Optimal permutation routing for low‐dimensional hypercubes

Abstract

AbstractWe consider the offline problem of routing a permutation of tokens on the nodes of a d‐dimensional hypercube, under a queueless MIMD communication model (in which we have the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d‐dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large d‐dimensional hypercubes to 2d‐ 4. We also find an interesting side‐result, that the number of possible communication steps in a d‐dimensional hypercube is the same as the number of perfect matchings in a (d + 1)‐dimensional hypercube, a combinatorial quantity for which there is no closed‐form expression. Finally we present some experimental observations which may lead to a proof of a more general result for hypercubes with arbitrarily large dimension d. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010