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(n2) 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/nc 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(n2logn) and another model where the average case upper bound
drops to O(nlog2n). 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(n3) bits on average. For worst-case static networks we
prove a Ω(n2logn) lower bound for shortest path routing and all
stretch factors <2 in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea