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95 lines
2.4 KiB
95 lines
2.4 KiB
// File: min_path_sum.go
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// Created Time: 2023-07-23
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// Author: Reanon (793584285@qq.com)
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package chapter_dynamic_programming
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import "math"
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/* 最小路徑和:暴力搜尋 */
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func minPathSumDFS(grid [][]int, i, j int) int {
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// 若為左上角單元格,則終止搜尋
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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,則返回 +∞ 代價
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if i < 0 || j < 0 {
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return math.MaxInt
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}
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// 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
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up := minPathSumDFS(grid, i-1, j)
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left := minPathSumDFS(grid, i, j-1)
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// 返回從左上角到 (i, j) 的最小路徑代價
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return int(math.Min(float64(left), float64(up))) + grid[i][j]
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}
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/* 最小路徑和:記憶化搜尋 */
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func minPathSumDFSMem(grid, mem [][]int, i, j int) int {
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// 若為左上角單元格,則終止搜尋
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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,則返回 +∞ 代價
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if i < 0 || j < 0 {
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return math.MaxInt
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}
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// 若已有記錄,則直接返回
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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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// 左邊和上邊單元格的最小路徑代價
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up := minPathSumDFSMem(grid, mem, i-1, j)
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left := minPathSumDFSMem(grid, mem, i, j-1)
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// 記錄並返回左上角到 (i, j) 的最小路徑代價
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mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]
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return mem[i][j]
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}
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/* 最小路徑和:動態規劃 */
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func minPathSumDP(grid [][]int) int {
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n, m := len(grid), len(grid[0])
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// 初始化 dp 表
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dp := make([][]int, n)
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for i := 0; i < n; i++ {
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dp[i] = make([]int, m)
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}
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dp[0][0] = grid[0][0]
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// 狀態轉移:首行
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for 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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// 狀態轉移:首列
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for 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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// 狀態轉移:其餘行和列
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for i := 1; i < n; i++ {
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for j := 1; j < m; j++ {
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dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(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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/* 最小路徑和:空間最佳化後的動態規劃 */
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func minPathSumDPComp(grid [][]int) int {
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n, m := len(grid), len(grid[0])
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// 初始化 dp 表
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dp := make([]int, m)
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// 狀態轉移:首行
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dp[0] = grid[0][0]
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for 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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// 狀態轉移:其餘行和列
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for i := 1; i < n; i++ {
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// 狀態轉移:首列
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dp[0] = dp[0] + grid[i][0]
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// 狀態轉移:其餘列
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for j := 1; j < m; j++ {
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dp[j] = int(math.Min(float64(dp[j-1]), float64(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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