Continuous routing and batch routing on the hypercube

In this paper we study two related routing problems on the n-dimensional hypercube: continuous routing, referring to the infinite routing process in which one new permutation is generated every constant r time steps, and batch routing, in which n permutations axe to be routed at the same time. We show that continuous routing with suitable value of r is feasible by giving a probabilistic routing algorithm that guarantees an 0 (n) completion time for any given permutation with overwhelming probability. The proof relies heavily on a probabilistic batch routing algorithm that has an 0 (n) completion time - an improvement over Valiant's classical 2-phase probabilistic routing algorithm when routing n permutations at the same time.