Distributed Computing of Efficient Routing Schemes in Generalized Chordal Graphs

Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. There are in fact at least two properties that any routing scheme must consider: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Therefore, we propose a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no chordless cycles of length more than k. We study the tradeoff between the length of routes and the time complexity for computing them. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k − 1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k − 1. In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses messages of size O(log n) and takes O(D) time. We then propose a slightly modified version of the algorithm for computing routing tables in time O(min{∆D,n}), where ∆ is the the maximum degree of the graph. Using these tables, our routing scheme achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). The routing scheme uses addresses of size logn bits and local memory of size 2(d− 1) logn bits per node with degree d.