Network discovery and verification with distance queries

The network discovery (verification) problem asks for a minimum subset Q subset of 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(root n log n) for graphs on n nodes. We also show lower bounds of Omega(root n) and Omega(log n) 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 NP-hard and present an O(log n)-approximation algorithm.