Efficient distributed source detection with limited bandwidth

Given a simple graph G = (V, E) and a set of sources SCV, denote for each node v ε V by L^(∞) the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ε S. For integers d, k ε NU{∞}, we consider the source detection, or (S, d, k)-detection task, requiring each node v to learn the first k entries of L^(∞) (if for all of them d(s, v) ≤ d) or all entries (d(s, v),s) ε L ^(∞), satisfying that d(s, v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k = ∞ requires each source s ε S to build a complete BFS tree rooted at s, whereas the special case of d = ∞ and S = V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most C(logn) bits, running in d + k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions.