Network Discovery and Verification

Abstract

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 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).

Network Discovery and Verification with Distance Queries

“…Nevertheless, it was proposed only recently [4] to study this problem from a combinatorial optimization point of view.…”

“…Beerliova et al [4] introduce the network discovery and verification problems, which ask to find a map of a network with a small number of queries (measurements). In the on-line network discovery problem only the nodes V of a graph G are known in the beginning.…”

“…In [4], a very strong query model was used: a query at a node v reveals all edges and non-edges whose endpoints have a different distance from v. That query model was motivated by the consideration that in certain scenarios, it is possible to identify all edges on shortest paths between the query node and all other nodes. In this paper, we study the network discovery problem and network verification problem in the model where a query q ∈ V gives all distances from q to any other node of the investigated graph G. We refer to the on-line problem as DIST-ALL-DISCOVERY and to the off-line problem as DIST-ALL-VERIFICATION.…”

“…In this paper, we study the network discovery problem and network verification problem in the model where a query q ∈ V gives all distances from q to any other node of the investigated graph G. We refer to the on-line problem as DIST-ALL-DISCOVERY and to the off-line problem as DIST-ALL-VERIFICATION. This distance query model is much weaker than the model used in [4], in the sense that typically a query reveals much less information about the network.…”

“…Beerliova et al [4] propose the general problem of network discovery (verification) and study it for the "layered graph" query model: a query q ∈ V returns all edges and non-edges between nodes of different distances from q. They present an o(log n) inapproximability result for the off-line version and give a randomized on-line algorithm with competitive ratio O( √ n log n).…”