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3988. Create Grid With Exactly K Paths I

Description

You are given three integers m, n, and k.

Construct any m x n grid consisting only of the characters '.' and '#', where:

  • '.' represents a free cell.
  • '#' represents an obstacle cell.

A valid path is a sequence of free cells that:

  • Starts at the top-left cell (0, 0).
  • Ends at the bottom-right cell (m - 1, n - 1).
  • Moves only:
    • Right, from (i, j) to (i, j + 1), or
    • Down, from (i, j) to (i + 1, j).

Return any grid such that there are exactly k valid paths from the top-left cell to the bottom-right cell. If no such grid exists, return an empty array.

Β 

Example 1:

Input: m = 2, n = 3, k = 2

Output: ["...","#.."]

Explanation:

There are exactly k = 2 valid paths from (0, 0) to (1, 2):

  • (0, 0) β†’ (0, 1) β†’ (0, 2) β†’ (1, 2)
  • (0, 0) β†’ (0, 1) β†’ (1, 1) β†’ (1, 2)

Example 2:

Input: m = 3, n = 3, k = 4

Output: ["..#","...","#.."]

Explanation:

There are exactly k = 4 valid paths from (0, 0) to (2, 2):

  • (0, 0) β†’ (0, 1) β†’ (1, 1) β†’ (1, 2) β†’ (2, 2)
  • (0, 0) β†’ (0, 1) β†’ (1, 1) β†’ (2, 1) β†’ (2, 2)
  • (0, 0) β†’ (1, 0) β†’ (1, 1) β†’ (1, 2) β†’ (2, 2)
  • (0, 0) β†’ (1, 0) β†’ (1, 1) β†’ (2, 1) β†’ (2, 2)

Example 3:

Input: m = 1, n = 4, k = 2

Output: []

Explanation:​

No grid exists with exactly k = 2 valid paths for a 1 x 4 grid, so the answer is an empty array.

Β 

Constraints:

  • 1 <= m, n <= 10
  • 1 <= k <= 4

Solutions

Solution 1

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