An Upper Bound Result for Multi-label Interval Routing on Planar Graphs

Interval routing is a space-efficient routing method for computer networks. In this paper, all graphs are assumed to be planar graphs, unless specified otherwise. We have four upper bound results in this paper. First, for D ≥ 3, we prove the existence of an O(D)-IRS on arbitrary graphs whose longest path is bounded by D, where D is the diameter not less than three. With a little modification, we can reduce the number of labels used to O(D) with the length of longest path being increased to (1 + α)D, where α is any constant in (0, 1). Together with the result in Theorem 4 of [14], this result implies an O(n 3 4 )-IRS on arbitrary graphs whose longest path is bounded by (1 + α)D, where n is the number of nodes in the graph. It was proved in [3] that for some non-planar graphs, there is a lower bound of 32D − 1 on the longest path for any M -IRS, M = O( n D log n D ). Comparing these two results, we conclude that interval routing can perform strictly better in planar graphs with D = O( 4 √n log n ). For larger diameters, the difference between planar and non-planar graphs has yet to be explored. For completeness, we also construct a 6-IRS for arbitrary graphs with D = 2.