Minimum path cost in a grid

We will use recursion with memoization to solve this problem. The function helper takes the current position (r, c) and the current cost cost. If the current position is the destination, we return the current cost. Otherwise, we iterate over the possible directions and return the minimum cost of the next position.

class Solution:
    def minPathCost(self, grid: List[List[int]], moveCost: List[List[int]]) -> int:
        ROWS, COLS = len(grid), len(grid[0])

        @cache
        def helper(r: int, c: int) -> int:
            if r == 0:
                return grid[r][c]
            else:
                return grid[r][c] + min(moveCost[grid[r-1][k]][c] + helper(r-1, k) for k in range(COLS))

        return min(helper(ROWS-1, c) for c in range(COLS))

Time complexity: O(nm)
Space complexity: O(nm)