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105 lines
3.2 KiB
105 lines
3.2 KiB
"""
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File: min_path_sum.py
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Created Time: 2023-07-04
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Author: krahets (krahets@163.com)
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"""
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from math import inf
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def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -> int:
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"""最小路径和：暴力搜索"""
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# 若为左上角单元格，则终止搜索
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if i == 0 and j == 0:
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return grid[0][0]
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# 若行列索引越界，则返回 +∞ 代价
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if i < 0 or j < 0:
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return inf
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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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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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# 返回从左上角到 (i, j) 的最小路径代价
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return min(left, up) + grid[i][j]
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def min_path_sum_dfs_mem(
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grid: list[list[int]], mem: list[list[int]], i: int, j: int
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) -> int:
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"""最小路径和：记忆化搜索"""
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# 若为左上角单元格，则终止搜索
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if i == 0 and j == 0:
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return grid[0][0]
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# 若行列索引越界，则返回 +∞ 代价
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if i < 0 or j < 0:
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return inf
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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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# 左边和上边单元格的最小路径代价
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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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# 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] = min(left, up) + grid[i][j]
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return mem[i][j]
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def min_path_sum_dp(grid: list[list[int]]) -> int:
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"""最小路径和：动态规划"""
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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 = [[0] * m for _ in range(n)]
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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 j in range(1, m):
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dp[0][j] = dp[0][j - 1] + grid[0][j]
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# 状态转移：首列
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for i in range(1, n):
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dp[i][0] = dp[i - 1][0] + grid[i][0]
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# 状态转移：其余行和列
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for i in range(1, n):
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for j in range(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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return dp[n - 1][m - 1]
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def min_path_sum_dp_comp(grid: list[list[int]]) -> int:
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"""最小路径和：空间优化后的动态规划"""
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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 = [0] * m
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# 状态转移：首行
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dp[0] = grid[0][0]
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for j in range(1, m):
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dp[j] = dp[j - 1] + grid[0][j]
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# 状态转移：其余行
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for i in range(1, n):
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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 j in range(1, m):
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dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
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return dp[m - 1]
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"""Driver Code"""
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if __name__ == "__main__":
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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 = len(grid), len(grid[0])
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# 暴力搜索
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res = min_path_sum_dfs(grid, n - 1, m - 1)
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print(f"从左上角到右下角的最小路径和为 {res}")
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# 记忆化搜索
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mem = [[-1] * m for _ in range(n)]
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res = min_path_sum_dfs_mem(grid, mem, n - 1, m - 1)
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print(f"从左上角到右下角的最小路径和为 {res}")
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# 动态规划
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res = min_path_sum_dp(grid)
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print(f"从左上角到右下角的最小路径和为 {res}")
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# 空间优化后的动态规划
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res = min_path_sum_dp_comp(grid)
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print(f"从左上角到右下角的最小路径和为 {res}")
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