# ---
# title: 1368. Minimum Cost to Make at Least One Valid Path in a Grid
# id: problem1368
# author: Tian Jun
# date: 2020-10-31
# difficulty: Hard
# categories: Breadth-first Search
# link: <https://leetcode.com/problems/minimum-cost-to-make-at-least-one-valid-path-in-a-grid/description/>
# hidden: true
# ---
# 
# 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`
# 
# 
## @lc code=start
using LeetCode

## add your code here:
## @lc code=end
