On the Space Requirement of Interval Routing

• Authors: Savio S. H. Tse;Francis C. M. Lau
• Venue: IEEE Transactions on Computers
• Year: 1999

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 $D=\Omega(n^{\frac{1}{3}})$ such that if $M\leq {\frac{n}{18D}}-O(\sqrt{\frac{n}{D}})$, the longest path is no shorter than $D+\Theta({\frac{D}{\sqrt{M}}})$. As a result, for any $M$-label IRS, if the longest path is to be shorter than $D+\Theta({\frac{D}{\sqrt{M}}})$, at least $M=\Omega({\frac{n}{D}})$ labels per edge would be necessary.