diff --git a/codes/dart/chapter_dynamic_programming/climbing_stairs_backtrack.dart b/codes/dart/chapter_dynamic_programming/climbing_stairs_backtrack.dart new file mode 100644 index 00000000..5bae93d3 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/climbing_stairs_backtrack.dart @@ -0,0 +1,39 @@ +/** + * File: climbing_stairs_backtrack.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 回溯 */ +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[0]++; + } + // 遍历所有选择 + for (int 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 + } +} + +/* 爬楼梯:回溯 */ +int climbingStairsBacktrack(int n) { + List choices = [1, 2]; // 可选择向上爬 1 或 2 阶 + int state = 0; // 从第 0 阶开始爬 + List res = []; + res.add(0); // 使用 res[0] 记录方案数量 + backtrack(choices, state, n, res); + return res[0]; +} + +/* Driver Code */ +void main() { + int n = 9; + + int res = climbingStairsBacktrack(n); + print("爬 $n 阶楼梯共有 $res 种方案"); +} diff --git a/codes/dart/chapter_dynamic_programming/climbing_stairs_constraint_dp.dart b/codes/dart/chapter_dynamic_programming/climbing_stairs_constraint_dp.dart new file mode 100644 index 00000000..1740b36a --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/climbing_stairs_constraint_dp.dart @@ -0,0 +1,33 @@ +/** + * File: climbing_stairs_constraint_dp.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 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 表,用于存储子问题的解 + List> dp = List.generate(n + 1, (index) => List.filled(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 */ +void main() { + int n = 9; + + int res = climbingStairsConstraintDP(n); + print("爬 $n 阶楼梯共有 $res 种方案"); +} diff --git a/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs.dart b/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs.dart new file mode 100644 index 00000000..cba40287 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs.dart @@ -0,0 +1,27 @@ +/** + * File: climbing_stairs_dfs.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 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 */ +void main() { + int n = 9; + + int res = climbingStairsDFS(n); + print("爬 $n 阶楼梯共有 $res 种方案"); +} diff --git a/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs_mem.dart b/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs_mem.dart new file mode 100644 index 00000000..0b64b9b2 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/climbing_stairs_dfs_mem.dart @@ -0,0 +1,33 @@ +/** + * File: climbing_stairs_dfs_mem.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 记忆化搜索 */ +int dfs(int i, List 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 + List mem = List.filled(n + 1, -1); + return dfs(n, mem); +} + +/* Driver Code */ +void main() { + int n = 9; + + int res = climbingStairsDFSMem(n); + print("爬 $n 阶楼梯共有 $res 种方案"); +} diff --git a/codes/dart/chapter_dynamic_programming/climbing_stairs_dp.dart b/codes/dart/chapter_dynamic_programming/climbing_stairs_dp.dart new file mode 100644 index 00000000..9bd9b24a --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/climbing_stairs_dp.dart @@ -0,0 +1,43 @@ +/** + * File: climbing_stairs_dp.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 爬楼梯:动态规划 */ +int climbingStairsDP(int n) { + if (n == 1 || n == 2) return n; + // 初始化 dp 表,用于存储子问题的解 + List dp = List.filled(n + 1, 0); + // 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 */ +void main() { + int n = 9; + + int res = climbingStairsDP(n); + print("爬 $n 阶楼梯共有 $res 种方案"); + + res = climbingStairsDPComp(n); + print("爬 $n 阶楼梯共有 $res 种方案"); +} diff --git a/codes/dart/chapter_dynamic_programming/coin_change.dart b/codes/dart/chapter_dynamic_programming/coin_change.dart new file mode 100644 index 00000000..31bb1c40 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/coin_change.dart @@ -0,0 +1,68 @@ +/** + * File: coin_change.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 零钱兑换:动态规划 */ +int coinChangeDP(List coins, int amt) { + int n = coins.length; + int MAX = amt + 1; + // 初始化 dp 表 + List> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0)); + // 状态转移:首行首列 + for (int a = 1; a <= amt; a++) { + dp[0][a] = MAX; + } + // 状态转移:其余行列 + for (int i = 1; i <= n; i++) { + for (int a = 1; a <= amt; a++) { + if (coins[i - 1] > a) { + // 若超过背包容量,则不选硬币 i + dp[i][a] = dp[i - 1][a]; + } else { + // 不选和选硬币 i 这两种方案的较小值 + dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1); + } + } + } + return dp[n][amt] != MAX ? dp[n][amt] : -1; +} + +/* 零钱兑换:状态压缩后的动态规划 */ +int coinChangeDPComp(List coins, int amt) { + int n = coins.length; + int MAX = amt + 1; + // 初始化 dp 表 + List dp = List.filled(amt + 1, MAX); + dp[0] = 0; + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int a = 1; a <= amt; a++) { + if (coins[i - 1] > a) { + // 若超过背包容量,则不选硬币 i + dp[a] = dp[a]; + } else { + // 不选和选硬币 i 这两种方案的较小值 + dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1); + } + } + } + return dp[amt] != MAX ? dp[amt] : -1; +} + +/* Driver Code */ +void main() { + List coins = [1, 2, 5]; + int amt = 4; + + // 动态规划 + int res = coinChangeDP(coins, amt); + print("凑到目标金额所需的最少硬币数量为 $res"); + + // 状态压缩后的动态规划 + res = coinChangeDPComp(coins, amt); + print("凑到目标金额所需的最少硬币数量为 $res"); +} diff --git a/codes/dart/chapter_dynamic_programming/coin_change_ii.dart b/codes/dart/chapter_dynamic_programming/coin_change_ii.dart new file mode 100644 index 00000000..ee4668e2 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/coin_change_ii.dart @@ -0,0 +1,64 @@ +/** + * File: coin_change_ii.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 零钱兑换 II:动态规划 */ +int coinChangeIIDP(List coins, int amt) { + int n = coins.length; + // 初始化 dp 表 + List> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0)); + // 初始化首列 + for (int i = 0; i <= n; i++) { + dp[i][0] = 1; + } + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int a = 1; a <= amt; a++) { + if (coins[i - 1] > a) { + // 若超过背包容量,则不选硬币 i + dp[i][a] = dp[i - 1][a]; + } else { + // 不选和选硬币 i 这两种方案之和 + dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]; + } + } + } + return dp[n][amt]; +} + +/* 零钱兑换 II:状态压缩后的动态规划 */ +int coinChangeIIDPComp(List coins, int amt) { + int n = coins.length; + // 初始化 dp 表 + List dp = List.filled(amt + 1, 0); + dp[0] = 1; + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int a = 1; a <= amt; a++) { + if (coins[i - 1] > a) { + // 若超过背包容量,则不选硬币 i + dp[a] = dp[a]; + } else { + // 不选和选硬币 i 这两种方案之和 + dp[a] = dp[a] + dp[a - coins[i - 1]]; + } + } + } + return dp[amt]; +} + +/* Driver Code */ +void main() { + List coins = [1, 2, 5]; + int amt = 5; + + // 动态规划 + int res = coinChangeIIDP(coins, amt); + print("凑出目标金额的硬币组合数量为 $res"); + + // 状态压缩后的动态规划 + res = coinChangeIIDPComp(coins, amt); + print("凑出目标金额的硬币组合数量为 $res"); +} diff --git a/codes/dart/chapter_dynamic_programming/edit_distance.dart b/codes/dart/chapter_dynamic_programming/edit_distance.dart new file mode 100644 index 00000000..f94b2bd7 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/edit_distance.dart @@ -0,0 +1,125 @@ +/** + * File: edit_distance.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 编辑距离:暴力搜索 */ +int editDistanceDFS(String s, String t, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) return i; + // 若两字符相等,则直接跳过此两字符 + if (s[i - 1] == t[j - 1]) return editDistanceDFS(s, t, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFS(s, t, i, j - 1); + int delete = editDistanceDFS(s, t, i - 1, j); + int replace = editDistanceDFS(s, t, i - 1, j - 1); + // 返回最少编辑步数 + return min(min(insert, delete), replace) + 1; +} + +/* 编辑距离:记忆化搜索 */ +int editDistanceDFSMem(String s, String t, List> mem, int i, int j) { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 && j == 0) return 0; + // 若 s 为空,则返回 t 长度 + if (i == 0) return j; + // 若 t 为空,则返回 s 长度 + if (j == 0) return i; + // 若已有记录,则直接返回之 + if (mem[i][j] != -1) return mem[i][j]; + // 若两字符相等,则直接跳过此两字符 + if (s[i - 1] == t[j - 1]) return editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + int insert = editDistanceDFSMem(s, t, mem, i, j - 1); + int delete = editDistanceDFSMem(s, t, mem, i - 1, j); + int replace = editDistanceDFSMem(s, t, mem, i - 1, j - 1); + // 记录并返回最少编辑步数 + mem[i][j] = min(min(insert, delete), replace) + 1; + return mem[i][j]; +} + +/* 编辑距离:动态规划 */ +int editDistanceDP(String s, String t) { + int n = s.length, m = t.length; + List> dp = List.generate(n + 1, (_) => List.filled(m + 1, 0)); + // 状态转移:首行首列 + for (int i = 1; i <= n; i++) { + dp[i][0] = i; + } + for (int j = 1; j <= m; j++) { + dp[0][j] = j; + } + // 状态转移:其余行列 + for (int i = 1; i <= n; i++) { + for (int j = 1; j <= m; j++) { + if (s[i - 1] == t[j - 1]) { + // 若两字符相等,则直接跳过此两字符 + dp[i][j] = dp[i - 1][j - 1]; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1; + } + } + } + return dp[n][m]; +} + +/* 编辑距离:状态压缩后的动态规划 */ +int editDistanceDPComp(String s, String t) { + int n = s.length, m = t.length; + List dp = List.filled(m + 1, 0); + // 状态转移:首行 + for (int j = 1; j <= m; j++) { + dp[j] = j; + } + // 状态转移:其余行 + for (int i = 1; i <= n; i++) { + // 状态转移:首列 + int leftup = dp[0]; // 暂存 dp[i-1, j-1] + dp[0] = i; + // 状态转移:其余列 + for (int j = 1; j <= m; j++) { + int temp = dp[j]; + if (s[i - 1] == t[j - 1]) { + // 若两字符相等,则直接跳过此两字符 + dp[j] = leftup; + } else { + // 最少编辑步数 = 插入、删除、替换这三种操作的最少编辑步数 + 1 + dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1; + } + leftup = temp; // 更新为下一轮的 dp[i-1, j-1] + } + } + return dp[m]; +} + +/* Driver Code */ +void main() { + String s = "bag"; + String t = "pack"; + int n = s.length, m = t.length; + + // 暴力搜索 + int res = editDistanceDFS(s, t, n, m); + print("将 " + s + " 更改为 " + t + " 最少需要编辑 $res 步"); + + // 记忆化搜索 + List> mem = List.generate(n + 1, (_) => List.filled(m + 1, -1)); + res = editDistanceDFSMem(s, t, mem, n, m); + print("将 " + s + " 更改为 " + t + " 最少需要编辑 $res 步"); + + // 动态规划 + res = editDistanceDP(s, t); + print("将 " + s + " 更改为 " + t + " 最少需要编辑 $res 步"); + + // 状态压缩后的动态规划 + res = editDistanceDPComp(s, t); + print("将 " + s + " 更改为 " + t + " 最少需要编辑 $res 步"); +} diff --git a/codes/dart/chapter_dynamic_programming/knapsack.dart b/codes/dart/chapter_dynamic_programming/knapsack.dart new file mode 100644 index 00000000..e9f58478 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/knapsack.dart @@ -0,0 +1,116 @@ +/** + * File: knapsack.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 0-1 背包:暴力搜索 */ +int knapsackDFS(List wgt, List val, int i, int c) { + // 若已选完所有物品或背包无容量,则返回价值 0 + if (i == 0 || c == 0) { + return 0; + } + // 若超过背包容量,则只能不放入背包 + if (wgt[i - 1] > c) { + return knapsackDFS(wgt, val, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + int no = knapsackDFS(wgt, val, i - 1, c); + int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]; + // 返回两种方案中价值更大的那一个 + return max(no, yes); +} + +/* 0-1 背包:记忆化搜索 */ +int knapsackDFSMem( + List wgt, + List val, + List> mem, + int i, + int c, +) { + // 若已选完所有物品或背包无容量,则返回价值 0 + if (i == 0 || c == 0) { + return 0; + } + // 若已有记录,则直接返回 + if (mem[i][c] != -1) { + return mem[i][c]; + } + // 若超过背包容量,则只能不放入背包 + if (wgt[i - 1] > c) { + return knapsackDFSMem(wgt, val, mem, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + int no = knapsackDFSMem(wgt, val, mem, i - 1, c); + int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]; + // 记录并返回两种方案中价值更大的那一个 + mem[i][c] = max(no, yes); + return mem[i][c]; +} + +/* 0-1 背包:动态规划 */ +int knapsackDP(List wgt, List val, int cap) { + int n = wgt.length; + // 初始化 dp 表 + List> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0)); + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int c = 1; c <= cap; c++) { + if (wgt[i - 1] > c) { + // 若超过背包容量,则不选物品 i + dp[i][c] = dp[i - 1][c]; + } else { + // 不选和选物品 i 这两种方案的较大值 + dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]); + } + } + } + return dp[n][cap]; +} + +/* 0-1 背包:状态压缩后的动态规划 */ +int knapsackDPComp(List wgt, List val, int cap) { + int n = wgt.length; + // 初始化 dp 表 + List dp = List.filled(cap + 1, 0); + // 状态转移 + for (int i = 1; i <= n; i++) { + // 倒序遍历 + for (int c = cap; c >= 1; c--) { + if (wgt[i - 1] <= c) { + // 不选和选物品 i 这两种方案的较大值 + dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]); + } + } + } + return dp[cap]; +} + +/* Driver Code */ +void main() { + List wgt = [10, 20, 30, 40, 50]; + List val = [50, 120, 150, 210, 240]; + int cap = 50; + int n = wgt.length; + + // 暴力搜索 + int res = knapsackDFS(wgt, val, n, cap); + print("不超过背包容量的最大物品价值为 $res"); + + // 记忆化搜索 + List> mem = + List.generate(n + 1, (index) => List.filled(cap + 1, -1)); + res = knapsackDFSMem(wgt, val, mem, n, cap); + print("不超过背包容量的最大物品价值为 $res"); + + // 动态规划 + res = knapsackDP(wgt, val, cap); + print("不超过背包容量的最大物品价值为 $res"); + + // 状态压缩后的动态规划 + res = knapsackDPComp(wgt, val, cap); + print("不超过背包容量的最大物品价值为 $res"); +} diff --git a/codes/dart/chapter_dynamic_programming/min_cost_climbing_stairs_dp.dart b/codes/dart/chapter_dynamic_programming/min_cost_climbing_stairs_dp.dart new file mode 100644 index 00000000..848fc223 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/min_cost_climbing_stairs_dp.dart @@ -0,0 +1,48 @@ +/** + * File: min_cost_climbing_stairs_dp.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 爬楼梯最小代价:动态规划 */ +int minCostClimbingStairsDP(List cost) { + int n = cost.length - 1; + if (n == 1 || n == 2) return cost[n]; + // 初始化 dp 表,用于存储子问题的解 + List dp = List.filled(n + 1, 0); + // 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(List 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 = min(a, tmp) + cost[i]; + a = tmp; + } + return b; +} + +/* Driver Code */ +void main() { + List cost = [0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1]; + print("输入楼梯的代价列表为 $cost"); + + int res = minCostClimbingStairsDP(cost); + print("爬完楼梯的最低代价为 $res"); + + res = minCostClimbingStairsDPComp(cost); + print("爬完楼梯的最低代价为 $res"); +} diff --git a/codes/dart/chapter_dynamic_programming/min_path_sum.dart b/codes/dart/chapter_dynamic_programming/min_path_sum.dart new file mode 100644 index 00000000..142eb6e3 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/min_path_sum.dart @@ -0,0 +1,120 @@ +/** + * File: min_path_sum.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 最小路径和:暴力搜索 */ +int minPathSumDFS(List> grid, int i, int j) { + // 若为左上角单元格,则终止搜索 + if (i == 0 && j == 0) { + return grid[0][0]; + } + // 若行列索引越界,则返回 +∞ 代价 + if (i < 0 || j < 0) { + // 在 Dart 中,int 类型是固定范围的整数,不存在表示“无穷大”的值 + return BigInt.from(2).pow(31).toInt(); + } + // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价 + int left = minPathSumDFS(grid, i - 1, j); + int up = minPathSumDFS(grid, i, j - 1); + // 返回从左上角到 (i, j) 的最小路径代价 + return min(left, up) + grid[i][j]; +} + +/* 最小路径和:记忆化搜索 */ +int minPathSumDFSMem(List> grid, List> mem, int i, int j) { + // 若为左上角单元格,则终止搜索 + if (i == 0 && j == 0) { + return grid[0][0]; + } + // 若行列索引越界,则返回 +∞ 代价 + if (i < 0 || j < 0) { + // 在 Dart 中,int 类型是固定范围的整数,不存在表示“无穷大”的值 + return BigInt.from(2).pow(31).toInt(); + } + // 若已有记录,则直接返回 + if (mem[i][j] != -1) { + return mem[i][j]; + } + // 左边和上边单元格的最小路径代价 + int left = minPathSumDFSMem(grid, mem, i - 1, j); + int up = minPathSumDFSMem(grid, mem, i, j - 1); + // 记录并返回左上角到 (i, j) 的最小路径代价 + mem[i][j] = min(left, up) + grid[i][j]; + return mem[i][j]; +} + +/* 最小路径和:动态规划 */ +int minPathSumDP(List> grid) { + int n = grid.length, m = grid[0].length; + // 初始化 dp 表 + List> dp = List.generate(n, (i) => List.filled(m, 0)); + dp[0][0] = grid[0][0]; + // 状态转移:首行 + for (int j = 1; j < m; j++) { + dp[0][j] = dp[0][j - 1] + grid[0][j]; + } + // 状态转移:首列 + for (int i = 1; i < n; i++) { + dp[i][0] = dp[i - 1][0] + grid[i][0]; + } + // 状态转移:其余行列 + for (int i = 1; i < n; i++) { + for (int j = 1; j < m; j++) { + dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]; + } + } + return dp[n - 1][m - 1]; +} + +/* 最小路径和:状态压缩后的动态规划 */ +int minPathSumDPComp(List> grid) { + int n = grid.length, m = grid[0].length; + // 初始化 dp 表 + List dp = List.filled(m, 0); + dp[0] = grid[0][0]; + for (int j = 1; j < m; j++) { + dp[j] = dp[j - 1] + grid[0][j]; + } + // 状态转移:其余行 + for (int i = 1; i < n; i++) { + // 状态转移:首列 + dp[0] = dp[0] + grid[i][0]; + // 状态转移:其余列 + for (int j = 1; j < m; j++) { + dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]; + } + } + return dp[m - 1]; +} + +/* Driver Code */ +void main() { + List> grid = [ + [1, 3, 1, 5], + [2, 2, 4, 2], + [5, 3, 2, 1], + [4, 3, 5, 2], + ]; + int n = grid.length, m = grid[0].length; + +// 暴力搜索 + int res = minPathSumDFS(grid, n - 1, m - 1); + print("从左上角到右下角的做小路径和为 $res"); + +// 记忆化搜索 + List> mem = List.generate(n, (i) => List.filled(m, -1)); + res = minPathSumDFSMem(grid, mem, n - 1, m - 1); + print("从左上角到右下角的做小路径和为 $res"); + +// 动态规划 + res = minPathSumDP(grid); + print("从左上角到右下角的做小路径和为 $res"); + +// 状态压缩后的动态规划 + res = minPathSumDPComp(grid); + print("从左上角到右下角的做小路径和为 $res"); +} diff --git a/codes/dart/chapter_dynamic_programming/unbounded_knapsack.dart b/codes/dart/chapter_dynamic_programming/unbounded_knapsack.dart new file mode 100644 index 00000000..ba9192c5 --- /dev/null +++ b/codes/dart/chapter_dynamic_programming/unbounded_knapsack.dart @@ -0,0 +1,62 @@ +/** + * File: unbounded_knapsack.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 完全背包:动态规划 */ +int unboundedKnapsackDP(List wgt, List val, int cap) { + int n = wgt.length; + // 初始化 dp 表 + List> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0)); + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int c = 1; c <= cap; c++) { + if (wgt[i - 1] > c) { + // 若超过背包容量,则不选物品 i + dp[i][c] = dp[i - 1][c]; + } else { + // 不选和选物品 i 这两种方案的较大值 + dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]); + } + } + } + return dp[n][cap]; +} + +/* 完全背包:状态压缩后的动态规划 */ +int unboundedKnapsackDPComp(List wgt, List val, int cap) { + int n = wgt.length; + // 初始化 dp 表 + List dp = List.filled(cap + 1, 0); + // 状态转移 + for (int i = 1; i <= n; i++) { + for (int c = 1; c <= cap; c++) { + if (wgt[i - 1] > c) { + // 若超过背包容量,则不选物品 i + dp[c] = dp[c]; + } else { + // 不选和选物品 i 这两种方案的较大值 + dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]); + } + } + } + return dp[cap]; +} + +/* Driver Code */ +void main() { + List wgt = [1, 2, 3]; + List val = [5, 11, 15]; + int cap = 4; + + // 动态规划 + int res = unboundedKnapsackDP(wgt, val, cap); + print("不超过背包容量的最大物品价值为 $res"); + + // 状态压缩后的动态规划 + int resComp = unboundedKnapsackDPComp(wgt, val, cap); + print("不超过背包容量的最大物品价值为 $resComp"); +} diff --git a/codes/dart/chapter_greedy/coin_change_greedy.dart b/codes/dart/chapter_greedy/coin_change_greedy.dart new file mode 100644 index 00000000..7b739f39 --- /dev/null +++ b/codes/dart/chapter_greedy/coin_change_greedy.dart @@ -0,0 +1,50 @@ +/** + * File: coin_change_greedy.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 零钱兑换:贪心 */ +int coinChangeGreedy(List coins, int amt) { + // 假设 coins 列表有序 + int i = coins.length - 1; + int count = 0; + // 循环进行贪心选择,直到无剩余金额 + while (amt > 0) { + // 找到小于且最接近剩余金额的硬币 + while (i > 0 && coins[i] > amt) { + i--; + } + // 选择 coins[i] + amt -= coins[i]; + count++; + } + // 若未找到可行方案,则返回 -1 + return amt == 0 ? count : -1; +} + +/* Driver Code */ +void main() { + // 贪心:能够保证找到全局最优解 + List coins = [1, 5, 10, 20, 50, 100]; + int amt = 186; + int res = coinChangeGreedy(coins, amt); + print("\ncoins = $coins, amt = $amt"); + print("凑到 $amt 所需的最少硬币数量为 $res"); + + // 贪心:无法保证找到全局最优解 + coins = [1, 20, 50]; + amt = 60; + res = coinChangeGreedy(coins, amt); + print("\ncoins = $coins, amt = $amt"); + print("凑到 $amt 所需的最少硬币数量为 $res"); + print("实际上需要的最少数量为 3 ,即 20 + 20 + 20"); + + // 贪心:无法保证找到全局最优解 + coins = [1, 49, 50]; + amt = 98; + res = coinChangeGreedy(coins, amt); + print("\ncoins = $coins, amt = $amt"); + print("凑到 $amt 所需的最少硬币数量为 $res"); + print("实际上需要的最少数量为 2 ,即 49 + 49"); +} diff --git a/codes/dart/chapter_greedy/fractional_knapsack.dart b/codes/dart/chapter_greedy/fractional_knapsack.dart new file mode 100644 index 00000000..61c5d501 --- /dev/null +++ b/codes/dart/chapter_greedy/fractional_knapsack.dart @@ -0,0 +1,47 @@ +/** + * File: fractional_knapsack.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +/* 物品 */ +class Item { + int w; // 物品重量 + int v; // 物品价值 + + Item(this.w, this.v); +} + +/* 分数背包:贪心 */ +double fractionalKnapsack(List wgt, List val, int cap) { + // 创建物品列表,包含两个属性:重量、价值 + List items = List.generate(wgt.length, (i) => Item(wgt[i], val[i])); + // 按照单位价值 item.v / item.w 从高到低进行排序 + items.sort((a, b) => (b.v / b.w).compareTo(a.v / a.w)); + // 循环贪心选择 + double res = 0; + for (Item item in items) { + if (item.w <= cap) { + // 若剩余容量充足,则将当前物品整个装进背包 + res += item.v; + cap -= item.w; + } else { + // 若剩余容量不足,则将当前物品的一部分装进背包 + res += item.v / item.w * cap; + // 已无剩余容量,因此跳出循环 + break; + } + } + return res; +} + +/* Driver Code */ +void main() { + List wgt = [10, 20, 30, 40, 50]; + List val = [50, 120, 150, 210, 240]; + int cap = 50; + + // 贪心算法 + double res = fractionalKnapsack(wgt, val, cap); + print("不超过背包容量的最大物品价值为 $res"); +} diff --git a/codes/dart/chapter_greedy/max_capacity.dart b/codes/dart/chapter_greedy/max_capacity.dart new file mode 100644 index 00000000..ce5c0633 --- /dev/null +++ b/codes/dart/chapter_greedy/max_capacity.dart @@ -0,0 +1,37 @@ +/** + * File: max_capacity.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 最大容量:贪心 */ +int maxCapacity(List ht) { + // 初始化 i, j 分列数组两端 + int i = 0, j = ht.length - 1; + // 初始最大容量为 0 + int res = 0; + // 循环贪心选择,直至两板相遇 + while (i < j) { + // 更新最大容量 + int cap = min(ht[i], ht[j]) * (j - i); + res = max(res, cap); + // 向内移动短板 + if (ht[i] < ht[j]) { + i++; + } else { + j--; + } + } + return res; +} + +/* Driver Code */ +void main() { + List ht = [3, 8, 5, 2, 7, 7, 3, 4]; + + // 贪心算法 + int res = maxCapacity(ht); + print("最大容量为 $res"); +} diff --git a/codes/dart/chapter_greedy/max_product_cutting.dart b/codes/dart/chapter_greedy/max_product_cutting.dart new file mode 100644 index 00000000..2df5c8e1 --- /dev/null +++ b/codes/dart/chapter_greedy/max_product_cutting.dart @@ -0,0 +1,37 @@ +/** + * File: max_product_cutting.dart + * Created Time: 2023-08-11 + * Author: liuyuxin (gvenusleo@gmail.com) + */ + +import 'dart:math'; + +/* 最大切分乘积:贪心 */ +int maxProductCutting(int n) { + // 当 n <= 3 时,必须切分出一个 1 + if (n <= 3) { + return 1 * (n - 1); + } + // 贪心地切分出 3 ,a 为 3 的个数,b 为余数 + int a = n ~/ 3; + int b = n % 3; + if (b == 1) { + // 当余数为 1 时,将一对 1 * 3 转化为 2 * 2 + return (pow(3, a - 1) * 2 * 2).toInt(); + } + if (b == 2) { + // 当余数为 2 时,不做处理 + return (pow(3, a) * 2).toInt(); + } + // 当余数为 0 时,不做处理 + return pow(3, a).toInt(); +} + +/* Driver Code */ +void main() { + int n = 58; + + // 贪心算法 + int res = maxProductCutting(n); + print("最大切分乘积为 $res"); +}