Optimal routing tables

Jaap-Henk Hoepman Paul Vitfinyi (2WI CWI and University of Amsterdam v # u an edge incident to u on a path from u to v. This way, The optimal space used to represent routing schemes in communication networks is established, both for worst-case static networks and on the average for all static networks. Several factors may influence the cost of representing a routing scheme for a particular network. It is therefore unavoidable that we first describe several reasonable models in which to measure this cost. Failure to do so in the past has obfuscated previous results. We show that, in most models, for almost all graphs @(nz) 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 – l/nc of all graphs on n nodes, where c ~ 3 is an arbitrary fixed constant. In contrsat, there is a model that rises the average case lower bound to $2(n2 log n) and another model where the average case upper bound drops to O(n logz n). This clearly exposes the sensitivity of such bounds to the model under consideration. Furthermore, if paths have to be short, but need not be shortest (i.e., if the stretch factor may be larger than 1), our other upper bounds indicate that much less space is needed on average, even in the more demanding models. For worst-case static networks we prove a f2(n2 log n) lower bound for shortest path routing, for those models where the nodes in the network are labelled 1,. ... n. This lower bound holds even for all stretch factors <2. Throughout, we use the incompressibility method baaed on Kolmogorov complexity.