在最小成本路径问题中找到解决方案数学

Question

def minCostPath(grid):

    dp = [[0 for i in range(len(grid))] for j in range(len(grid[0]))]
    dp[0][0] = grid[0][0]

    back = {}

    path = []

    for i in range(1, len(grid)):
        dp[0][i] = dp[0][i - 1] + grid[0][i]

    for i in range(1, len(grid[0])):
        dp[i][0] = dp[i - 1][0] + grid[i][0]

    if dp[1][0] < dp[0][1]:
        back[(1, 0)] = (0, 0)
    else:
        back[(0, 1)] = (0, 0)

    for i in range(1, len(grid)):
        for j in range(1, len(grid[0])):
            dp[i][j] = grid[i][j] + min(dp[i - 1][j], dp[i][j - 1])
            if dp[i - 1][j] < dp[i][j - 1]:
                back[(i, j)] = (i - 1, j)
            else:
                back[(i, j)] = (i, j - 1)

    curr = (len(grid) - 1, len(grid[0]) - 1)
    while curr != (0, 0):
        path.append(curr)
        curr = back[curr]

    return dp[len(grid) - 1][len(grid[0]) - 1], reversed(path)

我创建了一个DP解决方案，以查找从网格左上方到网格右下方的最小成本路径。然后，我尝试将所选单元格的前一个单元格存储到字典中，并使用它来计算最后的路径。

但是，我一直在获得

{(0, 1): (0, 0), (1, 1): (0, 1), (1, 2): (1, 1), (2, 1): (2, 0), (2, 2): (2, 1)}

其中grid = [[1, 7, 9], [8, 6, 3], [1, 6, 7]]

对于我的结果，这看起来很奇怪，并且缺少一些单元格。

我的逻辑有什么问题以及如何解决？

我解决的问题是：

Given a 2D-matrix where each cell has a cost to travel. You have to write an algorithm to find a path from left-top corner to bottom-right corner with minimum travel cost. You can move only right or down.

我有钱，但是我没有找到确切的解决方法。