A Scheme for Fast Parallel Communication

Consider $N = 2^n $ nodes connected by wires to make an n-dimensional binary cube. Suppose that initially the nodes contain one packet each addressed to distinct nodes of the cube. We show that there is a distributed randomized algorithm that can route every packet to its destination without two packets passing down the same wire at any one time, and finishes within time $O(\log N)$ with overwhelming probability for all such routing requests. Each packet carries with it $O(\log N)$ bits of bookkeeping information. No other communication among the nodes takes place. The algorithm offers the only scheme known for realizing arbitrary permutations in a sparse N node network in $O(\log N)$ time and has evident applications in the design of general purpose parallel computers.