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A Theory for Total Exchange in Multidimensional Interconnection Networks
July 1998 (vol. 9 no. 7)
pp. 639-649

Abstract—Total exchange (or multiscattering) is one of the important collective communication problems in multiprocessor interconnection networks. It involves the dissemination of distinct messages from every node to every other node. We present a novel theory for solving the problem in any multidimensional (cartesian product) network. These networks have been adopted as cost-effective interconnection structures for distributed-memory multiprocessors. We construct a general algorithm for single-port networks and provide conditions under which it behaves optimally. It is seen that many of the popular topologies, including hypercubes, k-ary n-cubes, and general tori satisfy these conditions. The algorithm is also extended to homogeneous networks with 2k dimensions and with multiport capabilities. Optimality conditions are also given for this model. In the course of our analysis, we also derive a formula for the average distance of nodes in multidimensional networks; it can be used to obtain almost closed-form results for many interesting networks.

[1] S.B. Akers and B. Krishnamurthy, “A Group-Theoretic Model for Symmetric Interconnection Networks,” IEEE Trans. Computers, vol. 38, no. 4, pp. 555-566, Apr. 1989.
[2] D. Bertsekas, C. Ozveren, G. Stamoulis, P. Tseng, and J. Tsitsiklis, "Optimal Communication Algorithms for Hypercubes," J. Parallel and Distributed Computing, vol. 11, pp. 263-275, 1991.
[3] D.P. Bertsekas and J.N. Tsitsiklis, Parallel and Distributed Computation.Englewood Cliffs, N.J.: Prentice Hall International, 1989.
[4] L.N. Bhuyan and D.P. Agrawal, "Generalized Hypercube and Hyperbus Structures for a Computer Network," IEEE Trans. Computers, vol. 33, no. 4, pp. 323-333, Apr. 1984.
[5] F. Buckley and F. Harary, Distance in Graphs.Reading, Mass.: Addison-Wesley, 1990.
[6] 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.
[7] V.V. Dimakopoulos and N.J. Dimopoulos, "Optimal Total Exchange in Linear Arrays and Rings," Proc. ISPAN'94 Int'l Symp. Parallel Architecture, Algorithms, and Networks, pp. 230-237,Kanazawa, Japan, Dec. 1994.
[8] V.V. Dimakopoulos and N.J. Dimopoulos, "Optimal Total Exchange in Cayley Networks," Proc. EUROPAR'96, European Conf. Parallel Processing, pp. 340-346,Lyon, France, Aug. 1996.
[9] K. Efe and A. Fernández, "Products of Networks with Logarithmic Diameter and Fixed Degree," IEEE Trans. Parallel and Distributed Systems, vol. 6, pp. 963-975, Sept. 1995.
[10] P. Fraigniaud and E. Lazard, "Methods and Problems of Communication in Usual Networks," Discrete Applied Math., vol. 53, pp. 79-133, 1994.
[11] D.B. Gannon and J. van Rosendale, "On the Impact of Communication Complexity on the Design of Parallel Numerical Algorithms," IEEE Trans. Computers, vol. 33, no. 12, pp. 1,180-1,194, Dec. 1984.
[12] E. Ganesan and D.K. Pradhan,“The hyper-de Bruijn multiprocessor networks: Scalable versatile architecture,” IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 9, pp. 962-978, Sept. 1993.
[13] S. Hiranandani, K. Kennedy, and C.-W. Tseng, "Compiling Fortran D for MIMD Distributed-Memory Machines," Comm. ACM, vol. 35, no. 8, pp. 66-80, Aug. 1992.
[14] S.L. Johnsson, "Communication Efficient Basic Linear Algebra Computations on Hypercube Architectures," J. Parallel and Distributed Computing, vol. 4, pp. 133-172, 1987.
[15] S.L. Johnsson and C.T. Ho,“Spanning graphs for optimum broadcasting and personalizedcommunication in hypercubes,” IEEE Trans. Computers, vol. 38, no. 9, pp. 1,249-1,268, Sept. 1989.
[16] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[17] D.B. Loveman, “High Performance Fortran,” IEEE Parallel and Distributed Technology, vol. 1, pp. 25-42, Feb. 1993.
[18] P.K. McKinley, Y.-J. Tsai, and D. Robinson, "Collective Communication in Wormhole-routed Massively Parallel Computers," Computer, vol. 28, no. 12, pp. 39-50, Dec. 1995.
[19] Message Passing Interface Forum, "MPI: A Message-Passing Interface Standard," Technical Report CS-93-214, Univ. of Tennessee, Apr. 1994.
[20] J. Misic and Z. Jovanovic, "Communication Aspects of the Star Graph Interconnection Network," IEEE Trans. Parallel and Distributed Systems, vol. 5, no. 7, pp. 678-687, July 1994.
[21] S.R. Öhring and S.K. Das, "The Folded Petersen Cube Network: New Competitors for the Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 2, pp. 151-168, Feb. 1996.
[22] D.A. Reed and D.C. Grunwalk, "The Performance of Multicomputer Interconnection Networks," Computer, vol. 20, no. 6, pp. 63-73, June 1987.
[23] A.L. Rosenberg,“Product-shuffle networks: Toward reconciling shuffles and butterflies,” Discrete Applied Mathematics, vol. 37/38, pp. 465-488, July 1992.
[24] Y. Saad and M.H. Schultz, "Data Communication in Hypercubes," J. Parallel and Distributed Computing, vol. 6, pp. 115-135, 1989.
[25] E.A. Varvarigos and D.P. Bertsekas, "Communication Algorithms for Isotropic Tasks in Hypercubes and Wraparound Meshes," Parallel Computing, vol. 18, pp. 1,233-1,257, 1992.
[26] A. Youssef and B. Narahari, "Banyan-Hypercube Networks," IEEE Trans. Parallel and Distributed Systems, vol. 1, no. 2, pp. 160-169, Apr. 1990.

Index Terms:
Collective communications, interconnection networks, multidimensional networks, packet-switched networks, total exchange.
Citation:
Vassilios V. Dimakopoulos, Nikitas J. Dimopoulos, "A Theory for Total Exchange in Multidimensional Interconnection Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 7, pp. 639-649, July 1998, doi:10.1109/71.707541
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