Average Size of Unstretched Remote-Spanners

Motivated by the optimization of link state routing in ad hoc networks, and the concept of multipoint relays, we introduce the notion of remote-spanner. Given an unweighted graph G, a remote spanner is a set of links H such that for any pair of nodes (u, v) there exists a shortest path in G for which all links in the path that are not adjacent to u belong to H. The remote spanner is a kind of minimal topology information beyond its neighborhood that any node would need in order to compute its shortest paths in a distributed way. This can be extended to k-connected graphs by considering minimum length sum over k disjoint paths as distance. In this paper, we give distributed algorithms for computing remote-spanners in order to obtain sparse remote-spanners with various properties. We provide a polynomial distributed algorithm that computes a k-connecting unstretched remote-spanner whose number of edges is at a factor 2(1 + log Δ) from optimal where Δ is the maximum degree of a node. Interestingly, its expected compression ratio in number of edges is O(k/n log n) in Erdos-Renyi graph model and O((k/n)2/3) in the unit disk graph model with a uniform Poisson distribution of nodes.