Rearrangeability of multistage shuffle/exchange networks

• Authors: A. Varma;C. S. Raghavendra
• Venue: ISCA '87 Proceedings of the 14th annual international symposium on Computer architecture
• Year: 1987

In this paper we study the rearrangeability of multistage shuffle/exchange networks. Although a theoretical lower bound of (2 log2N - 1) stages for rearrangeability of a network with N = 2n inputs and outputs has been known, the sufficiency of (2 log2N - 1) stages has neither been proved nor disproved. The best known upper bound for rearrangeability is (3 log2N - 3) stages. We prove that, if (2 log2R - 1) shuffle/exchange stages are sufficient for rearrangeability of a network with R = 2' inputs and outputs, then, for any N R, 3 log2N - (r + 1) stages are sufficient for a network with N inputs and outputs. This result is established by setting some of the middle stages of the network to realize a fixed permutation and showing the reduced network to be topologically equivalent to a member of the Benes class of rearrangeable networks. We first characterize equivalence to Benes networks in set-theoretic terms and use this to prove equivalence of the reduced shuffle/exchange network to the Benes network. From the known result that 5 stages are sufficient for rearrangeability when N = 8, we obtain an upper bound of (3 log2N - 4) stages for rearrangeability when N ≥ 8. Further, any increase in the network size R for which the rearrangeability of (2 log2R - 1) stages could be shown, results in a corresponding improvement in the upper bound for all N ≥ R. In addition, due to the one-to-one correspondence that exists between the switches in the reduced shuffle/exchange network and those in the Benes network, the former network can be controlled by the well-known looping algorithm.