Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

Abstract

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $Θ(n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $Ω(n^2 \log n)$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $Θ(n^3)$ bits on average. For worst-case static networks we prove a $Ω(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.