
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
Behrooz Parhami, Mikhail Rakov, "Perfect Difference Networks and Related Interconnection Structures for Parallel and Distributed Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 16, no. 8, pp. 714724, August, 2005.  
BibTex  x  
@article{ 10.1109/TPDS.2005.96, author = {Behrooz Parhami and Mikhail Rakov}, title = {Perfect Difference Networks and Related Interconnection Structures for Parallel and Distributed Systems}, journal ={IEEE Transactions on Parallel and Distributed Systems}, volume = {16}, number = {8}, issn = {10459219}, year = {2005}, pages = {714724}, doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2005.96}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Parallel and Distributed Systems TI  Perfect Difference Networks and Related Interconnection Structures for Parallel and Distributed Systems IS  8 SN  10459219 SP714 EP724 EPD  714724 A1  Behrooz Parhami, A1  Mikhail Rakov, PY  2005 KW  Bipartite graph KW  bisection width KW  chordal ring KW  degree KW  diameter KW  hyperstar KW  interconnection network KW  lowdiameter network KW  regular network KW  twohop connectivity. VL  16 JA  IEEE Transactions on Parallel and Distributed Systems ER   
Abstract—In view of their applicability to parallel and distributed computer systems, interconnection networks have been studied intensively by mathematicians, computer scientists, and computer designers. In this paper, we propose an asymptotically optimal method for connecting a set of nodes into a perfect difference network (PDN) with diameter 2, so that any node is reachable from any other node in one or two hops. The PDN interconnection scheme, which is based on the mathematical notion of perfect difference sets, is optimal in the sense that it can accommodate an asymptotically maximal number of nodes with smallest possible node degree under the constraint of the network diameter being 2. We present the network architecture in its basic and bipartite forms and show how the related multidimensional PDNs can be derived. We derive the exact average internode distance and tight upper and lower bounds for the bisection width of a PDN. We conclude that PDNs and their derivatives constitute worthy additions to the repertoire of network designers and may offer additional design points that can be exploited by current and emerging technologies, including wireless and optical interconnects. Performance, algorithmic, and robustness attributes of PDNs are analyzed in a companion paper.
[1] S.B. Akers and B. Krishnamurthy, “A GroupTheoretic Model for Symmetric Interconnection Networks,” IEEE Trans. Computers, vol. 38, no. 4, pp. 555566, Apr. 1989.
[2] AE. AlAyyoub and K. Day, “The Hyperstar Interconnection Network,” J. Parallel and Distributed Computing, vol. 48, no. 2, pp. 175199, Feb. 1998.
[3] B.W. Arden and H. Lee, “Analysis of Chordal Ring Networks,” IEEE Trans. Computers, vol. 30, no. 4, pp. 291295, Apr. 1981.
[4] W.C. Arlinghaus, “Block Designs and Latin Squares,” Applications of Discrete Math., J.G. Michaels and K.H. Rosen, eds., McGrawHill, 1991.
[5] A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, section 2.8, Cambridge, pp. 8188, 1998.
[6] L.N. Bhuyan and D.P. Agrawal, “Generalized Hypercube and Hyperbus Structures for a Computer Network,” IEEE Trans. Computers, vol. 33, no. 4, pp. 323333, Apr. 1984.
[7] N. Biggs, Algebraic Graph Theory. Cambridge Univ. Press, 1993.
[8] T.N. Bui, S. Chaudhuri, F.T. Leighton, and M. Sisper, “Graph Bisection Algorithms with Good Average Case Behavior,” Combinatorica, vol. 7, no. 2, pp. 171191, 1987.
[9] G.E. Carlson, J. Cruthirds, H. Section, and C. Wright, “Interconnection Networks Based on Generalization of CubeConnected Cycles,” IEEE Trans. Computers, vol. 34, pp. 769772, Aug. 1985.
[10] W.J. Dally, “Performance Analysis of $k{\hbox{}}{\rm{ary}}$ $n{\hbox{}}{\rm{cube}}$ Interconnection Networks,” IEEE Trans. Computers, vol. 39, no. 6, pp. 775785, June 1990.
[11] T.A. Evans and H.B. Mann, “On Simple Difference Sets,” Sankhya: Indian J. Statistics, vol. 11, pp. 357364, 1951.
[12] M.R. Garey, D.D. Johnson, and L. Stockmeyer, “Some Simplified NPComplete Graph Problems,” Theoretical Computer Science, vol. 1, pp. 237267, 1976.
[13] R.K. Guy, Unsolved Problems in Number Theory, second ed., Springer, pp. 118121, 1994.
[14] M. Hall Jr., “A Survey of Difference Sets,” Proc. Am. Math. Soc., vol. 7, pp. 975986, 1956.
[15] M. Hall, Combinatorial Theory. Blaisdell, 1967.
[16] T.P. Kirkman, “On the Perfect rPartitions of $r^2  r + 1$ ,” Trans. Historical Soc. of Lancashire and Cheshire, vol. 9, pp. 127142, 1857.
[17] D.M. Kwai and B. Parhami, “A Unified Formulation of Honeycomb and Diamond Networks,” IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 1, pp. 7480, Jan. 2001.
[18] S. Lakshmivarahan, J.S. Jwo, and S.K. Dahl, “Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Group: A Survey,” Parallel Computing, vol. 19, pp. 361401, 1993.
[19] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, and Hypercubes. Morgan Kaufmann, 1992.
[20] S. Lin and D.J. Costello Jr., Error Control Coding. PrenticeHall, 1983.
[21] P.K. McKinley, Y.j. Tsai, and D.F. Robinson, “Collective Communication in WormholeRouted Massively Parallel Computers,” Computer, vol. 28, no. 12, pp. 3950, Dec. 1995.
[22] L.M. Ni and P.K. McKinley, “A Survey of Wormhole Routing Techniques in Direct Networks,” Computer, vol. 26, no. 2, pp. 6276, Feb. 1993.
[23] B. Parhami, Introduction to Parallel Processing: Algorithms and Architectures. Plenum Press, 1999.
[24] B. Parhami and D.M. Kwai, “Periodically Regular Chordal Rings,” IEEE Trans. Parallel and Distributed Systems, vol. 10, no. 6, pp. 658672, June 1999.
[25] B. Parhami and D.M. Kwai, “Challenges in Interconnection Network Design in the Era of Multiprocessor and Massively Parallel Microchips,” Proc. Int'l Conf. Comm. in Computing, pp. 241246, June 2000.
[26] B. Parhami and D.M. Kwai, “Incomplete $k{\hbox{}}{\rm{ary}}$ $n{\hbox{}}{\rm{cube}}$ and Its Derivatives,” J. Parallel and Distributed Computing, vol. 64, no. 2, pp. 183190, Feb. 2004.
[27] B. Parhami and M. Rakov, “Performance, Algorithmic, and Robustness Attributes of Perfect Difference Networks,” IEEE. Trans. Parallel and Distributed Systems, vol. 16, no. 8, pp. 725736, 2005.
[28] B. Parhami and C.H. Yeh, “Why Network Diameter is Still Important,” Proc. Int'l Conf. Comm. in Computing, pp. 271274, June 2000.
[29] F.P. Preparata and J. Vuillemin, “The CubeConnected Cycles: A Versatile Network for Parallel Computation,” Comm. ACM, vol. 24, no. 5, pp. 300309, May 1981.
[30] M. Rakov, “Method of Interconnecting Nodes and a Hyperstar Interconnection Structure,” US Patent No. 5 734 580, Mar. 1998.
[31] M. Rakov, “Hyperstar: A New Interconnection Topology,” J. China Univ. of Posts and Telecomm., vol. 5, no. 2, pp. 1018, Dec. 1998.
[32] M. Rakov, “Multidimensional Hyperstar and Hyperhub Interconnection Methods and Structures,” US Patent Application No. 09/410 175, Sept. 1999.
[33] M. Rakov, “Hyperstar and Hyperhub Optical Networks Interconnection Methods and Structures,” US Patent Application No. 09/634 129, Aug. 2000.
[34] M. Rakov and J. Mackall, “Method of Interconnecting Functional Nodes and a Hyperstar Interconnection Structure,” US Patent No. 6 330 706, Dec. 2001.
[35] M. Rakov and O.A. Vakulskiy, “Computer Systems Design Using Apparatus of the Ideal Ring Proportions,” Theses of the Int'l Conf. New Information Technologies, Voronezh Polytechnic Inst., (in Russian), pp. 7778, 1992.
[36] M. Rakov, O. Vakulskiy, and I. Stetsko, “Using the Ideal Code Proportion Apparatus for Improving the Local Nets Main Characteristics,” Proc. Int'l Seminar on Local Area Networks, Riga, IEVT, (in Russian), pp. 3136, 1992.
[37] J.P. Robinson and A.J. Bernstein, “A Class of Binary Recurrent Codes with Limited Error Propagation,” IEEE Trans. Information Theory, vol. 13, no. 1, pp. 106113, Jan. 1967.
[38] J. Singer, “A Theorem in Finite Projective Geometry and Some Applications to Number Theory,” Trans. Am. Math. Soc., vol. 43, pp. 377385, 1938.
[39] M.B. Sverdlik and A.N. Meleshkevich, “Synthesis of Optimum Pulsed Sequences Having the Property of ‘No More than One Coincidence,’” Radio Eng. and Electronic Physics, vol. 19, no. 4, pp. 4654, 1974.
[40] M. Sverdlik, Optimal Discrete Signals, Moscow, Soviet Radio, (in Russian), 1975.
[41] M.B. Sverdlik and A.N. Meleshkevich, “Table of Optimal Sets with the Property of ‘No More than One Coincidence,’” Radio Eng. and Electronic Physics, vol. 20, no. 6, pp. 148150, 1975.
[42] M.B. Sverdlik and A.N. Meleshkevich, “Synthesis of Ensembles of Pulse Sequences with Properties of ‘No More than One Coincidence,’” Radio Eng. and Electronic Physics, vol. 21, no. 7, pp. 6168, 1976.
[43] J.H. van Lint and R.M. Wilson, A Course in Combinatorics, (see chapter 27 entitled “Difference Sets and Automorphisms”), Cambridge Univ. Press, 1992.
[44] C.H. Yeh and B. Parhami, “Swapped Networks: Unifying the Architectures and Algorithms of a Wide Class of Hierarchical Parallel Processors,” Proc. Int'l Conf. Parallel and Distributed Systems, pp. 230237, June 1996.
[45] A. Youssef, “Design and Analysis of Product Networks,” Proc. Symp. Frontiers of Massively Parallel Computation, pp. 521528, Feb. 1995.
[46] S.G. Ziavras, Q. Wang, and P. Papathanasiou, “Viable Architectures for HighPerformance Computing,” The Computer J., vol. 46, no. 1, pp. 3654, 2003.