1368. Minimum Cost to Make at Least One Valid Path in a Grid

1. Directions Mapping: Map each grid cell's direction (1 to 4) to the respective moves: right, left, down, up.
2. Priority Queue (Min-Heap): Use a priority queue to track (cost, x, y) where cost is the cost to reach (x, y).
3. Breadth-First Search (BFS): Start from (0, 0) with cost 0.
   • For each cell, try all four possible moves:
     • Move in the grid's indicated direction at no extra cost.
     • Move in other directions with a cost of 1.
   • Push the new states into the priority queue.
4. Termination: Stop when the bottom-right cell (m-1, n-1) is reached, returning the accumulated cost.

The priority queue ensures that the least costly path is explored first. If a move aligns with the grid’s direction, it incurs zero cost; otherwise, it costs 1 to change direction. The BFS continues until reaching the destination (m-1, n-1), ensuring the minimum cost path is found efficiently.

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