124 lines
3.6 KiB
Swift
124 lines
3.6 KiB
Swift
/**
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* File: min_path_sum.swift
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* Created Time: 2023-07-15
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* Author: nuomi1 (nuomi1@qq.com)
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*/
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/* 最小路徑和：暴力搜尋 */
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func minPathSumDFS(grid: [[Int]], i: Int, 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 .max
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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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let up = minPathSumDFS(grid: grid, i: i - 1, j: j)
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let left = minPathSumDFS(grid: grid, i: i, j: 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 min(left, 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: [[Int]], mem: inout [[Int]], i: Int, 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 .max
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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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let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)
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let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: 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] = min(left, up) + grid[i][j]
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return mem[i][j]
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}
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/* 最小路徑和：動態規劃 */
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func minPathSumDP(grid: [[Int]]) -> Int {
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let n = grid.count
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let m = grid[0].count
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// 初始化 dp 表
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var dp = Array(repeating: Array(repeating: 0, count: m), count: n)
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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 in 1 ..< m {
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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 in 1 ..< n {
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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 in 1 ..< n {
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for j in 1 ..< m {
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dp[i][j] = 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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let n = grid.count
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let m = grid[0].count
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// 初始化 dp 表
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var dp = Array(repeating: 0, count: 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 in 1 ..< m {
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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
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for i in 1 ..< n {
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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 in 1 ..< m {
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dp[j] = 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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@main
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enum MinPathSum {
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/* Driver Code */
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static func main() {
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let grid = [
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[1, 3, 1, 5],
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[2, 2, 4, 2],
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[5, 3, 2, 1],
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[4, 3, 5, 2],
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]
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let n = grid.count
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let m = grid[0].count
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// 暴力搜尋
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var res = minPathSumDFS(grid: grid, i: n - 1, j: m - 1)
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print("從左上角到右下角的做小路徑和為 \(res)")
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// 記憶化搜尋
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var mem = Array(repeating: Array(repeating: -1, count: m), count: n)
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res = minPathSumDFSMem(grid: grid, mem: &mem, i: n - 1, j: m - 1)
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print("從左上角到右下角的做小路徑和為 \(res)")
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// 動態規劃
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res = minPathSumDP(grid: grid)
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print("從左上角到右下角的做小路徑和為 \(res)")
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// 空間最佳化後的動態規劃
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res = minPathSumDPComp(grid: grid)
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print("從左上角到右下角的做小路徑和為 \(res)")
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}
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}
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