Solution: Minimum Cost to Make at Least One Valid Path in a Grid

Statement

You are given an $m \times n$ grid, where each cell contains a directional sign indicating which neighboring cell to move to next. The sign in a cell grid[i][j] can be:

• 1: Move right, i.e., from grid[i][j] to grid[i][j + 1].

• 2: Move left, i.e., from grid[i][j] to grid[i][j - 1].

• 3: Move down, i.e., from grid[i][j] to grid[i + 1][j].

• 4: Move up, i.e., from grid[i][j] to grid[i - 1][j].

Note: Some signs may point outside the boundaries of the grid.

Your starting position is the top-left cell $(0, 0)$. A valid path is any sequence of moves beginning at $(0,0)$ and ending at the bottom-right cell $(m-1$ ...