On Embedding Permutations in Hypercubes

Abstract

The interconnection network of a multiprocessor system should be able to embed an arbitrary permutation of nodes to map an arbitrary structure of a program graph and realize required communication paths. We show that distributed routing algorithms have high blocking probability to route permutations in binarycube-based systems. We further show that there exists no recursive algorithm to embed a permutation in binary n-cube for n 2 5 . We t.2en develop rearrangeable hypercube architectures and ro,uting algorithms to realize arbitra y permutations in circzlit switching. We show that if each connection between two neighboring nodes consists of 2 pairs of links, i.e., (2 full-duplex communication lines), the hypercube can embed 2 arbitrary permutations of nodes simultaneously. We also prove that a hypercube is rearrangeable i f one additional pair of links is provided in any one dimension of connections.