Quicksort and permutation routing on the hypercube and de Bruijn networks

Abstract

We consider the problems of sorting and routing on some unbounded interconnection networks, namely hypercube and de Bruijn network. We first present two efficient implementations of quicksort on the hypercube. The first algorithm sorts N items on an N-node hypercube, one item per node, in O((log/sup 2/ N)/(log log N)) time with high probability, while the other one sorts N items on an (N/log N)-node hypercube, log N items per node, in O(log/sup 2/ N) time with high probability, which achieves optimal speedup in the sense of PT product. Both algorithms beat the fastest previous quicksort that runs in O(log/sup 2/ N) expected time on a butterfly of N nodes. We also present a deterministic (nonoblivious) permutation routing algorithm which runs in O(d/spl middot/n/sup 2/) time on a d-ary de Bruijn network of N=d/sup n/ nodes. To the best of our knowledge, this routing algorithm is so far the fastest deterministic one for the de Bruijn network of arbitrary degree. The best previous one runs in O((log d)/spl middot/d/spl middot/n/sup 2/) time. All algorithms presented are simple, the constants hidden behind the big Oh being small. >