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93 lines
2.9 KiB
Ruby
93 lines
2.9 KiB
Ruby
=begin
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File: min_path_sum.rb
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Created Time: 2024-05-29
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Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)
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=end
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### 最小路徑和：暴力搜尋 ###
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def min_path_sum_dfs(grid, i, j)
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# 若為左上角單元格，則終止搜尋
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return grid[i][j] if i == 0 && j == 0
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# 若行列索引越界，則返回 +∞ 代價
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return Float::INFINITY if i < 0 || j < 0
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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 = min_path_sum_dfs(grid, i - 1, j)
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left = min_path_sum_dfs(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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[left, up].min + grid[i][j]
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end
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### 最小路徑和：記憶化搜尋 ###
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def min_path_sum_dfs_mem(grid, mem, i, j)
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# 若為左上角單元格，則終止搜尋
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return grid[0][0] if i == 0 && j == 0
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# 若行列索引越界，則返回 +∞ 代價
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return Float::INFINITY if i < 0 || j < 0
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# 若已有記錄，則直接返回
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return mem[i][j] if mem[i][j] != -1
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# 左邊和上邊單元格的最小路徑代價
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up = min_path_sum_dfs_mem(grid, mem, i - 1, j)
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left = min_path_sum_dfs_mem(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] = [left, up].min + grid[i][j]
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end
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### 最小路徑和：動態規劃 ###
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def min_path_sum_dp(grid)
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n, m = grid.length, grid.first.length
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# 初始化 dp 表
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dp = Array.new(n) { Array.new(m, 0) }
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dp[0][0] = grid[0][0]
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# 狀態轉移：首行
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(1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }
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# 狀態轉移：首列
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(1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }
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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] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]
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end
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end
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dp[n -1][m -1]
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end
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### 最小路徑和：空間最佳化後的動態規劃 ###
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def min_path_sum_dp_comp(grid)
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n, m = grid.length, grid.first.length
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# 初始化 dp 表
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dp = Array.new(m, 0)
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# 狀態轉移：首行
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dp[0] = grid[0][0]
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(1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }
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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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(1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }
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end
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dp[m - 1]
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end
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### Driver Code ###
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if __FILE__ == $0
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grid = [[1, 3, 1, 5], [2, 2, 4, 2], [5, 3, 2, 1], [4, 3, 5, 2]]
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n, m = grid.length, grid.first.length
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# 暴力搜尋
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res = min_path_sum_dfs(grid, n - 1, m - 1)
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puts "從左上角到右下角的最小路徑和為 #{res}"
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# 記憶化搜尋
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mem = Array.new(n) { Array.new(m, - 1) }
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res = min_path_sum_dfs_mem(grid, mem, n - 1, m -1)
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puts "從左上角到右下角的最小路徑和為 #{res}"
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# 動態規劃
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res = min_path_sum_dp(grid)
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puts "從左上角到右下角的最小路徑和為 #{res}"
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# 空間最佳化後的動態規劃
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res = min_path_sum_dp_comp(grid)
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puts "從左上角到右下角的最小路徑和為 #{res}"
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end
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