|Résumé||We address the problem of verifying the accuracy of a map of a network by making as few measurements as possible on its nodes. This task can be formalized as an optimization problem that, given a graph G=(V,E), and a query model specifying the information returned by a query at a node, asks for finding a minimum-size subset of nodes of G to be queried so as to univocally identify G. This problem has been faced w.r.t. a couple of query models assuming that a node had some global knowledge about the network. Here, we propose a new query model based on the local knowledge a node instead usually has. Quite naturally, we assume that a query at a given node returns the associated routing table, i.e., a set of entries which provides, for each destination node, a corresponding (set of) first-hop node(s) along an underlying shortest path. First, we show that any network of n nodes needs Omega(log log n) queries to be verified. Then, we prove that there is no o(log n)-approximation algorithm for the problem, unless P=NP, even for networks of diameter 2. On the positive side, we provide an O(log n)-approximation algorithm to verify a network of diameter 2, and we give exact polynomial-time algorithms for paths, trees, and cycles of even length.
This is joint work with Evangelos Bampas, Davide Bilo, Guido Drovandi, Luciano Guala and Guido Proietti.
Intervenant: Ralf Klasing |