"Firehouse" Problem

If you have an undirected, unweighted graph G, and a marked subset of points called 'Firehouses,' is there an O(n+m) algorithm to find every vertex's closes firehouse?

I can only come up with O(n^2) stuff. Although I do like the idea of repeatedly raising the adjacency matrix to a power n, until every vertex has a nonzero entry :P

For each firehouse assign the neighbors of that firehouse to it. Then for each firehouse assign the neighbors of the vertices that were assigned to that firehouse in the previous round to it. Repeat until all vertices have been assigned.