From 3f03663d2eb68e21b28df71611c7e6a4650f0475 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Fri, 30 Jun 2023 04:31:43 +0800 Subject: [PATCH] feat: add the section of the introduction to dynamic programming (#571) * add the section of the introduction to dynamic programming * add a code comments. --- codes/cpp/CMakeLists.txt | 1 + .../CMakeLists.txt | 5 + .../climbing_stairs_backtrack.cpp | 43 ++ .../climbing_stairs_dfs.cpp | 32 + .../climbing_stairs_dfs_mem.cpp | 39 ++ .../climbing_stairs_dp.cpp | 49 ++ .../min_cost_climbing_stairs_dp.cpp | 53 ++ .../climbing_stairs_backtrack.java | 44 ++ .../climbing_stairs_dfs.java | 31 + .../climbing_stairs_dfs_mem.java | 41 ++ .../climbing_stairs_dp.java | 48 ++ .../min_cost_climbing_stairs_dp.java | 53 ++ .../climbing_stairs_backtrack.py | 37 + .../climbing_stairs_dfs.py | 28 + .../climbing_stairs_dfs_mem.py | 35 + .../climbing_stairs_dp.py | 40 ++ .../min_cost_climbing_stairs_dp.py | 43 ++ .../space_complexity.md | 2 +- docs/chapter_dynamic_programming/index.md | 7 + .../climbing_stairs_dfs_memo_tree.png | Bin 0 -> 82404 bytes .../climbing_stairs_dfs_tree.png | Bin 0 -> 75442 bytes .../climbing_stairs_dp.png | Bin 0 -> 60680 bytes .../climbing_stairs_example.png | Bin 0 -> 60568 bytes .../climbing_stairs_state_transfer.png | Bin 0 -> 62905 bytes .../min_cost_cs_dp.png | Bin 0 -> 68335 bytes .../min_cost_cs_example.png | Bin 0 -> 70676 bytes .../intro_to_dynamic_programming.md | 632 ++++++++++++++++++ mkdocs.yml | 9 +- 28 files changed, 1268 insertions(+), 4 deletions(-) create mode 100644 codes/cpp/chapter_dynamic_programming/CMakeLists.txt create mode 100644 codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp create mode 100644 codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs.cpp create mode 100644 codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs_mem.cpp create mode 100644 codes/cpp/chapter_dynamic_programming/climbing_stairs_dp.cpp create mode 100644 codes/cpp/chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp create mode 100644 codes/java/chapter_dynamic_programming/climbing_stairs_backtrack.java create mode 100644 codes/java/chapter_dynamic_programming/climbing_stairs_dfs.java create mode 100644 codes/java/chapter_dynamic_programming/climbing_stairs_dfs_mem.java create mode 100644 codes/java/chapter_dynamic_programming/climbing_stairs_dp.java create mode 100644 codes/java/chapter_dynamic_programming/min_cost_climbing_stairs_dp.java create mode 100644 codes/python/chapter_dynamic_programming/climbing_stairs_backtrack.py create mode 100644 codes/python/chapter_dynamic_programming/climbing_stairs_dfs.py create mode 100644 codes/python/chapter_dynamic_programming/climbing_stairs_dfs_mem.py create mode 100644 codes/python/chapter_dynamic_programming/climbing_stairs_dp.py create mode 100644 codes/python/chapter_dynamic_programming/min_cost_climbing_stairs_dp.py create mode 100644 docs/chapter_dynamic_programming/index.md create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_dp.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_example.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/min_cost_cs_dp.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.assets/min_cost_cs_example.png create mode 100644 docs/chapter_dynamic_programming/intro_to_dynamic_programming.md diff --git a/codes/cpp/CMakeLists.txt b/codes/cpp/CMakeLists.txt index ad0e2f749..f3d16ae1a 100644 --- a/codes/cpp/CMakeLists.txt +++ b/codes/cpp/CMakeLists.txt @@ -15,3 +15,4 @@ add_subdirectory(chapter_graph) add_subdirectory(chapter_sorting) add_subdirectory(chapter_searching) add_subdirectory(chapter_backtracking) +add_subdirectory(chapter_dynamic_programming) diff --git a/codes/cpp/chapter_dynamic_programming/CMakeLists.txt b/codes/cpp/chapter_dynamic_programming/CMakeLists.txt new file mode 100644 index 000000000..ff0a7abdb --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/CMakeLists.txt @@ -0,0 +1,5 @@ +add_executable(climbing_stairs_backtrack climbing_stairs_backtrack.cpp) +add_executable(climbing_stairs_dfs climbing_stairs_dfs.cpp) +add_executable(climbing_stairs_dfs_mem climbing_stairs_dfs_mem.cpp) +add_executable(climbing_stairs_dp climbing_stairs_dp.cpp) +add_executable(min_cost_climbing_stairs_dp min_cost_climbing_stairs_dp.cpp) \ No newline at end of file diff --git a/codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp b/codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp new file mode 100644 index 000000000..167065109 --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp @@ -0,0 +1,43 @@ + +/** + * File: climbing_stairs_backtrack.cpp + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 回溯 */ +void backtrack(vector &choices, int state, int n, vector &res) { + // 当爬到第 n 阶时,方案数量加 1 + if (state == n) + res[0]++; + // 遍历所有选择 + for (auto &choice : choices) { + // 剪枝:不允许越过第 n 阶 + if (state + choice > n) + break; + // 尝试:做出选择,更新状态 + backtrack(choices, state + choice, n, res); + // 回退 + } +} + +/* 爬楼梯:回溯 */ +int climbingStairsBacktrack(int n) { + vector choices = {1, 2}; // 可选择向上爬 1 或 2 阶 + int state = 0; // 从第 0 阶开始爬 + vector res = {0}; // 使用 res[0] 记录方案数量 + backtrack(choices, state, n, res); + return res[0]; +} + +/* Driver Code */ +int main() { + int n = 9; + + int res = climbingStairsBacktrack(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + return 0; +} diff --git a/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs.cpp b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs.cpp new file mode 100644 index 000000000..0a1e062ee --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs.cpp @@ -0,0 +1,32 @@ +/** + * File: climbing_stairs_dfs.cpp + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 搜索 */ +int dfs(int i) { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 || i == 2) + return i; + // dp[i] = dp[i-1] + dp[i-2] + int count = dfs(i - 1) + dfs(i - 2); + return count; +} + +/* 爬楼梯:搜索 */ +int climbingStairsDFS(int n) { + return dfs(n); +} + +/* Driver Code */ +int main() { + int n = 9; + + int res = climbingStairsDFS(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + return 0; +} diff --git a/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs_mem.cpp b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs_mem.cpp new file mode 100644 index 000000000..3e7756036 --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dfs_mem.cpp @@ -0,0 +1,39 @@ +/** + * File: climbing_stairs_dfs_mem.cpp + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 记忆化搜索 */ +int dfs(int i, vector &mem) { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 || i == 2) + return i; + // 若存在记录 dp[i] ,则直接返回之 + if (mem[i] != -1) + return mem[i]; + // dp[i] = dp[i-1] + dp[i-2] + int count = dfs(i - 1, mem) + dfs(i - 2, mem); + // 记录 dp[i] + mem[i] = count; + return count; +} + +/* 爬楼梯:记忆化搜索 */ +int climbingStairsDFSMem(int n) { + // mem[i] 记录爬到第 i 阶的方案总数,-1 代表无记录 + vector mem(n + 1, -1); + return dfs(n, mem); +} + +/* Driver Code */ +int main() { + int n = 9; + + int res = climbingStairsDFSMem(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + return 0; +} diff --git a/codes/cpp/chapter_dynamic_programming/climbing_stairs_dp.cpp b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dp.cpp new file mode 100644 index 000000000..daa2f5b40 --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/climbing_stairs_dp.cpp @@ -0,0 +1,49 @@ +/** + * File: climbing_stairs_dp.cpp + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 爬楼梯:动态规划 */ +int climbingStairsDP(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); + // 初始状态:预设最小子问题的解 + dp[1] = 1; + dp[2] = 2; + // 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] = dp[i - 1] + dp[i - 2]; + } + return dp[n]; +} + +/* 爬楼梯:状态压缩后的动态规划 */ +int climbingStairsDPComp(int n) { + if (n == 1 || n == 2) + return n; + int a = 1, b = 2; + for (int i = 3; i <= n; i++) { + int tmp = b; + b = a + b; + a = tmp; + } + return b; +} + +/* Driver Code */ +int main() { + int n = 9; + + int res = climbingStairsDP(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + res = climbingStairsDPComp(n); + cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl; + + return 0; +} diff --git a/codes/cpp/chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp b/codes/cpp/chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp new file mode 100644 index 000000000..f24a7626a --- /dev/null +++ b/codes/cpp/chapter_dynamic_programming/min_cost_climbing_stairs_dp.cpp @@ -0,0 +1,53 @@ +/** + * File: min_cost_climbing_stairs_dp.cpp + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 爬楼梯最小代价:动态规划 */ +int minCostClimbingStairsDP(vector &cost) { + int n = cost.size() - 1; + if (n == 1 || n == 2) + return cost[n]; + // 初始化 dp 列表,用于存储子问题的解 + vector dp(n + 1); + // 初始状态:预设最小子问题的解 + dp[1] = cost[1]; + dp[2] = cost[2]; + // 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] = min(dp[i - 1], dp[i - 2]) + cost[i]; + } + return dp[n]; +} + +/* 爬楼梯最小代价:状态压缩后的动态规划 */ +int minCostClimbingStairsDPComp(vector &cost) { + int n = cost.size() - 1; + if (n == 1 || n == 2) + return cost[n]; + int a = cost[1], b = cost[2]; + for (int i = 3; i <= n; i++) { + int tmp = b; + b = min(a, tmp) + cost[i]; + a = tmp; + } + return b; +} + +/* Driver Code */ +int main() { + vector cost = {0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1}; + cout << "输入楼梯的代价列表为 "; + printVector(cost); + + int res = minCostClimbingStairsDP(cost); + cout << "爬完楼梯的最低代价为 " << res << endl; + + res = minCostClimbingStairsDPComp(cost); + cout << "爬完楼梯的最低代价为 " << res << endl; + + return 0; +} diff --git a/codes/java/chapter_dynamic_programming/climbing_stairs_backtrack.java b/codes/java/chapter_dynamic_programming/climbing_stairs_backtrack.java new file mode 100644 index 000000000..fa302e25e --- /dev/null +++ b/codes/java/chapter_dynamic_programming/climbing_stairs_backtrack.java @@ -0,0 +1,44 @@ +/** + * File: climbing_stairs_backtrack.java + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +import java.util.*; + +public class climbing_stairs_backtrack { + /* 回溯 */ + public static void backtrack(List choices, int state, int n, List res) { + // 当爬到第 n 阶时,方案数量加 1 + if (state == n) + res.set(0, res.get(0) + 1); + // 遍历所有选择 + for (Integer choice : choices) { + // 剪枝:不允许越过第 n 阶 + if (state + choice > n) + break; + // 尝试:做出选择,更新状态 + backtrack(choices, state + choice, n, res); + // 回退 + } + } + + /* 爬楼梯:回溯 */ + public static int climbingStairsBacktrack(int n) { + List choices = Arrays.asList(1, 2); // 可选择向上爬 1 或 2 阶 + int state = 0; // 从第 0 阶开始爬 + List res = new ArrayList<>(); + res.add(0); // 使用 res[0] 记录方案数量 + backtrack(choices, state, n, res); + return res.get(0); + } + + public static void main(String[] args) { + int n = 9; + + int res = climbingStairsBacktrack(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + } +} diff --git a/codes/java/chapter_dynamic_programming/climbing_stairs_dfs.java b/codes/java/chapter_dynamic_programming/climbing_stairs_dfs.java new file mode 100644 index 000000000..d634c8cb5 --- /dev/null +++ b/codes/java/chapter_dynamic_programming/climbing_stairs_dfs.java @@ -0,0 +1,31 @@ +/** + * File: climbing_stairs_dfs.java + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +public class climbing_stairs_dfs { + /* 搜索 */ + public static int dfs(int i) { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 || i == 2) + return i; + // dp[i] = dp[i-1] + dp[i-2] + int count = dfs(i - 1) + dfs(i - 2); + return count; + } + + /* 爬楼梯:搜索 */ + public static int climbingStairsDFS(int n) { + return dfs(n); + } + + public static void main(String[] args) { + int n = 9; + + int res = climbingStairsDFS(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + } +} diff --git a/codes/java/chapter_dynamic_programming/climbing_stairs_dfs_mem.java b/codes/java/chapter_dynamic_programming/climbing_stairs_dfs_mem.java new file mode 100644 index 000000000..7c2dae81d --- /dev/null +++ b/codes/java/chapter_dynamic_programming/climbing_stairs_dfs_mem.java @@ -0,0 +1,41 @@ +/** + * File: climbing_stairs_dfs_mem.java + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +import java.util.Arrays; + +public class climbing_stairs_dfs_mem { + /* 记忆化搜索 */ + public static int dfs(int i, int[] mem) { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 || i == 2) + return i; + // 若存在记录 dp[i] ,则直接返回之 + if (mem[i] != -1) + return mem[i]; + // dp[i] = dp[i-1] + dp[i-2] + int count = dfs(i - 1, mem) + dfs(i - 2, mem); + // 记录 dp[i] + mem[i] = count; + return count; + } + + /* 爬楼梯:记忆化搜索 */ + public static int climbingStairsDFSMem(int n) { + // mem[i] 记录爬到第 i 阶的方案总数,-1 代表无记录 + int[] mem = new int[n + 1]; + Arrays.fill(mem, -1); + return dfs(n, mem); + } + + public static void main(String[] args) { + int n = 9; + + int res = climbingStairsDFSMem(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + } +} \ No newline at end of file diff --git a/codes/java/chapter_dynamic_programming/climbing_stairs_dp.java b/codes/java/chapter_dynamic_programming/climbing_stairs_dp.java new file mode 100644 index 000000000..7249755a9 --- /dev/null +++ b/codes/java/chapter_dynamic_programming/climbing_stairs_dp.java @@ -0,0 +1,48 @@ +/** + * File: climbing_stairs_dp.java + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +public class climbing_stairs_dp { + /* 爬楼梯:动态规划 */ + public static int climbingStairsDP(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]; + // 初始状态:预设最小子问题的解 + dp[1] = 1; + dp[2] = 2; + // 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] = dp[i - 1] + dp[i - 2]; + } + return dp[n]; + } + + /* 爬楼梯:状态压缩后的动态规划 */ + public static int climbingStairsDPComp(int n) { + if (n == 1 || n == 2) + return n; + int a = 1, b = 2; + for (int i = 3; i <= n; i++) { + int tmp = b; + b = a + b; + a = tmp; + } + return b; + } + + public static void main(String[] args) { + int n = 9; + + int res = climbingStairsDP(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + + res = climbingStairsDPComp(n); + System.out.println(String.format("爬 %d 阶楼梯共有 %d 种方案", n, res)); + } +} diff --git a/codes/java/chapter_dynamic_programming/min_cost_climbing_stairs_dp.java b/codes/java/chapter_dynamic_programming/min_cost_climbing_stairs_dp.java new file mode 100644 index 000000000..ec6297e9f --- /dev/null +++ b/codes/java/chapter_dynamic_programming/min_cost_climbing_stairs_dp.java @@ -0,0 +1,53 @@ +/** + * File: min_cost_climbing_stairs_dp.java + * Created Time: 2023-06-30 + * Author: Krahets (krahets@163.com) + */ + +package chapter_dynamic_programming; + +import java.util.Arrays; + +public class min_cost_climbing_stairs_dp { + /* 爬楼梯最小代价:动态规划 */ + public static int minCostClimbingStairsDP(int[] cost) { + int n = cost.length - 1; + if (n == 1 || n == 2) + return cost[n]; + // 初始化 dp 列表,用于存储子问题的解 + int[] dp = new int[n + 1]; + // 初始状态:预设最小子问题的解 + dp[1] = cost[1]; + dp[2] = cost[2]; + // 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] = Math.min(dp[i - 1], dp[i - 2]) + cost[i]; + } + return dp[n]; + } + + /* 爬楼梯最小代价:状态压缩后的动态规划 */ + public static int minCostClimbingStairsDPComp(int[] cost) { + int n = cost.length - 1; + if (n == 1 || n == 2) + return cost[n]; + int a = cost[1], b = cost[2]; + for (int i = 3; i <= n; i++) { + int tmp = b; + b = Math.min(a, tmp) + cost[i]; + a = tmp; + } + return b; + } + + public static void main(String[] args) { + int[] cost = { 0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1 }; + System.out.println(String.format("输入楼梯的代价列表为 %s", Arrays.toString(cost))); + + int res = minCostClimbingStairsDP(cost); + System.out.println(String.format("爬完楼梯的最低代价为 %d", res)); + + res = minCostClimbingStairsDPComp(cost); + System.out.println(String.format("爬完楼梯的最低代价为 %d", res)); + } +} diff --git a/codes/python/chapter_dynamic_programming/climbing_stairs_backtrack.py b/codes/python/chapter_dynamic_programming/climbing_stairs_backtrack.py new file mode 100644 index 000000000..962df8514 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/climbing_stairs_backtrack.py @@ -0,0 +1,37 @@ +""" +File: climbing_stairs_backtrack.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int: + """回溯""" + # 当爬到第 n 阶时,方案数量加 1 + if state == n: + res[0] += 1 + # 遍历所有选择 + for choice in choices: + # 剪枝:不允许越过第 n 阶 + if state + choice > n: + break + # 尝试:做出选择,更新状态 + backtrack(choices, state + choice, n, res) + # 回退 + + +def climbing_stairs_backtrack(n: int) -> int: + """爬楼梯:回溯""" + choices = [1, 2] # 可选择向上爬 1 或 2 阶 + state = 0 # 从第 0 阶开始爬 + res = [0] # 使用 res[0] 记录方案数量 + backtrack(choices, state, n, res) + return res[0] + + +"""Driver Code""" +if __name__ == "__main__": + n = 9 + + res = climbing_stairs_backtrack(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") diff --git a/codes/python/chapter_dynamic_programming/climbing_stairs_dfs.py b/codes/python/chapter_dynamic_programming/climbing_stairs_dfs.py new file mode 100644 index 000000000..5e936990a --- /dev/null +++ b/codes/python/chapter_dynamic_programming/climbing_stairs_dfs.py @@ -0,0 +1,28 @@ +""" +File: climbing_stairs_dfs.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def dfs(i: int) -> int: + """搜索""" + # 已知 dp[1] 和 dp[2] ,返回之 + if i == 1 or i == 2: + return i + # dp[i] = dp[i-1] + dp[i-2] + count = dfs(i - 1) + dfs(i - 2) + return count + + +def climbing_stairs_dfs(n: int) -> int: + """爬楼梯:搜索""" + return dfs(n) + + +"""Driver Code""" +if __name__ == "__main__": + n = 9 + + res = climbing_stairs_dfs(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") diff --git a/codes/python/chapter_dynamic_programming/climbing_stairs_dfs_mem.py b/codes/python/chapter_dynamic_programming/climbing_stairs_dfs_mem.py new file mode 100644 index 000000000..6ee8836dd --- /dev/null +++ b/codes/python/chapter_dynamic_programming/climbing_stairs_dfs_mem.py @@ -0,0 +1,35 @@ +""" +File: climbing_stairs_dfs_mem.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def dfs(i: int, mem: list[int]) -> int: + """记忆化搜索""" + # 已知 dp[1] 和 dp[2] ,返回之 + if i == 1 or i == 2: + return i + # 若存在记录 dp[i] ,则直接返回之 + if mem[i] != -1: + return mem[i] + # dp[i] = dp[i-1] + dp[i-2] + count = dfs(i - 1, mem) + dfs(i - 2, mem) + # 记录 dp[i] + mem[i] = count + return count + + +def climbing_stairs_dfs_mem(n: int) -> int: + """爬楼梯:记忆化搜索""" + # mem[i] 记录爬到第 i 阶的方案总数,-1 代表无记录 + mem = [-1] * (n + 1) + return dfs(n, mem) + + +"""Driver Code""" +if __name__ == "__main__": + n = 9 + + res = climbing_stairs_dfs_mem(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") diff --git a/codes/python/chapter_dynamic_programming/climbing_stairs_dp.py b/codes/python/chapter_dynamic_programming/climbing_stairs_dp.py new file mode 100644 index 000000000..feb799f72 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/climbing_stairs_dp.py @@ -0,0 +1,40 @@ +""" +File: climbing_stairs_dp.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def climbing_stairs_dp(n: int) -> int: + """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] * (n + 1) + # 初始状态:预设最小子问题的解 + dp[1], dp[2] = 1, 2 + # 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] = dp[i - 1] + dp[i - 2] + return dp[n] + + +def climbing_stairs_dp_comp(n: int) -> int: + """爬楼梯:状态压缩后的动态规划""" + if n == 1 or n == 2: + return n + a, b = 1, 2 + for _ in range(3, n + 1): + a, b = b, a + b + return b + + +"""Driver Code""" +if __name__ == "__main__": + n = 9 + + res = climbing_stairs_dp(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") + + res = climbing_stairs_dp_comp(n) + print(f"爬 {n} 阶楼梯共有 {res} 种方案") diff --git a/codes/python/chapter_dynamic_programming/min_cost_climbing_stairs_dp.py b/codes/python/chapter_dynamic_programming/min_cost_climbing_stairs_dp.py new file mode 100644 index 000000000..020259bb4 --- /dev/null +++ b/codes/python/chapter_dynamic_programming/min_cost_climbing_stairs_dp.py @@ -0,0 +1,43 @@ +""" +File: min_cost_climbing_stairs_dp.py +Created Time: 2023-06-30 +Author: Krahets (krahets@163.com) +""" + + +def min_cost_climbing_stairs_dp(cost: list[int]) -> int: + """爬楼梯最小代价:动态规划""" + n = len(cost) - 1 + if n == 1 or n == 2: + return cost[n] + # 初始化 dp 列表,用于存储子问题的解 + dp = [0] * (n + 1) + # 初始状态:预设最小子问题的解 + dp[1], dp[2] = cost[1], cost[2] + # 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] = min(dp[i - 1], dp[i - 2]) + cost[i] + return dp[n] + + +def min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int: + """爬楼梯最小代价:状态压缩后的动态规划""" + n = len(cost) - 1 + if n == 1 or n == 2: + return cost[n] + a, b = cost[1], cost[2] + for i in range(3, n + 1): + a, b = b, min(a, b) + cost[i] + return b + + +"""Driver Code""" +if __name__ == "__main__": + cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1] + print(f"输入楼梯的代价列表为 {cost}") + + res = min_cost_climbing_stairs_dp(cost) + print(f"爬完楼梯的最低代价为 {res}") + + res = min_cost_climbing_stairs_dp_comp(cost) + print(f"爬完楼梯的最低代价为 {res}") diff --git a/docs/chapter_computational_complexity/space_complexity.md b/docs/chapter_computational_complexity/space_complexity.md index 6edc7e2a3..232a96517 100755 --- a/docs/chapter_computational_complexity/space_complexity.md +++ b/docs/chapter_computational_complexity/space_complexity.md @@ -318,7 +318,7 @@ ```python title="" def algorithm(n: int) -> None: - a = 0 # O(1) + a = 0 # O(1) b = [0] * 10000 # O(1) if n > 10: nums = [0] * n # O(n) diff --git a/docs/chapter_dynamic_programming/index.md b/docs/chapter_dynamic_programming/index.md new file mode 100644 index 000000000..cb24d496e --- /dev/null +++ b/docs/chapter_dynamic_programming/index.md @@ -0,0 +1,7 @@ +# 动态规划 + +
+ +![动态规划](../assets/covers/chapter_dynamic_programming.jpg){ width="70%" } + +
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