Efficient Routing in Networks with Long Range Contacts

Abstract

We investigate the notion of Long Range Contact graphs. Roughly speaking, such a graph is defined by (1) an underlying network topology G, and (2) one (or possibly more) extra link connecting every node u to a “long distance” neighbor, called the long range contact of u. This extra link represents the a priori knowledge that a node has about far nodes and is set up randomly according to some probability distributions p. To illustrate the claim that Long Range Contact graphs are a good model for the small world phenomenon, we study greedy routing in these graphs. Greedy routing is the distributed routing protocol in which a node u makes use of its long range contact to progress toward a target, if this contact is closer to the target, than the other neighbors. We give upper and lower bounds on greedy routing on the n-node ring C _n augmented with links chosen using the r-harmonic distributions. In particular, we show a tight Θ(log² n)-bound for the expected number of steps required for routing in C _n augmented using the 1-harmonic distribution. Hence, our study shows that the model of Kleinberg [11] can be simplified by using the ring rather than the mesh while preserving the main features of the model. Our study also demonstrates the signifi- cant difference (in term of both diameter and routing) between the ring augmented with long range contacts chosen with the harmonic distribution and the ring augmented with a random matching as introduced by Bollobas and Chung [3]. Finally, using epimorphisms of a graph onto another, for any network G, we show how to define a probability distribution p and study the performance of greedy routing in G augmented with p. For appropriate embeddings (if they exist), this performance turns out to be O(log² n).

Part of this work was done while the first and third author were visiting LRI at Orsay (additional support from Université Paris-Sud), and while the first and second authors were visiting Carleton University. Additional support from Carleton University, NATO, and MITACS(Mathematics of Information Technology and Complex Systems) grants.