Information Processing Letters 110 (2010) 211–215 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Fully symmetric swapped networks based on bipartite cluster connectivity ✩ Wenjun Xiao a , Behrooz Parhami b,∗ , Weidong Chen a , Mingxin He a , Wenhong Wei a a School of Software Engineering, South China University of Technology, Guangzhou 510641, PR China b Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560, USA article info abstract Article history: Received 15 August 2009 Received in revised form 15 December 2009 Accepted 15 December 2009 Available online 21 December 2009 Communicated by A.A. Bertossi Keywords: Bipartite graph Hierarchical network Interconnection network OTIS network Routing algorithm Swapped network Topological parameters The class of swapped or OTIS (optical transpose interconnect system) networks, built of n copies of an n-node cluster by connecting node i in cluster j to node j in cluster i for i = j , has been studied extensively. One problem with such networks is that node i of cluster i has no intercluster link. This slight asymmetry complicates a number of algorithms and hinders both theoretical investigations and practical pursuits, such as building parallel node-disjoint paths for fault tolerance. We introduce biswapped networks that are fully symmetric and have cluster connectivity very similar to swapped/OTIS networks. We derive basic topological parameters, present a simple distributed shortest-path routing algorithm, and point to a number of other interesting properties under investigation for biswapped networks. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Symmetry is a very desirable property of an intercon- nection network. For example, node symmetry (node tran- sitivity) allows one to develop a single generic algorithm for routing that is applicable to every node in the network. If the routing algorithm is to be fully distributed, so that each node makes its own decision as to how to forward an incoming message, the advantages of symmetry become even more pronounced. Symmetry also facilitates theoreti- cal proofs and certain derivations of practical significance, such as computing the average internode distance or con- structing a set of node-disjoint paths between a pair of nodes (to allow the routing of messages in parallel or to identify alternate paths in the event of node/link failures). ✩ Research of W. Xiao, W. Chen, M. He, and W. Wei was supported by the Natural Science Foundation of China (60973150). * Corresponding author. E-mail addresses: wjxiao@scut.edu.cn (W. Xiao), parhami@ece.ucsb.edu (B. Parhami). Our motivation for this study arose from a small, yet very important (in both theoretical and practical terms), asymmetry in swapped [8,6] or OTIS [5,3] networks, that have been found of interest by researchers in communi- cations and parallel computing. A swapped/OTIS network Sw(Ω) is built from n clusters, which are identical copies of an n-node basis network Ω, by connecting node i of cluster j to node j of cluster i , for all i = j . The lat- ter condition is what causes the asymmetry, because node i of cluster i , which has no intercluster link, maintains its original degree δ , whereas all other nodes have de- gree δ + 1. Consequently, many analyses and algorithms for swapped/OTIS networks become complicated by the need to treat node (i , i ) differently from a typical node (i , j ) having unequal cluster index i and node index j . This lack of full symmetry prevents us from using results that are applicable to classes of symmetric networks, such as the ubiquitous Cayley graphs [1,2,4,7]. We thus pondered the existence question for “an alter- nate or modified form of swapped network that is a Cay- ley graph when the basis network is a Cayley graph” [6]. 0020-0190/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2009.12.006