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

Explore how to determine the minimum cost to create at least one valid path in a grid where each cell has a directional sign. Understand graph traversal techniques, direction modifications, and cost management to solve complex grid path problems efficiently.

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Statement

You are given an $m \times n$ 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 ...