On the space requirement of interval routing

Interval routing is a space-efficient method for point-to-point networks. It is based on labeling the edges of a network with intervals of vertex numbers (called interval labels). An M-label scheme allows up to M labels to be attached on an edge. For arbitrary graphs of size m, n the number of vertices, the problem is to determine the minimum RP necessary for achieving optimality in the length of the longest routing path. The longest routing path resulted from a labeling is an important indicator of the performance of any algorithm that runs on the network. We prove that there exists a graph with D=/spl Omega/(n/sup 1/3/) such that if M/spl les/n/18D-O(/spl radic/n/D) the longest path is no shorter than D+/spl Theta/(D//spl radic/M). As a result, for any M-label 1RS, if the longest path is to be shorter than D+/spl Theta/(D//spl radic/M), at least M=/spl Theta/(n/D) labels per edge would be necessary.