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

TL;DR

This paper uses the incompressibility method to analyze the minimal routing information needed for almost all network topologies, revealing how bounds vary with different models and routing constraints.

Contribution

It introduces a novel application of Kolmogorov complexity to establish bounds on routing table sizes across various network models and routing requirements.

Findings

Almost all graphs require Θ(n^2) bits for shortest path routing.

Average case bounds vary significantly with the model, from Ω(n^2 log n) to O(n log^2 n).

Full-information routing needs Θ(n^3) bits on average.

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 $\Theta (n^2)$ 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 $\Omega(n^2 \log n)$ 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 $\Theta (n^3)$ bits on average. For worst-case static networks we prove a $\Omega(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.