The compactness of interval routing for almost all graphs

Interval routing is a compact way of representing routing tables on a graph. It is based on grouping together, in each node, destination addresses that use the same outgoing edge in the routing table. Such groups of addresses are represented by some intervals of consecutive integers. We show that almost all the graphs, i.e., a fraction of at least 1 - 1/n² of all the n-node graphs, support a shortest path interval routing with three intervals per outgoing edge, even if the addresses of the nodes are arbitrarily fixed in advance and cannot be chosen by the designer of the routing scheme. In case the addresses are initialized randomly, we show that two intervals per outgoing edge suffice, and, conversely, that two intervals are required for almost all graphs. Finally, if the node addresses can be chosen as desired, we show how to design in polynomial time a shortest path interval routing with a single interval per outgoing edge for all but at most O(log³ n) outgoing edges in each node. It follows that almost all graphs support a shortest path routing scheme which requires at most n + O(log⁴ n) bits of routing information per node, improving on the previous upper bound.