64. Minimum Path Sum -Medium

Question

Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Note: You can only move either down or right at any point in time.

给出一个由非负数组成的 m x n的表格，要求从左上角走到右下角的总和最小的路径。你只允许向下走或向右走

Example

None

Solution

• 动态规划解。这道题很明显从对每个表格选择左和上中较小的路径即可。定义dp[i][j]：从左上角到grid[i][j]的总和最小的路径。递推式为：dp[i][j] = min(dp[i][j - 1], dp[i - 1][j])

  class Solution(object):
      def minPathSum(self, grid):
          """
          :type grid: List[List[int]]
          :rtype: int
          """
          if len(grid) == 0 or len(grid[0]) == 0: return 0
          dp = [[0 for _ in range(len(grid[0]))] for _ in range(len(grid))]
          for index_r in range(len(grid)):
              for index_c in range(len(grid[0])):
                  # 第一行只能从左向右走
                  if index_r == 0:
                      dp[index_r][index_c] = grid[index_r][index_c] + dp[index_r][index_c - 1]
                  # 第一列只能从上向下走
                  elif index_c == 0:
                      dp[index_r][index_c] = grid[index_r][index_c] + dp[index_r - 1][index_c]
                  # 其他选择上和左中较小者
                  else:
                      dp[index_r][index_c] = min(dp[index_r - 1][index_c] + grid[index_r][index_c],
                                                 dp[index_r][index_c - 1] + grid[index_r][index_c])
          return dp[-1][-1]