New bounds for multi-label interval routing

• Authors: Savio S. H. Tse;Francis C. M. Lau
• Venue: Theoretical Computer Science
• Year: 2004

Interval routing (IR) is a space-efficient routing method for computer networks. For longest routing path analysis, researchers have focused on lower bounds for many years. For any n-node graph G of diameter D, there exists an upper bound of 2D for IR using one or more labels, and an upper bound of ⌈3/2D⌉ for IR using O(√nlogn) or more labels. We present two upper bounds in the first part of the paper. We show that for every integer i 0, every n-node graph of diameter D has a k-dominating set of size O(i+1√n) for k ≤ (1 - 1/3i)D. This result implies a new upper bound of ⌈(2 - 1/3i)D⌉ for IR using O(i+1√n) or more labels, where i is any positive integer constant. We apply the result by Kutten and Peleg [7] to achieve an upper bound of (1 + α)D for IR using O(n/D) or more labels, where α is any constant in (0, 1). The second part of the paper offers some lower bounds for planar graphs. For any M-label interval routing scheme (M-IRS), where M = O(3√n), we derive a lower bound of [(2M + 1)/(2M)]D - 1 on the longest path for M = O(√n), and a lower bound of [(2(1 + δ)M + 1)/(2(1 + δ)M)]D, where δ ∈ (0, 1], for M = O(√n). The latter result implies a lower bound of Ω(√n) on the number of labels needed to achieve optimality.