rust and zig : add codes for chapter_dynamic_programming (#606)

* rust : add codes for chapter_dynamic_programming

* zig : add codes for chapter_dynamic_programming

* rust : add codes for chapter_backtracking

* Update n_queens.rs

---------

Co-authored-by: Yudong Jin <krahets@163.com>

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<Vec<i32>>, 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<Vec<i32>>, mem: &mut Vec<Vec<i32>>, 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<Vec<i32>>) -> 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<Vec<i32>>) -> 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}");
+}