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Optimal Distance Networks of Low Degree for Parallel Computers
October 1991 (vol. 40 no. 10)
pp. 1109-1124

The authors introduce and study a family of interconnection schemes, the Midimew networks, based on circulant graphs of degree 4. A family of such circulants is determined and shown to be optimal with respect to two distance parameters simultaneously, namely maximum distance and average distance, among all circulants of degree 4.. These graphs are regular, point-symmetric, and maximally connected, and one such optimal graph exists for any given number of nodes. The proposed interconnection schemes consist of mesh-connected networks with wrap-around links, and are isomorphic to the optimal distance circulants previously considered. Ways to construct one such network for any number of nodes are shown, their good properties to build interconnection schemes for multicomputers are examined, and some interesting particular cases are discussed. The problem of routing is also addressed, and a basic algorithm is provided which is adequate for implementing the routing policy required to convey messages, traversing shortest paths between nodes.

[1] D. P. Agrawalet al., "Evaluating the performance of multicomputer configurations,"IEEE Comput. Mag., vol. 19, pp. 23-37, May 1986.
[2] S. B. Akers and B. Krishnamurthy, "A group-theoretic model for symmetric interconnection networks,"IEEE Trans. Comput., vol. 38, no. 4, pp. 555-565, Apr. 1989.
[3] W. C. Athas, "Fine grain concurrent computations," Caltech-5242:TR:87, Ph.D. dissertation, May 1987.
[4] W. H. Athas and C. L. Seitz, "Multicomputers: Message-passing concurrent computers,"IEEE Computer, vol. 21, no. 8, pp. 9-24, Aug. 1988.
[5] G. H. Barnes, R. M. Brown, M. Kato, D. J. Kuck, D. L. Slotnik, and R. A. Stokes, "The ILLIAC IV computer,"IEEE Trans. Comput., vol. C-17, pp. 746-757, 1968.
[6] R. Beivide, E. Herrada, J. L. Balcázar, and J. Labarta, "Optimized mesh-connected networks for SIMD and MIMD architectures," inProc. 14th Int. Symp. Comput. Architecture, June 1987, pp. 163-170.
[7] J.-C. Bermond, G. Illiades, and C. Peyrat, "An optimization problem in distributed loop computer networks," inProc. 3rd Int. Conf. on Combinatorial Mathematics, New York Academy of Sciences, 1985, pp. 1-13.
[8] F.T. Boesch and R. Tindell, "Circulants and their connectivities,"J. Graph Theory, vol. 8, pp. 487-499, 1984.
[9] F. T. Boesch and J. Wang, "Reliable circulant networks with minimum transmission delay,"IEEE Trans. Circuits Syst., vol. CAS-32, pp. 1286-1291, 1985.
[10] W. J. H. Bronnenberg, L. Nijman, E. A. M. Odijk, and R. A. H. van Twist, "DOOM: A decentralized object-oriented machine,"IEEE Micro, vol. 7, pp. 52-69, Oct. 1987.
[11] W. J. Dally, "On the performance ofK-aryn-cube interconnection networks," Caltech Comput. Sci., Tech. Rep. 5228-TR86, 1986.
[12] W.J. Dally and C.L. Seitz, "Deadlock-Free Message Routing in Multiprocessor Interconnection Networks,"IEEE Trans. Computers, Vol. C-36, No. 5, May 1987, pp. 547-553.
[13] W. J. Dally and C. L. Seitz, "The torus routing chip,"Distributed Comput.vol. 1, no. 3, 1986.
[14] P. W. Dowd and K. Jabbour, "Spanning multiaccess channel hypercube computer interconnection,"IEEE Trans. Comput., vol. 37, no. 9, pp. 1137-1142, Sept. 1988.
[15] M. A. Fiol, J. L. A. Yebra, I. Alegre, and M. Valero, "A discrete optimization problem in local networks and data alignment,"IEEE Trans. Comput., vol. C-36, no. 6, pp. 702-713, June 1987.
[16] F. Harary, "The maximum connectivity of a graph,"Proc. Nat. Acad. Sci. USA, vol. 48, pp. 1142-1146, 1962.
[17] F. K. Hwang, "Comments on 'Reliable loop topologies for large local computer networks',"IEEE Trans. Comput., vol. C-36, pp. 383-384, Mar. 1987.
[18] J. G. Kuhl and S. M. Reddy, "Fault-tolerance considerations in large, multiple-processor systems,"IEEE Computer, vol. 19, no. 3, pp. 56-67, Mar. 1986.
[19] A. J. Martin, "The Torus: and exercise in constructing a processor surface," inProc. 2nd Caltech Conf. VLSI, Jan. 1981.
[20] N. F. Maxemchuck, "Regular mesh topologies in local and metropolitan area networks,"AT&T Tech. J., vol. 64, pp. 1659-1685, Sept. 1985.
[21] J. Y. Ngai and Ch. L. Seitz, "A framework for adaptive routing in multicomputer networks," inProc. 1989 ACM Symp. on Parallel Algorithms and Architectures, June 1989, pp. 1-9.
[22] C. S. Raghavendra, M. Gerla, and A. Avizienis, "Reliable loop topologies for large local computer networks,"IEEE Trans. Comput., vol. C-34, no. 1, pp. 46-55, Jan. 1985.
[23] D. A. Reed and D. C. Grunwald, "The performance of multicomputer interconnection networks,"IEEE Comput. Mag., vol. 20, pp. 63-73, June 1987.
[24] D. A. Reed and H. D. Schwetman, "Cost-performance bounds for multimicrocomputer networks,"IEEE Trans. Comput., vol. C-32, no. 1, pp. 83-95, Jan. 1983.
[25] C. L. Seitz, "The Cosmic Cube,"Commun. ACM, pp. 22-33, Jan. 1985.
[26] C. L. Seitz, J. Seizovic, and W.-K. Su, "The C programmers abbreviated guide to multicomputer programming," Caltech Comput. Sci., Tech. Rep. Caltech-CS-Tr-88-1, Jan. 1988.
[27] C. H. Sequin, "Double twisted torus networks for VLSI processor arrays," inProc. 8th Int. Symp. Comput. Architecture, May 1981, pp. 471-550.
[28] R. S. Wilkov, "Analysis and design of reliable computer networks,"IEEE Trans. Comput., vol. C-20, no. 3, pp. 670-678, June 1972.
[29] L. D. Wittie, "Communication structures for large networks of microcomputers,"IEEE Trans. Comput., vol. C-30, no. 4, pp. 264-273, Apr. 1981.

Index Terms:
interconnection networks; distance networks; parallel computers; Midimew networks; circulant graphs; distance parameters; maximum distance; average distance; nodes; mesh-connected networks; wrap-around links; good properties; multicomputers; routing; graph theory; multiprocessor interconnection networks; parallel machines.
Citation:
R. Beivide, E. Herrada, J.L. Balcazar, A. Arruabarrena, "Optimal Distance Networks of Low Degree for Parallel Computers," IEEE Transactions on Computers, vol. 40, no. 10, pp. 1109-1124, Oct. 1991, doi:10.1109/12.93744
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