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 n, n the number of vertices, the problem is to determine the minimum M 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 = Ω(n1/3) such that if M ≤ n/18D - O(√n/D), the longest path is no shorter than D + Θ(D/√M). As a result, for any M-label IRS, if the longest path is to be shorter than D + Θ(D/√M), at least M = Ω(n/D) labels per edge would be necessary.published_or_final_versio

An Optimal Lower Bound for Interval Routing in General Networks

. Interval routing is a space-efficient (compact) routing method for point-to-point communication networks. It has already been shown that it is impossible to find labelings that would lead to all-shortest-paths for arbitrary graphs. In this paper, we prove the lower bound of 2D \Gamma 3 on the longest routing path for arbitrary graphs, where D = O( p n) is the graph's diameter and n is the number of nodes, as well as a lower bound of 2D \Gamma o(D) for D = O(n). Our results are very close to the best known upper bound which is 2D. 1 Introduction Interval routing was first proposed by Santoro and Khatib [2], and subsequently refined by van Leeuwen and Tan [6]. The idea is to label the nodes by integers (called node numbers) from a cyclicly ordered set, say, f0; 1; : : : ; n \Gamma 1g, where n is the number of nodes and the edges by intervals of the form hp; qi, where p; q are node numbers. hp; qi is the set fp; p + 1; : : : ; qg if p ! q, or fp; p + 1; : : : ; n \Gamma 1; 0; : : :..