Two Lower Bounds for Multi-Label Interval Routing

Interval routing is a space-efficient method for point-to-point networks. The method has been incorporated in the design of a commercially available routing chip [7], and is a basic element in some compact routing methods (e.g., [4]). With up to one interval label per edge, the method has been shown to be nonoptimal for arbitrary graphs [9, 13], where optimality is measured in terms of the longest (routing) path in a graph. It is intuitive to think that the longest path would depend on the number of interval labels used. In this paper, using some non-planar graphs, we prove that even with a relatively large number of labels, interval routing still falls short of being optimal for arbitrary graphs. The bounds on the longest path we prove are D, independent of any number of labels up to logn , where D is the diameter of the graph; and D, independent of any number of labels from logn to p n . Our lower bound results suggest that a large increase, a factor of O n for , in routing information will likely result in only a small decrease, a factor of O , in the longest path. Correspondence: Dr F.C.M. Lau, Department of Computer Science, The University of Hong Kong, Hong Kong / Email: [email protected] / Fax: (+852) 2559 8447