IEEE TRANSACTIONS ON COMPUTERS, VOL. 43, NO. 12, DECEMBER 1994 1439 ACKNOWLEDGMENT The authors thank S. Wei for his contribution at the initial phase of this work and P.-J. Chuang for his assistance in preparing figures. REFERENCES [I] D. A. Reed and zyxwvutsrqpon R. zyxwvutsrqponmlk M. Fujimoto, zyxwvutsrqpo Mulricompufer Networks: Message- Based Parallel Processing. [2] W.-J. Hsu, “Fibonacci cubes-- new interconnection topology,” IEEE Trans. Parallel Disrrib. zyxwvutsrqpo Sysr., vol. 4, pp. 3-12, Jan. 1993. 13) W. 3. Dally and C. L. Seitz, “IDeadlock-free message routing in multi- processor interconnection networks,” IEEE Trans. Compur., vol. C-36, pp. 547-553, May 1987. [SI M. Y. Chan and S.-J. Lee, “Fault-tolerant embedding of complete binary trees in hypercubes,” zyxwvutsrqpon IEEE Trans. Parallel Disrrih. zyxwvutsrqp Syst., vol. 4. pp. 277-288, Mar. 1993. [SI H. P. Katseff, “Incomplete hypercubes,” IEEE Trans. Compur., vol. 37, pp. zyxwvutsrqp 604-608, May 1988. [6] N.-F Tzeng, “Structural propen:ies of incomplete hypercube computers,” in Proc. 10th IEEE Inr. Con$ 13isrrib. Computing Syst., May 1990, pp. 262-269. 171 H.-L. Chen and N.-F. Tzeng, “Distributed identification of all maximal incomplete subcubes in a faulty hypercube,” in Proc. 8th Int. Parallel Processing Synip., Apr. 1994, pp. 723-728. 181 S. R. Deshpande and R. M. Jenevein, “Scalability of binary tree on a hypercube,” in Proo. 1986 Int. Cont Parallel Processing, zyxwvutsrq Aug. 1986, pp. 661-668. 191 A. Y. Wu, “Embedding of tree networks into hypercubes,” J. Parallel Distrih. Computing, vol. 2, pp, 238-249, 1985. [IO] N.-F. Tzeng, zyxwvutsrqp P.-J. Chuang, and H.-L. Chen, “Embeddings in incomplete hypercubes,” in Proc. 1990 Inr. Conf Purallel Processing, vol. 111, Aug. 1990, pp. 335-339. Cambridge, MA: The MIT Press, 1987. Hierarchical Classification of Permutation Classes in Multistage Interconnection Networks Nabanita Das, Bhargab B. Bhattacharya, and Jayasree Dattagupta Abstract- This brief contribution explores a new hierarchy among different permutation classes, that has many applications in multistage interconnection networks. The well-known LC (linear-complement) class is shown to be merely a subset of the closure set of the BP (bit-permute) class, known as the BPCL (bit-permute-closure) class; the closure is obtained by applying certain group-transformation rules on the BP- permutations. It indicates that for every permutation P of the LC class, there exists a permutation P* in the BP class, such that the conflict graphs of P and P* are isomorphic, for n-stage MIN’s. This obviates the practice of treating the LC class as a special case; the existing algorithm for optimal routing of BPC class in an 7)-stage MIN, can take care of optimal routing of the LC class as well. Finally, the relationships of BPCL with other classes of permutations, e.g., LIE (linear-input-equivalence), BPIE (hit-permute-input-equivalence), BPOE (bit-permute-output-equivalence) are also exposed. Apart from lending better understanding and an integral view of the universe of permutations, these results are found to be useful in accelerating routability in n-stage MIN’s as well as in (2n - lkstage Benes and shuffle-exchange networks. Index Terms- Multistage interconnection networks (MIN), BP(bit- permute) permutations, linear permutations, baseline network, Benes network, conflict graph, optimal routing. I. INTRODUCTION Design and analysis of multistage interconnection networks (MIN) play a crucial role in large scale parallel processing systems. Various topologies of such networks have been reported in the literature for use in SIMD and MIMD computers. An 3‘ x Ai unique-path, full- access MIN, e.g., the baseline or omega, has 11 (= log, zyx W) stages and (N.n/2) binary switches [I]-161. One fundamental criterion in the selection and design of a MIN is its permutation capability [7], Le., the ability to establish simultaneous one-to-one and onto communications among different modules, usually represented as a permutation. Routing an arbitrary permutation P through a unique-path full- access MIN, may require the usage of common links leading to a conflict, and therefore, may not be realizable in a single pass. The problem of optimal routing is to determine the minimum number of passes required to realize P. Equivalently. one needs to partition P into minimum number of subsets, such that transmissions included in each subset are conflict-free. and hence routable in a single pass. The conflict information is usually represented by a graph, called the conflict graph G(I-. E) [8], that consists of .V vertices each representing a transmission of P; two vertices are adjacent if and only if the corresponding transmissions are conflicting, Le., they demand a common link in the network. The optimal routing problem can then be mapped to the well-known graph-coloring problem [8], which for an arbitrary permutation, is NP-hard [9]. A heuristic algorithm of complexity O(:Y“) was reported in [IO] to tackle an arbitrary permutation. However, the BPC (bit-permute-complement) class of permutations, is optimally routable in omegddelta network [XI; the time complexity of the algorithm turns out to be linear in the number of switches. Later, it has been shown that the same algorithm can Manuscript received October 13, 1992; revised August 16, 1993. The authors are with the Electronics Unit, Indian Statistical Insti- tute, 203, Barrackpore Trunk Road, Calcutta 700 035, India; +mail: bhargab@ isical.ernet.in. IEEE Log Number 9404363. 0018-9340/94$04.00 0 1994 IEEE