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127 lines
4.1 KiB
C#
127 lines
4.1 KiB
C#
/**
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* File: min_path_sum.cs
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* Created Time: 2023-07-10
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* Author: hpstory (hpstory1024@163.com)
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*/
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namespace hello_algo.chapter_dynamic_programming;
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public class min_path_sum {
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/* 最小路径和:暴力搜索 */
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public int MinPathSumDFS(int[][] grid, int i, int j) {
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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 int.MaxValue;
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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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int up = MinPathSumDFS(grid, i - 1, j);
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int left = MinPathSumDFS(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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return Math.Min(left, up) + grid[i][j];
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}
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/* 最小路径和:记忆化搜索 */
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public int MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {
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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 int.MaxValue;
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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][j] != -1) {
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return mem[i][j];
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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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int up = MinPathSumDFSMem(grid, mem, i - 1, j);
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int left = MinPathSumDFSMem(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][j] = Math.Min(left, up) + grid[i][j];
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return mem[i][j];
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}
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/* 最小路径和:动态规划 */
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public int MinPathSumDP(int[][] grid) {
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int n = grid.Length, m = grid[0].Length;
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// 初始化 dp 表
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int[,] dp = new int[n, m];
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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 (int j = 1; j < m; j++) {
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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 (int i = 1; i < n; i++) {
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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:其余行列
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for (int i = 1; i < n; i++) {
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for (int j = 1; j < m; j++) {
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dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];
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}
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}
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return 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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public int MinPathSumDPComp(int[][] grid) {
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int n = grid.Length, m = grid[0].Length;
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// 初始化 dp 表
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int[] dp = new int[m];
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dp[0] = grid[0][0];
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// 状态转移:首行
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for (int j = 1; j < m; j++) {
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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 (int i = 1; i < n; i++) {
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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 (int j = 1; j < m; j++) {
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dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];
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}
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}
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return dp[m - 1];
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}
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[Test]
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public void Test() {
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int[][] grid =
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{
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new int[4] { 1, 3, 1, 5 },
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new int[4] { 2, 2, 4, 2 },
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new int[4] { 5, 3, 2, 1 },
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new int[4] { 4, 3, 5, 2 }
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};
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int n = grid.Length, m = grid[0].Length;
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// 暴力搜索
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int res = MinPathSumDFS(grid, n - 1, m - 1);
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Console.WriteLine("从左上角到右下角的做小路径和为 " + res);
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// 记忆化搜索
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int[][] mem = new int[n][];
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for (int i = 0; i < n; i++) {
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mem[i] = new int[m];
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Array.Fill(mem[i], -1);
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}
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res = MinPathSumDFSMem(grid, mem, n - 1, m - 1);
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Console.WriteLine("从左上角到右下角的做小路径和为 " + res);
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// 动态规划
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res = MinPathSumDP(grid);
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Console.WriteLine("从左上角到右下角的做小路径和为 " + res);
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// 空间优化后的动态规划
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res = MinPathSumDPComp(grid);
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Console.WriteLine("从左上角到右下角的做小路径和为 " + res);
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}
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}
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