题的解 + dp[1][1] = 1; + dp[1][2] = 0; + dp[2][1] = 0; + dp[2][2] = 1; + // 状态转移:从较小子问题逐步求解较大子问题 + for (int i = 3; i <= n; i++) { + dp[i][1] = dp[i - 1][2]; + dp[i][2] = dp[i - 2][1] + dp[i - 2][2]; + } + return dp[n][1] + dp[n][2]; + } + + public static void main(String[] args) { + int n = 9; + + int res = climbingStairsConstraintDP(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + } +} diff --git a/codes/python/chapter_dynamic_programming/climbing_stairs_constraint_dp.py b/codes/python/chapter_dynamic_programming/climbing_stairs_constraint_dp.py new file mode 100644 index 00000000..acbbfae6 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/climbing_stairs_constraint_dp.py @@ -0,0 +1,29 @@ +""" +File: climbing_stairs_constraint_dp.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def climbing_stairs_constraint_dp(n: int) -> int: + """带约束爬楼梯:动态规划""" + if n == 1 or n == 2: + return n + # 初始化 dp 列表,用于存储子问题的解 + dp = [[0] * 3 for _ in range(n + 1)] + # 初始状态:预设最小子问题的解 + dp[1][1], dp[1][2] = 1, 0 + dp[2][1], dp[2][2] = 0, 1 + # 状态转移:从较小子问题逐步求解较大子问题 + for i in range(3, n + 1): + dp[i][1] = dp[i - 1][2] + dp[i][2] = dp[i - 2][1] + dp[i - 2][2] + return dp[n][1] + dp[n][2] + + +"""Driver Code""" +if __name__ == "__main__": + n = 9 + + res = climbing_stairs_constraint_dp(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") diff --git a/docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_constraint_example.png b/docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_constraint_example.png new file mode 100644 index 0000000000000000000000000000000000000000..e73c32bbf86f8e629d48e1477fa476d71afad563 GIT binary patch literal 72034 zcmeFZWmr{P+XlK=E|l&L>5wjI>Fy5c2I&w`y1QFC6r@YKyChY*J0ztSa3=5G?)`py ze?QMTzdpUV)>P&ibBt#`dEd_%VM+>;$Or@oAP@*yT1rd>1cC)F!Q*gX;6sQ+1_t5*Ie_gA<(EfQ3 zFlLAq2=br%L7>;b=i?tk;OFr>e|$d4f%$vH9PsbEVL>@heqV#ffxAIIV#1!l2fTxn 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z-@rG2)|aiDITO6EKihZDbpCI1lO1uPuIhesF7SDFv3WC}W-ec0bu(A!)VSnJFUhJ}^){?>nZxe86;L=L+4XXy9q_w{M$TMYV5D=dGAsxXozC zCJFSs&%kKUqSWOjJj3cW@S^9UYZHNU_;csZoj32@@~EehO$^^6m2YAeGld=x{xdhG V3K(m)&k<$-0#8>zmvv4FO#nFfoh<+W diff --git a/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md b/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md index ff3b2615..8a7aa40c 100644 --- a/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md +++ b/docs/chapter_dynamic_programming/intro_to_dynamic_programming.md @@ -1,14 +1,16 @@ -# 初识动态规划 +# 初探动态规划 「动态规划 Dynamic Programming」是一种通过将复杂问题分解为更简单的子问题方式来求解问题的方法,通常用来求解最优方案的相关问题,例如寻找最短路径、最大利润、最少时间等。 -然而,并非所有的最优化问题都适合用动态规划来解决。**只有当问题具有重叠子问题和最优子结构时,动态规划才能发挥出其优势**。 +然而,并非所有的最优化问题都适合用动态规划来解决。**只有当问题具有重叠子问题、最优子结构、无后效性时,动态规划才能发挥出其优势**。 -在本节,我们先从两个经典例题入手,总览动态规划的主要特征,包括: +在本节,我们先从几个经典例题入手,总览动态规划的主要特征,包括: -1. 如何使用回溯算法(穷举)来求解动态规划问题。重叠子问题是什么,以及如何解决由它带来的时间复杂度过高的问题。 -2. 最优子结构的定义,以及它在动态规划问题中的表现形式。 -3. 动态规划中的主要术语,状态压缩的含义与实现方式。 +1. 如何使用回溯来暴力求解动态规划问题,其中为什么包含重叠子问题。 +2. 动态规划是如何通过引入“记忆化”来优化时间复杂度的,并给出从顶至底和从底至顶两种解法。 +3. 动态规划的常用术语,状态压缩的实现方式。 +4. 最优子结构在动态规划问题中的表现形式,动态规划与分治的区别是什么。 +5. 无后效性的含义,其对动态规划的意义是什么。 ## 重叠子问题 @@ -112,12 +114,9 @@ [class]{}-[func]{climbingStairsBacktrack} ``` -### 方法一:搜索 +### 方法一:暴力搜索 -然而,这道题并不是典型的回溯问题,而更适合从分治的角度进行解析: - -- 在分治算法中,原问题被分解为较小的子问题,通过组合子问题的解得到原问题的解。例如,归并排序将一个长数组从顶至底地划分为两个短数组,再从底至顶地将已排序的短数组进行排序。 -- 在动态规划中,原问题的解往往依赖于其子问题的解。这些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。 +然而,这道题并不是典型的回溯问题,而更适合从分治的角度进行解析:在分治算法中,原问题被分解为较小的子问题,通过组合子问题的解得到原问题的解。例如,归并排序将一个长数组从顶至底地划分为两个短数组,再从底至顶地将已排序的短数组进行排序。 对于本题,设爬到第 $i$ 阶共有 $dp[i]$ 种方案,那么 $dp[i]$ 就是原问题,其子问题包括 $dp[i-1]$ , $dp[i-2]$ , $\cdots$ , $dp[2]$ , $dp[1]$ 。 @@ -397,11 +396,15 @@ $$ [class]{}-[func]{climbingStairsDP} ``` +与回溯算法一样,动态规划也使用“状态”概念来表示问题求解的某个特定阶段,每个状态都对应一个子问题以及相应的局部最优解。例如对于爬楼梯问题,状态定义为当前所在楼梯阶数。动态规划的常用术语包括: + +- 将 $dp$ 数组称为「状态列表」,索引与状态逐个对应,每个元素对应一个子问题的解; +- 将最简单子问题对应的状态(即第 $1$ , $2$ 阶楼梯)称为「初始状态」; +- 将递推公式 $dp[i] = dp[i-1] + dp[i-2]$ 称为「状态转移方程」; + ![爬楼梯的动态规划过程](intro_to_dynamic_programming.assets/climbing_stairs_dp.png) -在动态规划中,我们通常将 $dp$ 数组称为「状态列表」,将最小子问题对应的状态(即 $dp[1]$ , $dp[2]$ )称为「初始状态」,将 $dp[i] = dp[i-1] + dp[i-2]$ 称为「状态转移方程」。这些名词出现频率很高,请你务必理解并记住。 - -细心的你可能发现,由于 $dp[i]$ 只与 $dp[i-1]$ 和 $dp[i-2]$ 有关,因此我们无需使用一个数组 `dp` 来存储所有状态,只需两个变量滚动前进即可。如以下代码所示,由于省去了数组 `dp` 占用的空间,因此空间复杂度从 $O(n)$ 降低至 $O(1)$ 。 +细心的你可能发现,由于 $dp[i]$ 只与 $dp[i-1]$ 和 $dp[i-2]$ 有关,因此我们无需使用一个数组 `dp` 来存储所有状态,而只需两个变量滚动前进即可。如以下代码所示,由于省去了数组 `dp` 占用的空间,因此空间复杂度从 $O(n)$ 降低至 $O(1)$ 。 === "Java" @@ -491,6 +494,8 @@ $$ 这便可以引出「最优子结构」的含义:**原问题的最优解是从子问题的最优解构建得来的**。对于本题,我们从两个子问题最优解 $dp[i-1]$ , $dp[i-2]$ 中挑选出较优的那一个,并用它构建出原问题 $dp[i]$ 的最优解。 +相较于分治算法问题,动态规划问题的解也是由其子问题的解构成的。不同的是,**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**。 + 那么,上道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。我们惊喜地发现,**虽然题目修改前后是等价的,但最优子结构浮现出来了**:第 $n$ 阶最大方案数量等于第 $n-1$ 阶和第 $n-2$ 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。 根据以上状态转移方程,以及初始状态 $dp[1] = cost[1]$ , $dp[2] = cost[2]$ ,我们可以得出动态规划解题代码。 @@ -630,3 +635,117 @@ $$ ```dart title="min_cost_climbing_stairs_dp.dart" [class]{}-[func]{minCostClimbingStairsDPComp} ``` + +## 无后效性 + +除了重叠子问题和最优子结构以外,「无后效性」也是动态规划能够有效解决问题的重要特性之一。我们先来看下无后效性定义:**给定一个确定的状态,它的未来发展只与当前状态有关,而与当前状态过去所经历过的所有状态无关**。 + +以爬楼梯问题为例,给定状态 $i$ ,它会发展出状态 $i+1$ 和状态 $i+2$ ,分别对应跳 $1$ 步和跳 $2$ 步。在做出这两种选择时,我们无需考虑状态 $i$ 之前的状态,即它们对状态 $i$ 的未来没有影响。 + +然而,如果我们向爬楼梯问题添加一个约束,情况就不一样了。 + +!!! question "带约束爬楼梯" + + 给定一个共有 $n$ 阶的楼梯,你每步可以上 $1$ 阶或者 $2$ 阶,**但不能连续两轮跳 $1$ 阶**,请问有多少种方案可以爬到楼顶。 + +例如,爬上第 $3$ 阶仅剩 $2$ 种可行方案,其中连续三次跳 $1$ 阶的方案不满足约束条件,因此被舍弃。 + +![带约束爬到第 3 阶的方案数量](intro_to_dynamic_programming.assets/climbing_stairs_constraint_example.png) + +在该问题中,**下一步选择不能由当前状态(当前楼梯阶数)独立决定,还和前一个状态(上轮楼梯阶数)有关**。如果上一轮是跳 $1$ 阶上来的,那么下一轮就必须跳 $2$ 阶。 + +不难发现,此问题已不满足无后效性,状态转移方程 $dp[i] = dp[i-1] + dp[i-2]$ 也随之失效,因为 $dp[i-1]$ 代表本轮跳 $1$ 阶,但其中包含了许多“上一轮跳 $1$ 阶上来的”方案,而为了满足约束,我们不能将 $dp[i-1]$ 直接计入 $dp[i]$ 中。 + +为了解决该问题,我们需要扩展状态定义:**状态 $[i, j]$ 表示处在第 $i$ 阶、并且上一轮跳了 $j$ 阶**,$dp[i, j]$ 表示该状态下的方案数量。此状态定义有效地区分了上一轮跳了 $1$ 阶还是 $2$ 阶,我们可以据此来决定下一步该怎么跳: + +- 当 $j$ 等于 $1$ ,即上一轮跳了 $1$ 阶时,这一轮只可选择跳 $2$ 阶; +- 当 $j$ 等于 $2$ ,即上一轮跳了 $2$ 阶时,这一步可选择跳 $1$ 阶或跳 $2$ 阶; + +![考虑约束下的递推关系](intro_to_dynamic_programming.assets/climbing_stairs_constraint_state_transfer.png) + +由此,我们便能推导出以下的状态转移方程: + +$$ +\begin{cases} +dp[i][1] = dp[i-1][2] \\ +dp[i][2] = dp[i-2][1] + dp[i-2][2] +\end{cases} +$$ + +最终,返回 $dp[n][1] + dp[n][2]$ 即可,两者之和代表爬到第 $n$ 阶的方案总数。 + +=== "Java" + + ```java title="climbing_stairs_constraint_dp.java" + [class]{climbing_stairs_constraint_dp}-[func]{climbingStairsConstraintDP} + ``` + +=== "C++" + + ```cpp title="climbing_stairs_constraint_dp.cpp" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "Python" + + ```python title="climbing_stairs_constraint_dp.py" + [class]{}-[func]{climbing_stairs_constraint_dp} + ``` + +=== "Go" + + ```go title="climbing_stairs_constraint_dp.go" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "JavaScript" + + ```javascript title="climbing_stairs_constraint_dp.js" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "TypeScript" + + ```typescript title="climbing_stairs_constraint_dp.ts" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "C" + + ```c title="climbing_stairs_constraint_dp.c" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "C#" + + ```csharp title="climbing_stairs_constraint_dp.cs" + [class]{climbing_stairs_constraint_dp}-[func]{climbingStairsConstraintDP} + ``` + +=== "Swift" + + ```swift title="climbing_stairs_constraint_dp.swift" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "Zig" + + ```zig title="climbing_stairs_constraint_dp.zig" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +=== "Dart" + + ```dart title="climbing_stairs_constraint_dp.dart" + [class]{}-[func]{climbingStairsConstraintDP} + ``` + +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: + +!!! question "爬楼梯与障碍生成" + + 给定一个共有 $n$ 阶的楼梯,你每步可以上 $1$ 阶或者 $2$ 阶。**规定当爬到第 $i$ 阶时,系统自动会给第 $2i$ 阶上放上障碍物,之后所有轮都不允许跳到第 $2i$ 阶上**。例如,前两轮分别跳到了第 $2, 3$ 阶上,则之后就不能跳到第 $4, 6$ 阶上。请问有多少种方案可以爬到楼顶。 + +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。 + +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。