mirror of
https://github.com/krahets/hello-algo.git
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e720aa2d24
* Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
120 lines
3.6 KiB
Rust
120 lines
3.6 KiB
Rust
/*
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* File: min_path_sum.rs
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* Created Time: 2023-07-09
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* Author: sjinzh (sjinzh@gmail.com)
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*/
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/* 最小路径和：暴力搜索 */
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fn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
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// 若为左上角单元格，则终止搜索
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if i == 0 && j == 0 {
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return grid[0][0];
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}
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// 若行列索引越界，则返回 +∞ 代价
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if i < 0 || j < 0 {
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return i32::MAX;
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}
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// 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
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let up = min_path_sum_dfs(grid, i - 1, j);
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let left = min_path_sum_dfs(grid, i, j - 1);
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// 返回从左上角到 (i, j) 的最小路径代价
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std::cmp::min(left, up) + grid[i as usize][j as usize]
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}
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/* 最小路径和：记忆化搜索 */
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fn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
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// 若为左上角单元格，则终止搜索
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if i == 0 && j == 0 {
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return grid[0][0];
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}
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// 若行列索引越界，则返回 +∞ 代价
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if i < 0 || j < 0 {
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return i32::MAX;
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}
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// 若已有记录，则直接返回
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if mem[i as usize][j as usize] != -1 {
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return mem[i as usize][j as usize];
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}
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// 左边和上边单元格的最小路径代价
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let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);
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let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);
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// 记录并返回左上角到 (i, j) 的最小路径代价
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mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];
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mem[i as usize][j as usize]
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}
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/* 最小路径和：动态规划 */
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fn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {
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let (n, m) = (grid.len(), grid[0].len());
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// 初始化 dp 表
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let mut dp = vec![vec![0; m]; n];
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dp[0][0] = grid[0][0];
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// 状态转移：首行
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for j in 1..m {
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dp[0][j] = dp[0][j - 1] + grid[0][j];
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}
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// 状态转移：首列
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for i in 1..n {
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dp[i][0] = dp[i - 1][0] + grid[i][0];
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}
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// 状态转移：其余行和列
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for i in 1..n {
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for j in 1..m {
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dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
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}
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}
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dp[n - 1][m - 1]
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}
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/* 最小路径和：空间优化后的动态规划 */
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fn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {
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let (n, m) = (grid.len(), grid[0].len());
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// 初始化 dp 表
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let mut dp = vec![0; m];
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// 状态转移：首行
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dp[0] = grid[0][0];
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for j in 1..m {
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dp[j] = dp[j - 1] + grid[0][j];
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}
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// 状态转移：其余行
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for i in 1..n {
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// 状态转移：首列
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dp[0] = dp[0] + grid[i][0];
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// 状态转移：其余列
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for j in 1..m {
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dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];
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}
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}
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dp[m - 1]
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}
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/* Driver Code */
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pub fn main() {
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let grid = vec![
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vec![ 1, 3, 1, 5 ],
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vec![ 2, 2, 4, 2 ],
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vec![ 5, 3, 2, 1 ],
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vec![ 4, 3, 5, 2 ]];
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let (n, m) = (grid.len(), grid[0].len());
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// 暴力搜索
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let res = min_path_sum_dfs(&grid, n as i32 - 1, m as i32 - 1);
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println!("从左上角到右下角的最小路径和为 {res}");
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// 记忆化搜索
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let mut mem = vec![vec![0; m]; n];
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for row in mem.iter_mut() {
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row.fill(-1);
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}
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let res = min_path_sum_dfs_mem(&grid, &mut mem, n as i32 - 1, m as i32 - 1);
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println!("从左上角到右下角的最小路径和为 {res}");
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// 动态规划
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let res = min_path_sum_dp(&grid);
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println!("从左上角到右下角的最小路径和为 {res}");
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// 空间优化后的动态规划
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let res = min_path_sum_dp_comp(&grid);
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println!("从左上角到右下角的最小路径和为 {res}");
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}
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