1368. Minimum Cost to Make at Least One Valid Path in a Grid

Hard

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15

Given an `m x n`

grid. Each cell of the grid has a sign pointing to the next cell you should visit if you are currently in this cell. The sign of `grid[i][j]`

can be:

`1`

which means go to the cell to the right. (i.e go from`grid[i][j]`

to`grid[i][j + 1]`

)`2`

which means go to the cell to the left. (i.e go from`grid[i][j]`

to`grid[i][j - 1]`

)`3`

which means go to the lower cell. (i.e go from`grid[i][j]`

to`grid[i + 1][j]`

)`4`

which means go to the upper cell. (i.e go from`grid[i][j]`

to`grid[i - 1][j]`

)

Notice that there could be some signs on the cells of the grid that point outside the grid.

You will initially start at the upper left cell `(0, 0)`

. A valid path in the grid is a path that starts from the upper left cell `(0, 0)`

and ends at the bottom-right cell `(m - 1, n - 1)`

following the signs on the grid. The valid path does not have to be the shortest.

You can modify the sign on a cell with `cost = 1`

. You can modify the sign on a cell **one time only**.

Return *the minimum cost to make the grid have at least one valid path*.

**Example 1:**

Input:grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]Output:3Explanation:You will start at point (0, 0). The path to (3, 3) is as follows. (0, 0) --> (0, 1) --> (0, 2) --> (0, 3) change the arrow to down with cost = 1 --> (1, 3) --> (1, 2) --> (1, 1) --> (1, 0) change the arrow to down with cost = 1 --> (2, 0) --> (2, 1) --> (2, 2) --> (2, 3) change the arrow to down with cost = 1 --> (3, 3) The total cost = 3.

**Example 2:**

Input:grid = [[1,1,3],[3,2,2],[1,1,4]]Output:0Explanation:You can follow the path from (0, 0) to (2, 2).

**Example 3:**

Input:grid = [[1,2],[4,3]]Output:1

**Constraints:**

`m == grid.length`

`n == grid[i].length`

`1 <= m, n <= 100`

`1 <= grid[i][j] <= 4`

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