feat(go/dp): support dynamic programming (#622)

* feat(go/dp): support climbing stairs

* feat(go/dp): support knapsack

* feat(go/dp): coin_change & edit_distance

codes/go/chapter_dynamic_programming/min_path_sum.go

@@ -0,0 +1,94 @@
+// File: min_path_sum.go
+// Created Time: 2023-07-23
+// Author: Reanon (793584285@qq.com)
+
+package chapter_dynamic_programming
+
+import "math"
+
+/* 最小路径和：暴力搜索 */
+func minPathSumDFS(grid [][]int, i, j int) int {
+	// 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 math.MaxInt
+	}
+	// 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
+	left := minPathSumDFS(grid, i-1, j)
+	up := minPathSumDFS(grid, i, j-1)
+	// 返回从左上角到 (i, j) 的最小路径代价
+	return int(math.Min(float64(left), float64(up))) + grid[i][j]
+}
+
+/* 最小路径和：记忆化搜索 */
+func minPathSumDFSMem(grid, mem [][]int, i, j int) int {
+	// 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 math.MaxInt
+	}
+	// 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
+	left := minPathSumDFSMem(grid, mem, i-1, j)
+	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] = int(math.Min(float64(left), float64(up))) + grid[i][j]
+	return mem[i][j]
+}
+
+/* 最小路径和：动态规划 */
+func minPathSumDP(grid [][]int) int {
+	n, m := len(grid), len(grid[0])
+	// 初始化 dp 表
+	dp := make([][]int, n)
+	for i := 0; i < n; i++ {
+		dp[i] = make([]int, m)
+	}
+	dp[0][0] = grid[0][0]
+	// 状态转移：首行
+	for j := 1; j < m; j++ {
+		dp[0][j] = dp[0][j-1] + grid[0][j]
+	}
+	// 状态转移：首列
+	for i := 1; i < n; i++ {
+		dp[i][0] = dp[i-1][0] + grid[i][0]
+	}
+	// 状态转移：其余行列
+	for i := 1; i < n; i++ {
+		for j := 1; j < m; j++ {
+			dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(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 */
+func minPathSumDPComp(grid [][]int) int {
+	n, m := len(grid), len(grid[0])
+	// 初始化 dp 表
+	dp := make([]int, m)
+	// 状态转移：首行
+	dp[0] = grid[0][0]
+	for j := 1; j < m; j++ {
+		dp[j] = dp[j-1] + grid[0][j]
+	}
+	// 状态转移：其余行列
+	for 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 j := 1; j < m; j++ {
+			dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]
+		}
+	}
+	return dp[m-1]
+}