REARRANGEABILITY OF (2n - 1)-STAGE SHUFFLE-EXCHANGE NETWORKS ∗ HASAN C ¸AM † SIAM J. COMPUT. c  2003 Society for Industrial and Applied Mathematics Vol. 32, No. 3, pp. 557–585 Abstract. Rearrangeablenetworkscanrealizeeachandeverypermutationinonepassthrough thenetwork. Shuffle-exchangenetworksprovideanefficientinterconnectionschemeforimplementing various types of parallel processes. Whether (2n - 1)-stage shuffle-exchange networks with N =2 n inputs/outputs are rearrangeable has remained an open question for approximately three decades. Thisquestionhasbeenansweredaffirmativelyinthispaper. Animportantcorollaryofthemainresult istheproofthattwopassesthroughanOmeganetworkaresufficientandnecessarytoimplementany permutation. In obtaining the main results of this paper, frames thatlooklikegridswithhorizontal links of different lengths are shown to be remarkable tools for identifying and characterizing the binary matrix representations of permutations. Key words. shuffle-exchangenetwork,rearrangeablenetwork,permutations,balancedmatrices, frames AMS subject classifications. 68M07, 68M10, 05A05, 05C70 PII. S0097539798344847 1. Introduction. Aninterconnectionnetwork(IN)with N =2 n inputs/outputs iscalledarearrangeablenetworkifitrealizeseachandeveryoneof N ! permutations in a single pass [9]. It is known that the lower bound for the number of stages of a symmetrical multistage IN with 2 × 2 switching boxes (SBs) to be rearrangeable is 2n - 1 [10]. The validity of this lower bound for shuffle-exchange (SE) networks can beestablishedbyshowingtheexistenceofapermutationwhichcannotbeperformed with less than 2n - 1 stages [2]. The question of whether or not a (2n - 1)-stage SE network is rearrangeable has remained open for three decades [3, 4, 7, 8, 12, 13, 16, 18,19,21,25,26,27].Thispaperprovesthata(2n - 1)-stageSEnetworkwith2 × 2 SBs is rearrangeable. SE networks, initially proposed by Stone [7], provide an efficient interconnection schemeforimplementingvarioustypesofparallelprocesses[7,1,6,11,14,20]. These networksareconstructedofrepeatedcopiesofanSEstagewhichconsistsofa“perfect shuffle”interconnectionpatternfollowedbyacolumnof2×2SBs[7,8].Themostused SEnetworkistheOmeganetworkconsistingof n SEstages. Foraboutthreedecades, severalresearchershavebeeninterestedinthenumberofSEstagesneededtorealize all N ! permutations. The algorithm proposed by Stone [7] can be used to realize any permutationonanSEnetworkwith n 2 stages. Siegel[12]alsodescribedanalgorithm forrealizinganypermutationonasingleSEstagein2n 2 passes. Parker[8]showedthat three passes through the Omega network are sufficient to generate any permutation, and two passes are necessary. Wu and Feng [18] have shown that a (3n - 1)-stage SE network implements all N ! permutations. The upper bound of (3n - 1) stages waslaterreducedto(3n - 3)stagesbydifferentresearchers(HuangandTripathi[19] and Babu and Raghavendra [16]). Using a constructive approach, Raghavendra and Varma [3] showed that an SE network of five stages with N = 8 inputs/outputs is ∗ Received by the editors September 16, 1998; accepted for publication (in revised form) August 12, 2002; published electronically March 5, 2003. http://www.siam.org/journals/sicomp/32-3/34484.html † Computer Science and Engineering Department, Arizona State University, Tempe, AZ 85287 (hasan.cam@asu.edu). 557