This Article 
 Bibliographic References 
 Add to: 
Products of Networks with Logarithmic Diameter and Fixed Degree
September 1995 (vol. 6 no. 9)
pp. 963-975

Abstract—This paper first analyzes some general properties of product networks pertinent to parallel architectures and then focuses on three case studies. These are products of complete binary trees, shuffle-exchange, and de Bruijn networks. It is shown that all of these are powerful architectures for parallel computation, as evidenced by their ability to efficiently emulate numerous other architectures. In particular, r-dimensional grids, and r-dimensional meshes of trees can be embedded efficiently in products of these graphs, i.e. either as a subgraph or with small constant dilation and congestion. In addition, the shuffle-exchange network can be embedded in r-dimensional product of shuffle exchange networks with dilation cost 2r and congestion cost 2. Similarly, the de Bruijn network can be embedded in r-dimensional product of de Bruijn networks with dilation cost r and congestion cost 4. Moreover, it is well known that shuffle-exchange and de Bruijn graphs can emulate the hypercube with a small constant slowdown for “normal” algorithms. This means that their product versions can also emulate these hypercube algorithms with constant slowdown. Conclusions include a discussion of many open research areas.

[1] M. Baumslag and F. Annextein,“A unified framework for off-line permutation routing in product petworks,” Math. Systems Theory, vol. 24, no. 4, pp. 233-251, 1991.
[2] S. Bhatt,F. Chung,J.-W. Hong,T. Leighton,, and A. Rosenberg,“Optimal simulations by butterfly networks,” Proc. 20th Ann. ACM Symp. Theory of Computing, pp. 192-204, May 1988.
[3] 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.
[4] M.Y. Chan,“Embedding of grids into optimal hypercubes,” SIAM J. Computing, vol. 20, pp. 834-864, Oct. 1991.
[5] K. Efe,“Embedding mesh of trees in the hypercube,” J. Parallel and Distributed Computing, vol. 11, no. 3, pp. 222-230, Mar. 1991.
[6] K. Efe, “The Crossed Cube Architecture for Parallel Computing,” IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 5, pp. 513-524, Sept.-Oct. 1992.
[7] K. Efe,“Embedding large mesh of trees and related networks on smaller hypercubes with load balancing,” Proc. Int’l Conf. Parallel Processing, vol. 3, pp. 311-315, 1993.
[8] T. El-Ghazawi and A. Youssef,“A unified approach to fault tolerant routing,” Proc. 12th Int’l Conf. Distributed Computing Systems, pp. 210-217,Yokohama, Japan, June 1992.
[9] R. Feldmann and P. Mysliwietz,“The shuffle exchange network has a Hamiltonian path,” Proc. Mathematical Foundations of Computer Science, pp. 246-254, 1992.
[10] A. Fernandez and K. Efe,“Efficient VLSI layouts for homogeneous product networks,” Technical Report 94-8-4, Center for Advanced Computer Studies, Univ. ofSouthwestern Louisiana (submitted for publication).
[11] J.P. Fishburn and R.A. Finkel,“Quotient networks,” IEEE Trans. Computers, vol. 31 no. 4, pp. 288-295, Apr. 1982.
[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] I. Havel and P. Kiebl,“Embedding the polytomic tree into the n-cube,” Casopis pro Pestovan i Matematiky, vol. 98, pp. 307-314, 1973.
[14] M. Imase,T. Soneko,, and K. Okada,“Connectivity of regular directed graphs with small diameters,” IEEE Trans. Computers, vol. 34, no. 3, pp. 267-273, Mar. 1985.
[15] R. Koch,T. Leighton,B. Maggs,S. Rao,, and A. L. Rosenberg,“Work-preserving emulations for fixed-connection networks,” Proc. 21st Ann. ACM Symp. Theory of Computing, pp. 227-240,Seattle, May 1989.
[16] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[17] A.L. Rosenberg,“Product-shuffle networks: Toward reconciling shuffles and butterflies,” Discrete Applied Mathematics, vol. 37/38, pp. 465-488, July 1992.
[18] M. Yoeli,“Binary ring sequences,” American Math. Monthly, vol. 69, pp. 852-855, 1962.
[19] C.D. Thompson,“A complexity theory for VLSI,” PhD thesis, Carnegie-Mellon Univ., Aug. 1980.
[20] A.K. Gupta,A.J. Boals,N.A. Sherwani,, and S.E. Hambrusch,“A lower bound on embedding large hypercubes into small hypercubes,” Congressus Numerantium, vol. 78, pp. 141-151, 1990.

Index Terms:
Product networks, interconnection networks, parallel architectures, multiprocessors, graph embedding, application specific array processors, emulation, embedded architectures.
Kemal Efe, Antonio Fernández, "Products of Networks with Logarithmic Diameter and Fixed Degree," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 9, pp. 963-975, Sept. 1995, doi:10.1109/71.466633
Usage of this product signifies your acceptance of the Terms of Use.