Network Discovery and Verification with Distance Queries

The network discovery (verification) problem asks for a minimum subset q ? v of queries in an undirected graph g = (v,e) such that these queries discover all edges and non-edges of the graph. In the distance query model, a query at node q returns the distances from q to all other nodes in the graph. In the on-line network discovery problem, the graph is initially unknown, and the algorithm has to select queries one by one based only on the results of previous queries. We give a randomized on-line algorithm with competitive ratio o(\sqrt{n\log{n}})o(\sqrt{n\log{n}}) for graphs on n nodes. We also show lower bounds of \omega(\sqrt{n})\omega(\sqrt{n}) and ?(logn) on the competitive ratio of deterministic and randomized on-line algorithms, respectively. In the off-line network verification problem, the graph is known in advance and the problem is to compute a minimum number of queries that verify all edges and non-edges. We show that the problem is \mathcal{np}\mathcal{np}-hard and present an o(logn)-approximation algorithm.