Python 为什么将 DP 添加到我的递归中会使其停止正常工作

谁能告诉我为什么这两个代码给出不同的输出.它们不是具有两种不同实现的相同代码吗(一种是强力实现，另一种是使用动态编程).

程序描述:给出一个用非负数填充的m x n网格，找到一条从左上角到右下角的路径，它使路径上所有数字的和最小化.

def minPathSum_dp(grid):
        def dfs(i, j, pathSum):
            if (i, j) in dp:
                return dp[(i, j)]
            
            if i == len(grid)-1 and j == len(grid[i])-1:
                return pathSum + grid[i][j]

            if i == len(grid)-1:
                return dfs(i, j+1, pathSum + grid[i][j])
            
            if j == len(grid[i])-1:
                return dfs(i+1, j, pathSum + grid[i][j])
            
            path1 = dfs(i+1, j, pathSum + grid[i][j])
            path2 = dfs(i, j+1, pathSum + grid[i][j])
            dp[(i, j)] = min(path1, path2)
            return dp[(i, j)]
        
        dp = {}
        return dfs(0, 0, 0) 

def minPathSum_bf(grid):
        def dfs(i, j, pathSum):          
            if i == len(grid)-1 and j == len(grid[i])-1:
                return pathSum + grid[i][j]
            
            if i == len(grid)-1:
                return dfs(i, j+1, pathSum + grid[i][j])
            
            if j == len(grid[i])-1:
                return dfs(i+1, j, pathSum + grid[i][j])
            
            path1 = dfs(i+1, j, pathSum + grid[i][j])
            path2 = dfs(i, j+1, pathSum + grid[i][j])
            return min(path1, path2)
        
        return dfs(0, 0, 0) 

grid = [[1,4,8,6,2,2,1,7],[4,7,3,1,4,5,5,1],[8,8,2,1,1,8,0,1],[8,9,2,9,8,0,8,9],[5,7,5,7,1,8,5,5],[7,0,9,4,5,6,5,6],[4,9,9,7,9,1,9,0]]

UPDATED: WORKING SOLUTION

def minPathSum(self, grid: List[List[int]]) -> int:
        def dfs(i, j):
            if (i, j) in dp:
                return dp[(i, j)]
            
            if i == len(grid)-1 and j == len(grid[i])-1:
                return grid[i][j]
            
            if i == len(grid)-1:
                dp[(i, j)] = grid[i][j] + dfs(i, j+1)
                return dp[(i, j)]
            
            if j == len(grid[i])-1:
                dp[(i, j)] = grid[i][j] + dfs(i+1, j)
                return dp[(i, j)]
            
            path1 = grid[i][j] + dfs(i+1, j)
            path2 = grid[i][j] + dfs(i, j+1)
            dp[(i, j)] = min(path1, path2)
            return dp[(i, j)]
        
        dp = {}
        return dfs(0, 0)

print(minPathSum_dp(grid))
print(minPathSum_bf(grid))