Fill rooms with nearest gate distances

You are given an m×n integer grid representing rooms, where -1 is a wall, 0 is a gate, and INF (a large sentinel such as 2^31− 1) is an empty room. For each empty room, fill it with the minimum number of steps to the nearest gate using four-directional movement; if unreachable, leave it as INF. Implement an in-place algorithm, justify your traversal strategy (e.g., multi-source BFS), and analyze time and space complexity. Cover edge cases such as multiple gates, enclosed areas, and very large grids.