Minimal adaptive routing on the mesh with bounded queue size

Abstract

An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters. Such algorithms potentially avoid network bottlenecks by routing packets around “hot spots.” Minimal adaptive routing algorithms have the additional advantage that the path each packet takes is a shortest one.For a large class of minimal adptive routing algorithms, we present an O(n2/k2) bound on the worst case time to route a static permutation of packets on an n × n mesh or torus with nodes that can hold up to k ≥ 1 packets each. This is the first nontrivial lower bound on adaptive routing algorithms. The argument extends to more general routing problems, such as the h-h routing problem. It also extends to a large class of dimension order routing algorithms, yielding an O(n2/k) time bound.To complement these lower bounds, we present two upper bounds. One is an O(n2/k) time dimension order routing algorithm that matches the lower bound. The other is the first instance of a minimal adaptive routing algorithm that achieves O(n) time with constant sized queues per node. We point out why the latter algorithm is outside the model of our lower bounds.