An Optimal Lower Bound for Interval Routing in General Networks

Interval routing is a space-efficient (compact) routing method for point-topoint communication networks. The method is based on proper labeling of edges of the graph with intervals. An optimal labeling would result in routing of messages through the shortest paths. Optimal labelings exist for regular as well as some of the common topologies, but not for arbitrary graphs. It has been shown that it is impossible to find optimal labelings for arbitrary graphs [4]. In this paper, we prove the lower bound of 2D − 3 on the longest routing path for arbitrary graphs, where D = O( √ n) is the graph’s diameter and n is the number of nodes, as well as a lower bound of 2D − o(D) for D = O(n). Our results are very close to the best known upper bound which is 2D.