Network Discovery and Verification

Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffmann, M. Mihal'ak, L.S. Ram
2006 IEEE Journal on Selected Areas in Communications  
Consider the problem of discovering (or verifying) the edges and nonedges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and nonedges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive
more » ... g competitive analysis and give a randomized on-line algorithm with competitive ratio O( √ n log n) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P = N P. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Θ(d/ log d).
doi:10.1109/jsac.2006.884015 fatcat:yfawcmkdlnfmffxico7oceaaku