### ON THE REARRANGEABILITY OF SHUFFLE-EXCHANGE NETWORKS

HUNG Q. NGO, DING-ZHU DU
2000 Algorithms and Architectures for Parallel Processing
Let m(n) be the minimum positive integer k so that the Shuffle-Exchange network with k stages, N = 2 n inputs and N outputs is rearrangeable. Beneš conjectured that m(n) = 2n ? 1. The best bounds known so far are 2n ? 1 m(n) 3n ? 4. In this paper, we verify Beneš conjecture for n = 4, and use this result to show that m(n) 3n ? 5. The n = 4 case is considerably more complex than the n = 3 case, which have been done in the literature. We believe that hidden in our proof there is some general
more » ... ique that would help improve the bound further. : The 7-stage SE network for N = 16, i.e. (SE 4 ) 7 which (SE n ) k can not route if k 2n ? 2, in effect showing 2n ? 1 m(n). With a different formulation, Linial and Tarsi (1989, ) also verified the conjecture for N = 8 and showed m(n) 3n ? 4. From their formulation it is easy to see that at least 2n ? 1 stages are needed to route all permutations. Feng and Seo (1994, ) gave a proof of the conjecture, which was incomplete as pointed out by Kim, Yoon, and Maeng (1997, ). In this paper, we give a proof that m(4) = 7 using a new method, and then adapting Linial and Tarsi's results to show that m(n) 3n ? 5. As we shall see, the n = 4 case is considerably more difficult than the n = 3 case. We believe that hidden in our proof there is some general technique(s) that would help improve the bound further. Preliminaries This section presents related concepts and previous results on the problem. Throughout the paper, we shall assume that n 2 N and N = 2 n . The following definitions and lemmas are from Linial and Tarsi . Definition 2.