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

https://leetcode.com/problems/minimum-cost-to-make-at-least-one-valid-path-in-a-grid

## Description

Given a _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 **invalid signs** on the cells of the `grid` which points outside the `grid`.

You will initially start at the upper left cell `(0,0)`. A valid path in the grid is a path which 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 **doesn't 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:**

![](https://assets.leetcode.com/uploads/2020/02/13/grid1.png)

```
**Input:** grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]
**Output:** 3
**Explanation:** 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:**

![](https://assets.leetcode.com/uploads/2020/02/13/grid2.png)

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

**Example 3:**

![](https://assets.leetcode.com/uploads/2020/02/13/grid3.png)

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

**Example 4:**

```
**Input:** grid = [[2,2,2],[2,2,2]]
**Output:** 3
```

**Example 5:**

```
**Input:** grid = [[4]]
**Output:** 0
```

**Constraints:**

* `m == grid.length`
* `n == grid[i].length`
* `1 <= m, n <= 100`

## ac

```java
```
