Solve BFS grid delivery routing

You are given an m×n grid representing a city: 'S' are DashMart stores, 'H' are homes, '.' are roads, and '#' are obstacles. Movement is allowed in four directions on roads only. For each home cell, compute the length of the shortest path to any store; if unreachable, return -1 for that home. Output an array of distances aligned to the list of homes. Provide the algorithm, complexity, and code. Follow-up: if new obstacles are added frequently between queries, explain how you would update distances efficiently without recomputing everything from scratch.