diff --git a/codes/rust/Cargo.toml b/codes/rust/Cargo.toml index 206578d22..358ab3421 100644 --- a/codes/rust/Cargo.toml +++ b/codes/rust/Cargo.toml @@ -249,5 +249,50 @@ path = "chapter_graph/graph_bfs.rs" name = "graph_dfs" path = "chapter_graph/graph_dfs.rs" +# Run Command: cargo run --bin coin_change +[[bin]] +name = "coin_change" +path = "chapter_dynamic_programming/coin_change.rs" + +# Run Command: cargo run --bin coin_change_ii +[[bin]] +name = "coin_change_ii" +path = "chapter_dynamic_programming/coin_change_ii.rs" + +# Run Command: cargo run --bin unbounded_knapsack +[[bin]] +name = "unbounded_knapsack" +path = "chapter_dynamic_programming/unbounded_knapsack.rs" + +# Run Command: cargo run --bin knapsack +[[bin]] +name = "knapsack" +path = "chapter_dynamic_programming/knapsack.rs" + +# Run Command: cargo run --bin min_path_sum +[[bin]] +name = "min_path_sum" +path = "chapter_dynamic_programming/min_path_sum.rs" + +# Run Command: cargo run --bin edit_distance +[[bin]] +name = "edit_distance" +path = "chapter_dynamic_programming/edit_distance.rs" + +# Run Command: cargo run --bin n_queens +[[bin]] +name = "n_queens" +path = "chapter_backtracking/n_queens.rs" + +# Run Command: cargo run --bin permutations_i +[[bin]] +name = "permutations_i" +path = "chapter_backtracking/permutations_i.rs" + +# Run Command: cargo run --bin permutations_ii +[[bin]] +name = "permutations_ii" +path = "chapter_backtracking/permutations_ii.rs" + [dependencies] rand = "0.8.5" diff --git a/codes/rust/chapter_backtracking/n_queens.rs b/codes/rust/chapter_backtracking/n_queens.rs new file mode 100644 index 000000000..ffa5b890f --- /dev/null +++ b/codes/rust/chapter_backtracking/n_queens.rs @@ -0,0 +1,72 @@ +/* + * File: n_queens.rs + * Created Time: 2023-07-15 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 回溯算法:N 皇后 */ +fn backtrack(row: usize, n: usize, state: &mut Vec>, res: &mut Vec>>, + cols: &mut [bool], diags1: &mut [bool], diags2: &mut [bool]) { + // 当放置完所有行时,记录解 + if row == n { + let mut copy_state: Vec> = Vec::new(); + for s_row in state.clone() { + copy_state.push(s_row); + } + res.push(copy_state); + return; + } + // 遍历所有列 + for col in 0..n { + // 计算该格子对应的主对角线和副对角线 + let diag1 = row + n - 1 - col; + let diag2 = row + col; + // 剪枝:不允许该格子所在列、主对角线、副对角线存在皇后 + if !cols[col] && !diags1[diag1] && !diags2[diag2] { + // 尝试:将皇后放置在该格子 + state.get_mut(row).unwrap()[col] = "Q".into(); + (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true); + // 放置下一行 + backtrack(row + 1, n, state, res, cols, diags1, diags2); + // 回退:将该格子恢复为空位 + state.get_mut(row).unwrap()[col] = "#".into(); + (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false); + } + } +} + +/* 求解 N 皇后 */ +fn n_queens(n: usize) -> Vec>> { + // 初始化 n*n 大小的棋盘,其中 'Q' 代表皇后,'#' 代表空位 + let mut state: Vec> = Vec::new(); + for _ in 0..n { + let mut row: Vec = Vec::new(); + for _ in 0..n { + row.push("#".into()); + } + state.push(row); + } + let mut cols = vec![false; n]; // 记录列是否有皇后 + let mut diags1 = vec![false; 2 * n - 1]; // 记录主对角线是否有皇后 + let mut diags2 = vec![false; 2 * n - 1]; // 记录副对角线是否有皇后 + let mut res: Vec>> = Vec::new(); + + backtrack(0, n, &mut state, &mut res, &mut cols, &mut diags1, &mut diags2); + + res +} + +/* Driver Code */ +pub fn main() { + let n: usize = 4; + let res = n_queens(n); + + println!("输入棋盘长宽为 {n}"); + println!("皇后放置方案共有 {} 种", res.len()); + for state in res.iter() { + println!("--------------------"); + for row in state.iter() { + println!("{:?}", row); + } + } +} diff --git a/codes/rust/chapter_backtracking/permutations_i.rs b/codes/rust/chapter_backtracking/permutations_i.rs new file mode 100644 index 000000000..3c41a458c --- /dev/null +++ b/codes/rust/chapter_backtracking/permutations_i.rs @@ -0,0 +1,46 @@ +/* + * File: permutations_i.rs + * Created Time: 2023-07-15 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 回溯算法:全排列 I */ +fn backtrack(mut state: Vec, choices: &[i32], selected: &mut [bool], res: &mut Vec>) { + // 当状态长度等于元素数量时,记录解 + if state.len() == choices.len() { + res.push(state); + return; + } + // 遍历所有选择 + for i in 0..choices.len() { + let choice = choices[i]; + // 剪枝:不允许重复选择元素 且 不允许重复选择相等元素 + if !selected[i] { + // 尝试:做出选择,更新状态 + selected[i] = true; + state.push(choice); + // 进行下一轮选择 + backtrack(state.clone(), choices, selected, res); + // 回退:撤销选择,恢复到之前的状态 + selected[i] = false; + state.remove(state.len() - 1); + } + } +} + +/* 全排列 I */ +fn permutations_i(nums: &mut [i32]) -> Vec> { + let mut res = Vec::new(); // 状态(子集) + backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res); + res +} + +/* Driver Code */ +pub fn main() { + let mut nums = [ 1, 2, 3 ]; + + let res = permutations_i(&mut nums); + + println!("输入数组 nums = {:?}", &nums); + println!("所有排列 res = {:?}", &res); +} diff --git a/codes/rust/chapter_backtracking/permutations_ii.rs b/codes/rust/chapter_backtracking/permutations_ii.rs new file mode 100644 index 000000000..a93fe2e1f --- /dev/null +++ b/codes/rust/chapter_backtracking/permutations_ii.rs @@ -0,0 +1,50 @@ +/* + * File: permutations_ii.rs + * Created Time: 2023-07-15 + * Author: sjinzh (sjinzh@gmail.com) + */ + +use std::collections::HashSet; + +/* 回溯算法:全排列 II */ +fn backtrack(mut state: Vec, choices: &[i32], selected: &mut [bool], res: &mut Vec>) { + // 当状态长度等于元素数量时,记录解 + if state.len() == choices.len() { + res.push(state); + return; + } + // 遍历所有选择 + let mut duplicated = HashSet::::new(); + for i in 0..choices.len() { + let choice = choices[i]; + // 剪枝:不允许重复选择元素 且 不允许重复选择相等元素 + if !selected[i] && !duplicated.contains(&choice) { + // 尝试:做出选择,更新状态 + duplicated.insert(choice); // 记录选择过的元素值 + selected[i] = true; + state.push(choice); + // 进行下一轮选择 + backtrack(state.clone(), choices, selected, res); + // 回退:撤销选择,恢复到之前的状态 + selected[i] = false; + state.remove(state.len() - 1); + } + } +} + +/* 全排列 II */ +fn permutations_ii(nums: &mut [i32]) -> Vec> { + let mut res = Vec::new(); + backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res); + res +} + +/* Driver Code */ +pub fn main() { + let mut nums = [ 1, 2, 2 ]; + + let res = permutations_ii(&mut nums); + + println!("输入数组 nums = {:?}", &nums); + println!("所有排列 res = {:?}", &res); +} diff --git a/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs.rs b/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs.rs index 7f33cb43a..c5e9b1ac1 100644 --- a/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs.rs +++ b/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs.rs @@ -7,7 +7,7 @@ /* 搜索 */ fn dfs(i: usize) -> i32 { // 已知 dp[1] 和 dp[2] ,返回之 - if i == 1 || i == 2 { return i as i32 }; + if i == 1 || i == 2 { return i as i32; } // dp[i] = dp[i-1] + dp[i-2] let count = dfs(i - 1) + dfs(i - 2); count diff --git a/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs_mem.rs b/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs_mem.rs index b7ef74a12..cb02c623a 100644 --- a/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs_mem.rs +++ b/codes/rust/chapter_dynamic_programming/climbing_stairs_dfs_mem.rs @@ -7,9 +7,9 @@ /* 记忆化搜索 */ fn dfs(i: usize, mem: &mut [i32]) -> i32 { // 已知 dp[1] 和 dp[2] ,返回之 - if i == 1 || i == 2 { return i as i32}; + if i == 1 || i == 2 { return i as i32; } // 若存在记录 dp[i] ,则直接返回之 - if mem[i] != -1 { return mem[i] }; + if mem[i] != -1 { return mem[i]; } // dp[i] = dp[i-1] + dp[i-2] let count = dfs(i - 1, mem) + dfs(i - 2, mem); // 记录 dp[i] diff --git a/codes/rust/chapter_dynamic_programming/climbing_stairs_dp.rs b/codes/rust/chapter_dynamic_programming/climbing_stairs_dp.rs index 2fdee62ea..75bbe79df 100644 --- a/codes/rust/chapter_dynamic_programming/climbing_stairs_dp.rs +++ b/codes/rust/chapter_dynamic_programming/climbing_stairs_dp.rs @@ -7,7 +7,7 @@ /* 爬楼梯:动态规划 */ fn climbing_stairs_dp(n: usize) -> i32 { // 已知 dp[1] 和 dp[2] ,返回之 - if n == 1 || n == 2 { return n as i32 }; + if n == 1 || n == 2 { return n as i32; } // 初始化 dp 表,用于存储子问题的解 let mut dp = vec![-1; n + 1]; // 初始状态:预设最小子问题的解 @@ -22,7 +22,7 @@ fn climbing_stairs_dp(n: usize) -> i32 { /* 爬楼梯:状态压缩后的动态规划 */ fn climbing_stairs_dp_comp(n: usize) -> i32 { - if n == 1 || n == 2 { return n as i32 }; + if n == 1 || n == 2 { return n as i32; } let (mut a, mut b) = (1, 2); for _ in 3..=n { let tmp = b; diff --git a/codes/rust/chapter_dynamic_programming/coin_change.rs b/codes/rust/chapter_dynamic_programming/coin_change.rs new file mode 100644 index 000000000..4a976887d --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/coin_change.rs @@ -0,0 +1,67 @@ +/* + * File: coin_change.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 零钱兑换:动态规划 */ +fn coin_change_dp(coins: &[i32], amt: usize) -> i32 { + let n = coins.len(); + let max = amt + 1; + // 初始化 dp 表 + let mut dp = vec![vec![0; amt + 1]; n + 1]; + // 状态转移:首行首列 + for a in 1..= amt { + dp[0][a] = max; + } + // 状态转移:其余行列 + for i in 1..=n { + for a in 1..=amt { + if coins[i - 1] > a as i32 { + // 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] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1); + } + } + } + if dp[n][amt] != max { return dp[n][amt] as i32; } else { -1 } +} + +/* 零钱兑换:状态压缩后的动态规划 */ +fn coin_change_dp_comp(coins: &[i32], amt: usize) -> i32 { + let n = coins.len(); + let max = amt + 1; + // 初始化 dp 表 + let mut dp = vec![0; amt + 1]; + dp.fill(max); + dp[0] = 0; + // 状态转移 + for i in 1..=n { + for a in 1..=amt { + if coins[i - 1] > a as i32 { + // 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] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1); + } + } + } + if dp[amt] != max { return dp[amt] as i32; } else { -1 } +} + +/* Driver Code */ +pub fn main() { + let coins = [ 1, 2, 5 ]; + let amt: usize = 4; + + // 动态规划 + let res = coin_change_dp(&coins, amt); + println!("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 + let res = coin_change_dp_comp(&coins, amt); + println!("凑到目标金额所需的最少硬币数量为 {res}"); +} diff --git a/codes/rust/chapter_dynamic_programming/coin_change_ii.rs b/codes/rust/chapter_dynamic_programming/coin_change_ii.rs new file mode 100644 index 000000000..6e8090515 --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/coin_change_ii.rs @@ -0,0 +1,64 @@ +/* + * File: coin_change_ii.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 零钱兑换 II:动态规划 */ +fn coin_change_ii_dp(coins: &[i32], amt: usize) -> i32 { + let n = coins.len(); + // 初始化 dp 表 + let mut dp = vec![vec![0; amt + 1]; n + 1]; + // 初始化首列 + for i in 0..= n { + dp[i][0] = 1; + } + // 状态转移 + for i in 1..=n { + for a in 1..=amt { + if coins[i - 1] > a as i32 { + // 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] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize]; + } + } + } + dp[n][amt] +} + +/* 零钱兑换 II:状态压缩后的动态规划 */ +fn coin_change_dp_ii_comp(coins: &[i32], amt: usize) -> i32 { + let n = coins.len(); + // 初始化 dp 表 + let mut dp = vec![0; amt + 1]; + dp[0] = 1; + // 状态转移 + for i in 1..=n { + for a in 1..=amt { + if coins[i - 1] > a as i32 { + // 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] = dp[a] + dp[a - coins[i - 1] as usize]; + } + } + } + dp[amt] +} + +/* Driver Code */ +pub fn main() { + let coins = [ 1, 2, 5 ]; + let amt: usize = 5; + + // 动态规划 + let res = coin_change_ii_dp(&coins, amt); + println!("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 + let res = coin_change_dp_ii_comp(&coins, amt); + println!("凑出目标金额的硬币组合数量为 {res}"); +} diff --git a/codes/rust/chapter_dynamic_programming/edit_distance.rs b/codes/rust/chapter_dynamic_programming/edit_distance.rs new file mode 100644 index 000000000..2bef9fa34 --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/edit_distance.rs @@ -0,0 +1,130 @@ +/* + * File: edit_distance.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 编辑距离:暴力搜索 */ +fn edit_distance_dfs(s: &str, t: &str, i: usize, j: usize) -> i32 { + // 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 as i32; } + // 若 t 为空,则返回 s 长度 + if j == 0 {return i as i32; } + // 若两字符相等,则直接跳过此两字符 + if s.chars().nth(i - 1) == t.chars().nth(j - 1) { + return edit_distance_dfs(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 + let insert = edit_distance_dfs(s, t, i, j - 1); + let delete = edit_distance_dfs(s, t, i - 1, j); + let replace = edit_distance_dfs(s, t, i - 1, j - 1); + // 返回最少编辑步数 + std::cmp::min(std::cmp::min(insert, delete), replace) + 1 +} + +/* 编辑距离:记忆化搜索 */ +fn edit_distance_dfs_mem(s: &str, t: &str, mem: &mut Vec>, i: usize, j: usize) -> i32 { + // 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 as i32; } + // 若 t 为空,则返回 s 长度 + if j == 0 {return i as i32; } + // 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.chars().nth(i - 1) == t.chars().nth(j - 1) { + return edit_distance_dfs_mem(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 + let insert = edit_distance_dfs_mem(s, t, mem, i, j - 1); + let delete = edit_distance_dfs_mem(s, t, mem, i - 1, j); + let replace = edit_distance_dfs_mem(s, t, mem, i - 1, j - 1); + // 记录并返回最少编辑步数 + mem[i][j] = std::cmp::min(std::cmp::min(insert, delete), replace) + 1; + mem[i][j] +} + +/* 编辑距离:动态规划 */ +fn edit_distance_dp(s: &str, t: &str) -> i32 { + let (n, m) = (s.len(), t.len()); + let mut dp = vec![vec![0; m + 1]; n + 1]; + // 状态转移:首行首列 + for i in 1..= n { + dp[i][0] = i as i32; + } + for j in 1..m { + dp[0][j] = j as i32; + } + // 状态转移:其余行列 + for i in 1..=n { + for j in 1..=m { + if s.chars().nth(i - 1) == t.chars().nth(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] = std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1; + } + } + } + dp[n][m] +} + +/* 编辑距离:状态压缩后的动态规划 */ +fn edit_distance_dp_comp(s: &str, t: &str) -> i32 { + let (n, m) = (s.len(), t.len()); + let mut dp = vec![0; m + 1]; + // 状态转移:首行 + for j in 1..m { + dp[j] = j as i32; + } + // 状态转移:其余行 + for i in 1..=n { + // 状态转移:首列 + let mut leftup = dp[0]; // 暂存 dp[i-1, j-1] + dp[0] = i as i32; + // 状态转移:其余列 + for j in 1..=m { + let temp = dp[j]; + if s.chars().nth(i - 1) == t.chars().nth(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] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1; + } + leftup = temp; // 更新为下一轮的 dp[i-1, j-1] + } + } + dp[m] +} + +/* Driver Code */ +pub fn main() { + let s = "bag"; + let t = "pack"; + let (n, m) = (s.len(), t.len()); + + // 暴力搜索 + let res = edit_distance_dfs(s, t, n, m); + println!("将 {s} 更改为 {t} 最少需要编辑 {res} 步"); + + // 记忆搜索 + let mut mem = vec![vec![0; m + 1]; n + 1]; + for row in mem.iter_mut() { + row.fill(-1); + } + let res = edit_distance_dfs_mem(s, t, &mut mem, n, m); + println!("将 {s} 更改为 {t} 最少需要编辑 {res} 步"); + + // 动态规划 + let res = edit_distance_dp(s, t); + println!("将 {s} 更改为 {t} 最少需要编辑 {res} 步"); + + // 状态压缩后的动态规划 + let res = edit_distance_dp_comp(s, t); + println!("将 {s} 更改为 {t} 最少需要编辑 {res} 步"); +} diff --git a/codes/rust/chapter_dynamic_programming/knapsack.rs b/codes/rust/chapter_dynamic_programming/knapsack.rs new file mode 100644 index 000000000..d31c8fc28 --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/knapsack.rs @@ -0,0 +1,110 @@ +/* + * File: knapsack.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 0-1 背包:暴力搜索 */ +fn knapsack_dfs(wgt: &[i32], val: &[i32], i: usize, c: usize) -> i32 { + // 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 as i32 { + return knapsack_dfs(wgt, val, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + let no = knapsack_dfs(wgt, val, i - 1, c); + let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1]; + // 返回两种方案中价值更大的那一个 + std::cmp::max(no, yes) +} + +/* 0-1 背包:记忆化搜索 */ +fn knapsack_dfs_mem(wgt: &[i32], val: &[i32], mem: &mut Vec>, i: usize, c: usize) -> i32 { + // 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 as i32 { + return knapsack_dfs_mem(wgt, val, mem, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c); + let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1]; + // 记录并返回两种方案中价值更大的那一个 + mem[i][c] = std::cmp::max(no, yes); + mem[i][c] +} + +/* 0-1 背包:动态规划 */ +fn knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 { + let n = wgt.len(); + // 初始化 dp 表 + let mut dp = vec![vec![0; cap + 1]; n + 1]; + // 状态转移 + for i in 1..=n { + for c in 1..=cap { + if wgt[i - 1] > c as i32 { + // 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] = std::cmp::max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1]); + } + } + } + dp[n][cap] +} + +/* 0-1 背包:状态压缩后的动态规划 */ +fn knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 { + let n = wgt.len(); + // 初始化 dp 表 + let mut dp = vec![0; cap + 1]; + // 状态转移 + for i in 1..=n { + // 倒序遍历 + for c in (1..=cap).rev() { + if wgt[i - 1] <= c as i32 { + // 不选和选物品 i 这两种方案的较大值 + dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]); + } + } + } + dp[cap] +} + +/* Driver Code */ +pub fn main() { + let wgt = [ 10, 20, 30, 40, 50 ]; + let val = [ 50, 120, 150, 210, 240 ]; + let cap: usize = 50; + let n = wgt.len(); + + // 暴力搜索 + let res = knapsack_dfs(&wgt, &val, n, cap); + println!("不超过背包容量的最大物品价值为 {res}"); + + // 记忆搜索 + let mut mem = vec![vec![0; cap + 1]; n + 1]; + for row in mem.iter_mut() { + row.fill(-1); + } + let res = knapsack_dfs_mem(&wgt, &val, &mut mem, n, cap); + println!("不超过背包容量的最大物品价值为 {res}"); + + // 动态规划 + let res = knapsack_dp(&wgt, &val, cap); + println!("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 + let res = knapsack_dp_comp(&wgt, &val, cap); + println!("不超过背包容量的最大物品价值为 {res}"); +} diff --git a/codes/rust/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rs b/codes/rust/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rs index a92a0b45e..c6b9c029c 100644 --- a/codes/rust/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rs +++ b/codes/rust/chapter_dynamic_programming/min_cost_climbing_stairs_dp.rs @@ -9,7 +9,7 @@ /* 爬楼梯最小代价:动态规划 */ fn min_cost_climbing_stairs_dp(cost: &[i32]) -> i32 { let n = cost.len() - 1; - if n == 1 || n == 2 { return cost[n] }; + if n == 1 || n == 2 { return cost[n]; } // 初始化 dp 表,用于存储子问题的解 let mut dp = vec![-1; n + 1]; // 初始状态:预设最小子问题的解 diff --git a/codes/rust/chapter_dynamic_programming/min_path_sum.rs b/codes/rust/chapter_dynamic_programming/min_path_sum.rs new file mode 100644 index 000000000..8dcd36704 --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/min_path_sum.rs @@ -0,0 +1,119 @@ +/* + * File: min_path_sum.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 最小路径和:暴力搜索 */ +fn min_path_sum_dfs(grid: &Vec>, i: i32, j: i32) -> i32 { + // 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 { + return i32::MAX; + } + // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价 + let left = min_path_sum_dfs(grid, i - 1, j); + let up = min_path_sum_dfs(grid, i, j - 1); + // 返回从左上角到 (i, j) 的最小路径代价 + std::cmp::min(left, up) + grid[i as usize][j as usize] +} + +/* 最小路径和:记忆化搜索 */ +fn min_path_sum_dfs_mem(grid: &Vec>, mem: &mut Vec>, i: i32, j: i32) -> i32 { + // 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 { + return i32::MAX; + } + // 若已有记录,则直接返回 + if mem[i as usize][j as usize] != -1 { + return mem[i as usize][j as usize]; + } + // 左边和上边单元格的最小路径代价 + let left = min_path_sum_dfs_mem(grid, mem, i - 1, j); + let up = min_path_sum_dfs_mem(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 as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize]; + mem[i as usize][j as usize] +} + +/* 最小路径和:动态规划 */ +fn min_path_sum_dp(grid: &Vec>) -> i32 { + let (n, m) = (grid.len(), grid[0].len()); + // 初始化 dp 表 + let mut dp = vec![vec![0; m]; n]; + dp[0][0] = grid[0][0]; + // 状态转移:首行 + for j in 1..m { + dp[0][j] = dp[0][j - 1] + grid[0][j]; + } + // 状态转移:首列 + for i in 1..n { + dp[i][0] = dp[i - 1][0] + grid[i][0]; + } + // 状态转移:其余行列 + for i in 1..n { + for j in 1..m { + dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]; + } + } + dp[n - 1][m - 1] +} + +/* 最小路径和:状态压缩后的动态规划 */ +fn min_path_sum_dp_comp(grid: &Vec>) -> i32 { + let (n, m) = (grid.len(), grid[0].len()); + // 初始化 dp 表 + let mut dp = vec![0; m]; + // 状态转移:首行 + dp[0] = grid[0][0]; + for j in 1..m { + dp[j] = dp[j - 1] + grid[0][j]; + } + // 状态转移:其余行 + for i in 1..n { + // 状态转移:首列 + dp[0] = dp[0] + grid[i][0]; + // 状态转移:其余列 + for j in 1..m { + dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j]; + } + } + dp[m - 1] +} + +/* Driver Code */ +pub fn main() { + let grid = vec![ + vec![ 1, 3, 1, 5 ], + vec![ 2, 2, 4, 2 ], + vec![ 5, 3, 2, 1 ], + vec![ 4, 3, 5, 2 ]]; + let (n, m) = (grid.len(), grid[0].len()); + + // 暴力搜索 + let res = min_path_sum_dfs(&grid, n as i32 - 1, m as i32 - 1); + println!("从左上角到右下角的最小路径和为 {res}"); + + // 记忆化搜索 + let mut mem = vec![vec![0; m]; n]; + for row in mem.iter_mut() { + row.fill(-1); + } + let res = min_path_sum_dfs_mem(&grid, &mut mem, n as i32 - 1, m as i32 - 1); + println!("从左上角到右下角的最小路径和为 {res}"); + + // 动态规划 + let res = min_path_sum_dp(&grid); + println!("从左上角到右下角的最小路径和为 {res}"); + + // 状态压缩后的动态规划 + let res = min_path_sum_dp_comp(&grid); + println!("从左上角到右下角的最小路径和为 {res}"); +} diff --git a/codes/rust/chapter_dynamic_programming/unbounded_knapsack.rs b/codes/rust/chapter_dynamic_programming/unbounded_knapsack.rs new file mode 100644 index 000000000..1f92cb056 --- /dev/null +++ b/codes/rust/chapter_dynamic_programming/unbounded_knapsack.rs @@ -0,0 +1,60 @@ +/* + * File: unbounded_knapsack.rs + * Created Time: 2023-07-09 + * Author: sjinzh (sjinzh@gmail.com) + */ + +/* 完全背包:动态规划 */ +fn unbounded_knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 { + let n = wgt.len(); + // 初始化 dp 表 + let mut dp = vec![vec![0; cap + 1]; n + 1]; + // 状态转移 + for i in 1..=n { + for c in 1..=cap { + if wgt[i - 1] > c as i32 { + // 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] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]); + } + } + } + return dp[n][cap]; +} + +/* 完全背包:状态压缩后的动态规划 */ +fn unbounded_knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 { + let n = wgt.len(); + // 初始化 dp 表 + let mut dp = vec![0; cap + 1]; + // 状态转移 + for i in 1..=n { + for c in 1..=cap { + if wgt[i - 1] > c as i32 { + // 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] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]); + } + } + } + dp[cap] +} + +/* Driver Code */ +pub fn main() { + let wgt = [ 1, 2, 3 ]; + let val = [ 5, 11, 15 ]; + let cap: usize = 4; + + // 动态规划 + let res = unbounded_knapsack_dp(&wgt, &val, cap); + println!("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 + let res = unbounded_knapsack_dp_comp(&wgt, &val, cap); + println!("不超过背包容量的最大物品价值为 {res}"); +} diff --git a/codes/zig/build.zig b/codes/zig/build.zig index 025107f6e..bff1a25f9 100644 --- a/codes/zig/build.zig +++ b/codes/zig/build.zig @@ -143,6 +143,54 @@ pub fn build(b: *std.Build) void { // Source File: "chapter_sorting/radix_sort.zig" // Run Command: zig build run_radix_sort -Doptimize=ReleaseFast .{ .name = "radix_sort", .path = "chapter_sorting/radix_sort.zig" }, + + // Source File: "chapter_dynamic_programming/climbing_stairs_backtrack.zig" + // Run Command: zig build run_climbing_stairs_backtrack -Doptimize=ReleaseFast + .{ .name = "climbing_stairs_backtrack", .path = "chapter_dynamic_programming/climbing_stairs_backtrack.zig" }, + + // Source File: "chapter_dynamic_programming/climbing_stairs_constraint_dp.zig" + // Run Command: zig build run_climbing_stairs_constraint_dp -Doptimize=ReleaseFast + .{ .name = "climbing_stairs_constraint_dp", .path = "chapter_dynamic_programming/climbing_stairs_constraint_dp.zig" }, + + // Source File: "chapter_dynamic_programming/climbing_stairs_dfs_mem.zig" + // Run Command: zig build run_climbing_stairs_dfs_mem -Doptimize=ReleaseFast + .{ .name = "climbing_stairs_dfs_mem", .path = "chapter_dynamic_programming/climbing_stairs_dfs_mem.zig" }, + + // Source File: "chapter_dynamic_programming/climbing_stairs_dfs.zig" + // Run Command: zig build run_climbing_stairs_dfs -Doptimize=ReleaseFast + .{ .name = "climbing_stairs_dfs", .path = "chapter_dynamic_programming/climbing_stairs_dfs.zig" }, + + // Source File: "chapter_dynamic_programming/climbing_stairs_dp.zig" + // Run Command: zig build run_climbing_stairs_dp -Doptimize=ReleaseFast + .{ .name = "climbing_stairs_dp", .path = "chapter_dynamic_programming/climbing_stairs_dp.zig" }, + + // Source File: "chapter_dynamic_programming/coin_change_ii.zig" + // Run Command: zig build run_coin_change_ii -Doptimize=ReleaseFast + .{ .name = "coin_change_ii", .path = "chapter_dynamic_programming/coin_change_ii.zig" }, + + // Source File: "chapter_dynamic_programming/coin_change.zig" + // Run Command: zig build run_coin_change -Doptimize=ReleaseFast + .{ .name = "coin_change", .path = "chapter_dynamic_programming/coin_change.zig" }, + + // Source File: "chapter_dynamic_programming/edit_distance.zig" + // Run Command: zig build run_edit_distance -Doptimize=ReleaseFast + .{ .name = "edit_distance", .path = "chapter_dynamic_programming/edit_distance.zig" }, + + // Source File: "chapter_dynamic_programming/knapsack.zig" + // Run Command: zig build run_knapsack -Doptimize=ReleaseFast + .{ .name = "knapsack", .path = "chapter_dynamic_programming/knapsack.zig" }, + + // Source File: "chapter_dynamic_programming/min_cost_climbing_stairs_dp.zig" + // Run Command: zig build run_min_cost_climbing_stairs_dp -Doptimize=ReleaseFast + .{ .name = "min_cost_climbing_stairs_dp", .path = "chapter_dynamic_programming/min_cost_climbing_stairs_dp.zig" }, + + // Source File: "chapter_dynamic_programming/min_path_sum.zig" + // Run Command: zig build run_min_path_sum -Doptimize=ReleaseFast + .{ .name = "min_path_sum", .path = "chapter_dynamic_programming/min_path_sum.zig" }, + + // Source File: "chapter_dynamic_programming/unbounded_knapsack.zig" + // Run Command: zig build run_unbounded_knapsack -Doptimize=ReleaseFast + .{ .name = "unbounded_knapsack", .path = "chapter_dynamic_programming/unbounded_knapsack.zig" }, }; inline for (group_name_path) |name_path| { diff --git a/codes/zig/chapter_dynamic_programming/climbing_stairs_backtrack.zig b/codes/zig/chapter_dynamic_programming/climbing_stairs_backtrack.zig new file mode 100644 index 000000000..0fc5ef107 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/climbing_stairs_backtrack.zig @@ -0,0 +1,44 @@ +// File: climbing_stairs_backtrack.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 回溯 +fn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void { + // 当爬到第 n 阶时,方案数量加 1 + if (state == n) { + res.items[0] = res.items[0] + 1; + } + // 遍历所有选择 + for (choices) |choice| { + // 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 +fn climbing_stairs_backtrack(n: usize) !i32 { + var choices = [_]i32{ 1, 2 }; // 可选择向上爬 1 或 2 阶 + var state: i32 = 0; // 从第 0 阶开始爬 + var res = std.ArrayList(i32).init(std.heap.page_allocator); + defer res.deinit(); + try res.append(0); // 使用 res[0] 记录方案数量 + backtrack(&choices, state, @intCast(n), res); + return res.items[0]; +} + +// Driver Code +pub fn main() !void { + var n: usize = 9; + + var res = try climbing_stairs_backtrack(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/climbing_stairs_constraint_dp.zig b/codes/zig/chapter_dynamic_programming/climbing_stairs_constraint_dp.zig new file mode 100644 index 000000000..cd4e43ee9 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/climbing_stairs_constraint_dp.zig @@ -0,0 +1,35 @@ +// File: climbing_stairs_constraint_dp.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 带约束爬楼梯:动态规划 +fn climbing_stairs_constraint_dp(comptime n: usize) i32 { + if (n == 1 or n == 2) { + return @intCast(n); + } + // 初始化 dp 表,用于存储子问题的解 + var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1); + // 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 (3..n + 1) |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 +pub fn main() !void { + comptime var n: usize = 9; + + var res = climbing_stairs_constraint_dp(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs.zig b/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs.zig new file mode 100644 index 000000000..2126a508f --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs.zig @@ -0,0 +1,31 @@ +// File: climbing_stairs_dfs.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 搜索 +fn dfs(i: usize) i32 { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 or i == 2) { + return @intCast(i); + } + // dp[i] = dp[i-1] + dp[i-2] + var count = dfs(i - 1) + dfs(i - 2); + return count; +} + +// 爬楼梯:搜索 +fn climbing_stairs_dfs(comptime n: usize) i32 { + return dfs(n); +} + +// Driver Code +pub fn main() !void { + comptime var n: usize = 9; + + var res = climbing_stairs_dfs(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs_mem.zig b/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs_mem.zig new file mode 100644 index 000000000..e29d1a29d --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/climbing_stairs_dfs_mem.zig @@ -0,0 +1,39 @@ +// File: climbing_stairs_dfs_mem.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 记忆化搜索 +fn dfs(i: usize, mem: []i32) i32 { + // 已知 dp[1] 和 dp[2] ,返回之 + if (i == 1 or i == 2) { + return @intCast(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] + var 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 +fn climbing_stairs_dfs_mem(comptime n: usize) i32 { + // mem[i] 记录爬到第 i 阶的方案总数,-1 代表无记录 + var mem = [_]i32{ -1 } ** (n + 1); + return dfs(n, &mem); +} + +// Driver Code +pub fn main() !void { + comptime var n: usize = 9; + + var res = climbing_stairs_dfs_mem(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/climbing_stairs_dp.zig b/codes/zig/chapter_dynamic_programming/climbing_stairs_dp.zig new file mode 100644 index 000000000..49641273a --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/climbing_stairs_dp.zig @@ -0,0 +1,51 @@ +// File: climbing_stairs_dp.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 爬楼梯:动态规划 +fn climbing_stairs_dp(comptime n: usize) i32 { + // 已知 dp[1] 和 dp[2] ,返回之 + if (n == 1 or n == 2) { + return @intCast(n); + } + // 初始化 dp 表,用于存储子问题的解 + var dp = [_]i32{-1} ** (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 (3..n + 1) |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 +fn climbing_stairs_dp_comp(comptime n: usize) i32 { + if (n == 1 or n == 2) { + return @intCast(n); + } + var a: i32 = 1; + var b: i32 = 2; + for (3..n + 1) |_| { + var tmp = b; + b = a + b; + a = tmp; + } + return b; +} + +// Driver Code +pub fn main() !void { + comptime var n: usize = 9; + + var res = climbing_stairs_dp(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + res = climbing_stairs_dp_comp(n); + std.debug.print("爬 {} 阶楼梯共有 {} 种方案\n", .{ n, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/coin_change.zig b/codes/zig/chapter_dynamic_programming/coin_change.zig new file mode 100644 index 000000000..280237c6d --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/coin_change.zig @@ -0,0 +1,77 @@ +// File: coin_change.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 零钱兑换:动态规划 +fn coin_change_dp(comptime coins: []i32, comptime amt: usize) i32 { + comptime var n = coins.len; + comptime var max = amt + 1; + // 初始化 dp 表 + var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1); + // 状态转移:首行首列 + for (1..amt + 1) |a| { + dp[0][a] = max; + } + // 状态转移:其余行列 + for (1..n + 1) |i| { + for (1..amt + 1) |a| { + if (coins[i - 1] > @as(i32, @intCast(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 - @as(usize, @intCast(coins[i - 1]))] + 1); + } + } + } + if (dp[n][amt] != max) { + return @intCast(dp[n][amt]); + } else { + return -1; + } +} + +// 零钱兑换:状态压缩后的动态规划 +fn coin_change_dp_comp(comptime coins: []i32, comptime amt: usize) i32 { + comptime var n = coins.len; + comptime var max = amt + 1; + // 初始化 dp 表 + var dp = [_]i32{0} ** (amt + 1); + @memset(&dp, max); + dp[0] = 0; + // 状态转移 + for (1..n + 1) |i| { + for (1..amt + 1) |a| { + if (coins[i - 1] > @as(i32, @intCast(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 - @as(usize, @intCast(coins[i - 1]))] + 1); + } + } + } + if (dp[amt] != max) { + return @intCast(dp[amt]); + } else { + return -1; + } +} + +// Driver Code +pub fn main() !void { + comptime var coins = [_]i32{ 1, 2, 5 }; + comptime var amt: usize = 4; + + // 动态规划 + var res = coin_change_dp(&coins, amt); + std.debug.print("凑到目标金额所需的最少硬币数量为 {}\n", .{res}); + + // 状态压缩后的动态规划 + res = coin_change_dp_comp(&coins, amt); + std.debug.print("凑到目标金额所需的最少硬币数量为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/coin_change_ii.zig b/codes/zig/chapter_dynamic_programming/coin_change_ii.zig new file mode 100644 index 000000000..f556db1f0 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/coin_change_ii.zig @@ -0,0 +1,66 @@ +// File: coin_change_ii.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 零钱兑换 II:动态规划 +fn coin_change_ii_dp(comptime coins: []i32, comptime amt: usize) i32 { + comptime var n = coins.len; + // 初始化 dp 表 + var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1); + // 初始化首列 + for (0..n + 1) |i| { + dp[i][0] = 1; + } + // 状态转移 + for (1..n + 1) |i| { + for (1..amt + 1) |a| { + if (coins[i - 1] > @as(i32, @intCast(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] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))]; + } + } + } + return dp[n][amt]; +} + +// 零钱兑换 II:状态压缩后的动态规划 +fn coin_change_dp_ii_comp(comptime coins: []i32, comptime amt: usize) i32 { + comptime var n = coins.len; + // 初始化 dp 表 + var dp = [_]i32{0} ** (amt + 1); + dp[0] = 1; + // 状态转移 + for (1..n + 1) |i| { + for (1..amt + 1) |a| { + if (coins[i - 1] > @as(i32, @intCast(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] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))]; + } + } + } + return dp[amt]; +} + +// Driver Code +pub fn main() !void { + comptime var coins = [_]i32{ 1, 2, 5 }; + comptime var amt: usize = 5; + + // 动态规划 + var res = coin_change_ii_dp(&coins, amt); + std.debug.print("凑出目标金额的硬币组合数量为 {}\n", .{res}); + + // 状态压缩后的动态规划 + res = coin_change_dp_ii_comp(&coins, amt); + std.debug.print("凑出目标金额的硬币组合数量为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/edit_distance.zig b/codes/zig/chapter_dynamic_programming/edit_distance.zig new file mode 100644 index 000000000..0adbb95d8 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/edit_distance.zig @@ -0,0 +1,146 @@ +// File: edit_distance.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 编辑距离:暴力搜索 +fn edit_distance_dfs(comptime s: []const u8, comptime t: []const u8, i: usize, j: usize) i32 { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 and j == 0) { + return 0; + } + // 若 s 为空,则返回 t 长度 + if (i == 0) { + return @intCast(j); + } + // 若 t 为空,则返回 s 长度 + if (j == 0) { + return @intCast(i); + } + // 若两字符相等,则直接跳过此两字符 + if (s[i - 1] == t[j - 1]) { + return edit_distance_dfs(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 + var insert = edit_distance_dfs(s, t, i, j - 1); + var delete = edit_distance_dfs(s, t, i - 1, j); + var replace = edit_distance_dfs(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 +fn edit_distance_dfs_mem(comptime s: []const u8, comptime t: []const u8, mem: anytype, i: usize, j: usize) i32 { + // 若 s 和 t 都为空,则返回 0 + if (i == 0 and j == 0) { + return 0; + } + // 若 s 为空,则返回 t 长度 + if (i == 0) { + return @intCast(j); + } + // 若 t 为空,则返回 s 长度 + if (j == 0) { + return @intCast(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 edit_distance_dfs_mem(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 + var insert = edit_distance_dfs_mem(s, t, mem, i, j - 1); + var delete = edit_distance_dfs_mem(s, t, mem, i - 1, j); + var replace = edit_distance_dfs_mem(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 +fn edit_distance_dp(comptime s: []const u8, comptime t: []const u8) i32 { + comptime var n = s.len; + comptime var m = t.len; + var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1); + // 状态转移:首行首列 + for (1..n + 1) |i| { + dp[i][0] = @intCast(i); + } + for (1..m + 1) |j| { + dp[0][j] = @intCast(j); + } + // 状态转移:其余行列 + for (1..n + 1) |i| { + for (1..m + 1) |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 +fn edit_distance_dp_comp(comptime s: []const u8, comptime t: []const u8) i32 { + comptime var n = s.len; + comptime var m = t.len; + var dp = [_]i32{0} ** (m + 1); + // 状态转移:首行 + for (1..m + 1) |j| { + dp[j] = @intCast(j); + } + // 状态转移:其余行 + for (1..n + 1) |i| { + // 状态转移:首列 + var leftup = dp[0]; // 暂存 dp[i-1, j-1] + dp[0] = @intCast(i); + // 状态转移:其余列 + for (1..m + 1) |j| { + var 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 +pub fn main() !void { + const s = "bag"; + const t = "pack"; + comptime var n = s.len; + comptime var m = t.len; + + // 暴力搜索 + var res = edit_distance_dfs(s, t, n, m); + std.debug.print("将 {s} 更改为 {s} 最少需要编辑 {} 步\n", .{ s, t, res }); + + // 记忆搜索 + var mem = [_][m + 1]i32{[_]i32{-1} ** (m + 1)} ** (n + 1); + res = edit_distance_dfs_mem(s, t, @constCast(&mem), n, m); + std.debug.print("将 {s} 更改为 {s} 最少需要编辑 {} 步\n", .{ s, t, res }); + + // 动态规划 + res = edit_distance_dp(s, t); + std.debug.print("将 {s} 更改为 {s} 最少需要编辑 {} 步\n", .{ s, t, res }); + + // 状态压缩后的动态规划 + res = edit_distance_dp_comp(s, t); + std.debug.print("将 {s} 更改为 {s} 最少需要编辑 {} 步\n", .{ s, t, res }); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/knapsack.zig b/codes/zig/chapter_dynamic_programming/knapsack.zig new file mode 100644 index 000000000..e591e4a47 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/knapsack.zig @@ -0,0 +1,110 @@ +// File: knapsack.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 0-1 背包:暴力搜索 +fn knapsack_dfs(wgt: []i32, val: []i32, i: usize, c: usize) i32 { + // 若已选完所有物品或背包无容量,则返回价值 0 + if (i == 0 or 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 knapsack_dfs(wgt, val, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + var no = knapsack_dfs(wgt, val, i - 1, c); + var yes = knapsack_dfs(wgt, val, i - 1, c - @as(usize, @intCast(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 +fn knapsack_dfs_mem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 { + // 若已选完所有物品或背包无容量,则返回价值 0 + if (i == 0 or 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 knapsack_dfs_mem(wgt, val, mem, i - 1, c); + } + // 计算不放入和放入物品 i 的最大价值 + var no = knapsack_dfs_mem(wgt, val, mem, i - 1, c); + var yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - @as(usize, @intCast(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 +fn knapsack_dp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 { + comptime var n = wgt.len; + // 初始化 dp 表 + var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1); + // 状态转移 + for (1..n + 1) |i| { + for (1..cap + 1) |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 - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]); + } + } + } + return dp[n][cap]; +} + +// 0-1 背包:状态压缩后的动态规划 +fn knapsack_dp_comp(wgt: []i32, val: []i32, comptime cap: usize) i32 { + var n = wgt.len; + // 初始化 dp 表 + var dp = [_]i32{0} ** (cap + 1); + // 状态转移 + for (1..n + 1) |i| { + // 倒序遍历 + var c = cap; + while (c > 0) : (c -= 1) { + 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 - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]); + } + } + } + return dp[cap]; +} + +// Driver Code +pub fn main() !void { + comptime var wgt = [_]i32{ 10, 20, 30, 40, 50 }; + comptime var val = [_]i32{ 50, 120, 150, 210, 240 }; + comptime var cap = 50; + comptime var n = wgt.len; + + // 暴力搜索 + var res = knapsack_dfs(&wgt, &val, n, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + // 记忆搜索 + var mem = [_][cap + 1]i32{[_]i32{-1} ** (cap + 1)} ** (n + 1); + res = knapsack_dfs_mem(&wgt, &val, @constCast(&mem), n, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + // 动态规划 + res = knapsack_dp(&wgt, &val, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + // 状态压缩后的动态规划 + res = knapsack_dp_comp(&wgt, &val, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/min_cost_climbing_stairs_dp.zig b/codes/zig/chapter_dynamic_programming/min_cost_climbing_stairs_dp.zig new file mode 100644 index 000000000..29bdb746e --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/min_cost_climbing_stairs_dp.zig @@ -0,0 +1,54 @@ +// File: min_cost_climbing_stairs_dp.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 爬楼梯最小代价:动态规划 +fn min_cost_climbing_stairs_dp(comptime cost: []i32) i32 { + comptime var n = cost.len - 1; + if (n == 1 or n == 2) { + return cost[n]; + } + // 初始化 dp 表,用于存储子问题的解 + var dp = [_]i32{-1} ** (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 (3..n + 1) |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 +fn min_cost_climbing_stairs_dp_comp(cost: []i32) i32 { + var n = cost.len - 1; + if (n == 1 or n == 2) { + return cost[n]; + } + var a = cost[1]; + var b = cost[2]; + // 状态转移:从较小子问题逐步求解较大子问题 + for (3..n + 1) |i| { + var tmp = b; + b = @min(a, tmp) + cost[i]; + a = tmp; + } + return b; +} + +// Driver Code +pub fn main() !void { + comptime var cost = [_]i32{ 0, 1, 10, 1, 1, 1, 10, 1, 1, 10, 1 }; + std.debug.print("输入楼梯的代价列表为 {any}\n", .{cost}); + + var res = min_cost_climbing_stairs_dp(&cost); + std.debug.print("输入楼梯的代价列表为 {}\n", .{res}); + + res = min_cost_climbing_stairs_dp_comp(&cost); + std.debug.print("输入楼梯的代价列表为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/min_path_sum.zig b/codes/zig/chapter_dynamic_programming/min_path_sum.zig new file mode 100644 index 000000000..4efb35fde --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/min_path_sum.zig @@ -0,0 +1,122 @@ +// File: min_path_sum.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 最小路径和:暴力搜索 +fn min_path_sum_dfs(grid: anytype, i: i32, j: i32) i32 { + // 若为左上角单元格,则终止搜索 + if (i == 0 and j == 0) { + return grid[0][0]; + } + // 若行列索引越界,则返回 +∞ 代价 + if (i < 0 or j < 0) { + return std.math.maxInt(i32); + } + // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价 + var left = min_path_sum_dfs(grid, i - 1, j); + var up = min_path_sum_dfs(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[@as(usize, @intCast(i))][@as(usize, @intCast(j))]; +} + +// 最小路径和:记忆化搜索 +fn min_path_sum_dfs_mem(grid: anytype, mem: anytype, i: i32, j: i32) i32 { + // 若为左上角单元格,则终止搜索 + if (i == 0 and j == 0) { + return grid[0][0]; + } + // 若行列索引越界,则返回 +∞ 代价 + if (i < 0 or j < 0) { + return std.math.maxInt(i32); + } + // 若已有记录,则直接返回 + if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) { + return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))]; + } + // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价 + var left = min_path_sum_dfs_mem(grid, mem, i - 1, j); + var up = min_path_sum_dfs_mem(grid, mem, i, j - 1); + // 返回从左上角到 (i, j) 的最小路径代价 + // 记录并返回左上角到 (i, j) 的最小路径代价 + mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))]; + return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))]; +} + +// 最小路径和:动态规划 +fn min_path_sum_dp(comptime grid: anytype) i32 { + comptime var n = grid.len; + comptime var m = grid[0].len; + // 初始化 dp 表 + var dp = [_][m]i32{[_]i32{0} ** m} ** n; + dp[0][0] = grid[0][0]; + // 状态转移:首行 + for (1..m) |j| { + dp[0][j] = dp[0][j - 1] + grid[0][j]; + } + // 状态转移:首列 + for (1..n) |i| { + dp[i][0] = dp[i - 1][0] + grid[i][0]; + } + // 状态转移:其余行列 + for (1..n) |i| { + for (1..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 +fn min_path_sum_dp_comp(comptime grid: anytype) i32 { + comptime var n = grid.len; + comptime var m = grid[0].len; + // 初始化 dp 表 + var dp = [_]i32{0} ** m; + // 状态转移:首行 + dp[0] = grid[0][0]; + for (1..m) |j| { + dp[j] = dp[j - 1] + grid[0][j]; + } + // 状态转移:其余行 + for (1..n) |i| { + // 状态转移:首列 + dp[0] = dp[0] + grid[i][0]; + for (1..m) |j| { + dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j]; + } + } + return dp[m - 1]; +} + +// Driver Code +pub fn main() !void { + comptime var grid = [_][4]i32{ + [_]i32{ 1, 3, 1, 5 }, + [_]i32{ 2, 2, 4, 2 }, + [_]i32{ 5, 3, 2, 1 }, + [_]i32{ 4, 3, 5, 2 }, + }; + comptime var n = grid.len; + comptime var m = grid[0].len; + + // 暴力搜索 + var res = min_path_sum_dfs(&grid, n - 1, m - 1); + std.debug.print("从左上角到右下角的最小路径和为 {}\n", .{res}); + + // 记忆化搜索 + var mem = [_][m]i32{[_]i32{-1} ** m} ** n; + res = min_path_sum_dfs_mem(&grid, &mem, n - 1, m - 1); + std.debug.print("从左上角到右下角的最小路径和为 {}\n", .{res}); + + // 动态规划 + res = min_path_sum_dp(&grid); + std.debug.print("从左上角到右下角的最小路径和为 {}\n", .{res}); + + // 状态压缩后的动态规划 + res = min_path_sum_dp_comp(&grid); + std.debug.print("从左上角到右下角的最小路径和为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +} diff --git a/codes/zig/chapter_dynamic_programming/unbounded_knapsack.zig b/codes/zig/chapter_dynamic_programming/unbounded_knapsack.zig new file mode 100644 index 000000000..855d25048 --- /dev/null +++ b/codes/zig/chapter_dynamic_programming/unbounded_knapsack.zig @@ -0,0 +1,62 @@ +// File: unbounded_knapsack.zig +// Created Time: 2023-07-15 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); + +// 完全背包:动态规划 +fn unbounded_knapsack_dp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 { + comptime var n = wgt.len; + // 初始化 dp 表 + var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1); + // 状态转移 + for (1..n + 1) |i| { + for (1..cap + 1) |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 - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]); + } + } + } + return dp[n][cap]; +} + +// 完全背包:状态压缩后的动态规划 +fn unbounded_knapsack_dp_comp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 { + comptime var n = wgt.len; + // 初始化 dp 表 + var dp = [_]i32{0} ** (cap + 1); + // 状态转移 + for (1..n + 1) |i| { + for (1..cap + 1) |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 - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]); + } + } + } + return dp[cap]; +} + +// Driver Code +pub fn main() !void { + comptime var wgt = [_]i32{ 1, 2, 3 }; + comptime var val = [_]i32{ 5, 11, 15 }; + comptime var cap = 4; + + // 动态规划 + var res = unbounded_knapsack_dp(&wgt, &val, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + // 状态压缩后的动态规划 + res = unbounded_knapsack_dp_comp(&wgt, &val, cap); + std.debug.print("不超过背包容量的最大物品价值为 {}\n", .{res}); + + _ = try std.io.getStdIn().reader().readByte(); +}