980 - Unique Paths III\.

Hard

You are given an m x n integer array grid where grid[i][j] could be:

• 1 representing the starting square. There is exactly one starting square.

• 2 representing the ending square. There is exactly one ending square.

• 0 representing empty squares we can walk over.

• -1 representing obstacles that we cannot walk over.

Return the number of 4-directional walks from the starting square to the ending square, that walk over every non-obstacle square exactly once.

Example 1:

Input: grid = \[\[1,0,0,0],0,0,0,0,0,0,2,-1]

Output: 2

Explanation: We have the following two paths:

• (0,0),(0,1),(0,2),(0,3),(1,3),(1,2),(1,1),(1,0),(2,0),(2,1),(2,2)

• (0,0),(1,0),(2,0),(2,1),(1,1),(0,1),(0,2),(0,3),(1,3),(1,2),(2,2)

Example 2:

Input: grid = \[\[1,0,0,0],0,0,0,0,0,0,0,2]

Output: 4

Explanation: We have the following four paths:

• (0,0),(0,1),(0,2),(0,3),(1,3),(1,2),(1,1),(1,0),(2,0),(2,1),(2,2),(2,3)

• (0,0),(0,1),(1,1),(1,0),(2,0),(2,1),(2,2),(1,2),(0,2),(0,3),(1,3),(2,3)

• (0,0),(1,0),(2,0),(2,1),(2,2),(1,2),(1,1),(0,1),(0,2),(0,3),(1,3),(2,3)

• (0,0),(1,0),(2,0),(2,1),(1,1),(0,1),(0,2),(0,3),(1,3),(1,2),(2,2),(2,3)

Example 3:

Input: grid = \[\[0,1],2,0]

Output: 0

Explanation: There is no path that walks over every empty square exactly once. Note that the starting and ending square can be anywhere in the grid.

Constraints:

• m == grid.length

• n == grid[i].length

• 1 <= m, n <= 20

• 1 <= m * n <= 20

• -1 <= grid[i][j] <= 2

• There is exactly one starting cell and one ending cell.