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. This is motivated by the common approach of combining local measurements in order to obtain maps of the Internet or other dynamically growing networks. In the distance query model, a query at node q returns the distances from q to all other nodes in the graph. We describe how the existence of an individual edge or non-edge in G can be deduced by potentially combining the results of several queries. This leads to a characterization of when a set of queries Q "discovers" the graph G. 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 the previous ones. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O( √ n log n) for graphs on n nodes. We also show lower bounds Ω( √ n) and Ω(log n) on competitive ratios of deterministic on-line algorithms and randomized on-line algorithms, respectively. In the off-line network verification problem, the graph is known in the beginning and the problem asks for a minimum number of queries to verify all edges and non-edges. We show that the problem is N P-hard and present an O(log n)-approximation algorithm.

Near-Linear Query Complexity for Graph Inference

“…return A Algorithm 2 is a generalization of the algorithm Center in [17]; and Lemma 16 is a trivial extension of Theorem 3.1 in [17]. 7 Using a set of centers A, we define, for each a ∈ A, its extended Voronoi cell D a ⊆ U as follows:…”

“…Yet, this topology can be extremely difficult to find, due to the dynamic structure of the network and to the lack of centralized control. The network reconstruction problem has been studied extensively [1,2,6,7,11,16]. Sometimes we have some idea of what the network should be like, based perhaps on its state at some past time, and we want to check whether our image of the network is correct.…”

“…Beerliova et al [2] studied network verification and reconstruction using an oracle, which, upon receiving a node q, returns all shortest paths from q to all other nodes, instead of one shortest path between a pair of nodes as in our first model. Erlebach et al [7] studied network verification and reconstruction using an oracle which, upon receiving a node q, returns the distances from q to all other nodes in the graph, instead of the distance between a pair of nodes as in our second model. They showed that minimizing the number of queries for verification is NP-hard and admits an O(log n)-approximation algorithm.…”

Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

“…Learning one hidden star (similar to a hub) was studied in [1] in a different model. We also mention a series of articles ( [3,4,9,10] and others) dealing with the discovery and verification of unknown graphs by a complicated type of distance queries, motivated by routing and communication in networks. Random sampling was used in [11] for finding approximations to (1 − )-dominating sets.…”

Graph Reconstruction via Distance Oracles

“…To reconstruct graphs of bounded degree, we apply some algorithmic ideas previously developed for compact routing [21] and ideas for Voronoi cells [15]. A closely related model in network discovery and verification provides queries which, upon receiving a node q, returns the distances from q to all other nodes in the graph [12], instead of the distance between a pair of nodes in our model. The problem of minimizing the number of queries is NP-hard and admits an O(log n)-approximation algorithm (see [12]).…”