Network Discovery and Verification with Distance Queries

Abstract

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 $O(\sqrt{n\log{n}})$ for graphs on n nodes. We also show lower bounds of $\Omega(\sqrt{n})$ and Ω(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 $\mathcal{NP}$-hard and present an O(logn)-approximation algorithm.

Work partially supported by European Commission – Fet Open project DELIS IST-001907 Dynamically Evolving Large Scale Information Systems, for which funding in Switzerland is provided by SBF grant 03.0378-1.