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Issue No.08 - August (2008 vol.57)
pp: 1046-1056
Carmen Martínez , University of Cantabria, Santander
Ramón Beivide , University of Cantabria, Santander
Esteban Stafford , University of Cantabria, Santander
Miquel Moretó , UPC UPC , Barcelona Barcelona
Ernst M. Gabidulin , Moscow Institute of Physics and Technology, Moscow
ABSTRACT
In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian Integers, the subset of the Complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.
INDEX TERMS
Packet-switching networks, Network topology, Graph theory, Graphs and networks, Packet routing
CITATION
Carmen Martínez, Ramón Beivide, Esteban Stafford, Miquel Moretó, Ernst M. Gabidulin, "Modeling Toroidal Networks with the Gaussian Integers", IEEE Transactions on Computers, vol.57, no. 8, pp. 1046-1056, August 2008, doi:10.1109/TC.2008.57
REFERENCES
[1] N.R. Adiga et al., “An Overview of the BlueGene/L Supercomputer,” Supercomputing 2002 Technical Papers, http://sc-2002.org/paperpdfspap.pap207.pdf , Nov. 2002.
[2] A. Agarwal, “Limits on Interconnection Network Performance,” IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 4, pp. 398-412, Oct. 1991.
[3] R. Beivide, E. Herrada, J.L. Balcázar, and A. Arruabarrena, “Optimal Distance Networks of Low Degree for Parallel Computers,” IEEE Trans. Computers, vol. 40, no. 10, pp. 1109-1124, Oct. 1991.
[4] R. Beivide, E. Herrada, J.L. Balcazar, and J. Labarta, “Optimized Mesh-Connected Networks for SIMD and MIMD Architectures,” Proc. 14th Ann. Int'l Symp. Computer Architecture, pp. 163-169, 1987.
[5] E.R. Berlekamp, Algebraic Coding Theory. Aegean Park Press, 1984.
[6] B. Bose, B. Broeg, Y. Known, and Y. Ashir, “Lee Distance and Topological Properties of $k\hbox{-}{\rm Ary}$ $n\hbox{-}{\rm Cubes}$ ,” IEEE Trans. Computers, vol. 44, no. 8, pp. 1021-1030, Aug. 1995.
[7] J.-Y. Cai, G. Havas, B. Mans, A. Nerurkar, J.-P. Seifert, and I. Shparlinski, “On Routing in Circulant Graphs,” Proc. Fifth Ann. Int'l Computing and Combinatorics Conf., 1999.
[8] J. Camara, M. Moreto, E. Vallejo, R. Bievide, J. Miguel-Alonso, C. Martinez, and J. Navaridas, “Mixed-Radix Twisted Torus Interconnection Networks,” Proc. 21st IEEE Int'l Parallel and Distributed Processing Symp., Mar. 2007.
[9] Cray X1E Datasheet, http://www.cray.com/downloadsX1E_ datasheet.pdf , 2008.
[10] Z. Cvetanovic, “Performance Analysis of the Alpha 21364-Based HP GS1280 Multiprocessor,” Proc. 30th Ann. Int'l Symp. Computer Architecture, pp. 218-228, 2003.
[11] M.A. Fiol, J.L. Yebra, I. Alegre, and M. Valero, “A Discrete Optimization Problem in Local Networks and Data Alignment,” IEEE Trans. Computers, vol. 36, no. 6, pp. 702-713, June 1987.
[12] E. Gabidulin, C. Martínez, R. Beivide, and J. Gutierrez, “On the Weight Distribution of Gaussian Graphs with an Application to Coding Theory,” Proc. Eighth Int'l Symp. Comm. Theory and Applications, pp. 17-22, July 2005.
[13] D. Gómez, J. Gutierrez, A. Ibeas, C. Martínez, and R. Beivide, “On Finding a Shortest Path in Circulant Graphs with Two Jumps,” Lecture Notes in Computer Science, vol. 3595, pp. 777-786, Springer-Verlag, 2005.
[14] K. Huber, “Codes over Tori,” IEEE Trans. Information Theory, vol. 43, no. 2, pp. 740-744, Mar. 1997.
[15] J.H. Jordan and C.J. Potratz, “Complete Residue Systems in the Gaussian Integers,” Math. Magazine, pp. 1-12, 1965.
[16] C. Martínez, E. Vallejo, R. Beivide, C. Izu, and M. Moretó, “Dense Gaussian Networks: Suitable Topologies for On-Chip Multiprocessors,” Int'l J. Parallel Programming, vol. 34, no. 3, June 2006.
[17] C. Martínez, M. Moretó, R. Beivide, and E. Gabidulin, “A Generalization of Perfect Lee Codes over Gaussian Integers,” Proc. IEEE Int'l Symp. Information Theory, July 2006.
[18] C. Martínez, R. Beivide, and E. Gabidulin, “Perfect Codes for Metrics Induced by Circulant Graphs,” IEEE Trans. Information Theory, vol. 53, no. 9, pp. 3042-3052, Sept. 2007.
[19] M.M.K. Martin, M.D. Hill, and D.A. Wood, “Token Coherence: Decoupling Performance and Correctness,” Proc. 30th Ann. Int'l Symp. Computer Architecture, pp. 182-193, 2003.
[20] J. Mellor-Crummey and M. Scott, “Algorithms for Scalable Synchronization on Shared-Memory Multiprocessors,” ACM Trans. Computer Systems, vol. 9, no. 1, pp. 21-65, Feb. 1991.
[21] B. Robic, “Optimal Routing in 2-Jump Circulant Networks,” TR397, Univ. of Cambridge Computer Laboratory, 1996.
[22] S.L. Scott and G.M. Thorson, “The Cray T3E Network: Adaptive Routing in a High Performance 3D Torus,” Proc. IEEE Hot Interconnects IV Symp., Aug. 1996.
[23] C.H. Sequin, “Doubly Twisted Torus Networks for VLSI Processor Arrays,” Proc. Eighth Ann. Int'l Symp. Computer Architecture, pp.471-480, 1981.
[24] C.K. Wong and D. Coppersmith, “A Combinatorial Problem Related to Multimodule Memory Organizations,” J. ACM, vol. 21, no. 3, pp. 392-402, 1974.
[25] Y. Yang, A. Funashi, A. Jouraku, H. Nishi, H. Amano, and T. Sueyoshi, “Recursive Diagonal Torus: An Interconnection Network for Massively Parallel Computers,” IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 7, July 2001.
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