mirror of
https://github.com/krahets/hello-algo.git
synced 2024-12-27 16:26:29 +08:00
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
94 lines
2.4 KiB
Go
94 lines
2.4 KiB
Go
// 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
|
|
up := minPathSumDFS(grid, i-1, j)
|
|
left := 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
|
|
up := minPathSumDFSMem(grid, mem, i-1, j)
|
|
left := 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：其余行和列
|
|
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：空间优化后的动态规划 */
|
|
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：其余行和列
|
|
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]
|
|
}
|