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Grokking the Coding Interview Patterns

Solution: Minimum Cost to Make at Least One Valid Path in a Grid

Understand how to determine the minimum total cost to form a valid path from the top-left to the bottom-right cell in a grid with directional signs. Explore the use of graph theory and 0-1 BFS to efficiently update costs and find the optimal path by modifying cell directions as needed.

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Statement

You are given an m×nm \times n grid, where each cell contains a directional sign indicating which neighboring cell to move to next. The sign in a cell grid[i][j] can be:

  • 1: Move right, i.e., from grid[i][j] to grid[i][j + 1].

  • 2: Move left, i.e., from grid[i][j] to grid[i][j - 1].

  • 3: Move down, i.e., from grid[i][j] to grid[i + 1][j].

  • 4: Move up, i.e., from grid[i][j] to grid[i - 1][j].

Note: Some signs may point outside the boundaries of the grid.

Your starting position is the top-left cell (0,0)(0, 0). A valid path is any sequence of moves beginning at (0,0)(0,0) and ending at the bottom-right cell (m−1 ...