The Community for Technology Leaders
RSS Icon
Issue No.03 - March (2014 vol.25)
pp: 570-580
Jordi Arjona Aroca , Institute IMDEA Networks, Leganés
Antonio Fernandez Anta , Institute IMDEA Networks, Madrid
The bisection width of interconnection networks has always been important in parallel computing, since it bounds the speed at which information can be moved from one side of a network to another, i.e., the bisection bandwidth. Finding its exact value has proven to be challenging for some network families. For instance, the problem of finding the exact bisection width of the multidimensional torus was posed by Leighton [1, Problem 1.281] and has remained open for almost 20 years. We provide two general results that allow us to obtain upper and lower bounds on the bisection width of any product graph as a function of some properties of its factor graphs. The power of these results is shown by deriving the exact value of the bisection width of the torus, as well as of several $(d)$-dimensional classical parallel topologies that can be obtained by the application of the Cartesian product of graphs. We also apply these results to data centers, by obtaining bounds for the bisection bandwidth of the $(d)$-dimensional BCube network, a recently proposed topology for data centers.
mesh-connected trees, Bisection bandwidth, bisection width, torus, BCube, product graphs, complete binary trees, extended trees,
Jordi Arjona Aroca, Antonio Fernandez Anta, "Bisection (Band)Width of Product Networks with Application to Data Centers", IEEE Transactions on Parallel & Distributed Systems, vol.25, no. 3, pp. 570-580, March 2014, doi:10.1109/TPDS.2013.95
[1] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Array, Trees, Hypercubes. Morgan Kaufmann, 1992.
[2] J. Duato, S. Yalamanchili, and N. Lionel, Interconnection Networks: An Engineering Approach. Morgan Kaufmann, 2002.
[3] W. Dally and B. Towles, Principles and Practices of Interconnection Networks. Morgan Kaufmann, 2003.
[4] D.N. Jayasimha, B. Zafar, and Y. Hoskote, "On Chip Interconnection Networks Why They Are Different and How to Compare Them," Intel, 2006.
[5] M. Mirza-Aghatabar, S. Koohi, S. Hessabi, and M. Pedram, "An Empirical Investigation of Mesh and Torus NoC Topologies under Different Routing Algorithms and Traffic Models," Proc. 10th Euromicro Conf. Digital System Design Architectures, Methods and Tools, pp. 19-26, 2007.
[6] D. Zydek and H. Selvaraj, "Fast and Efficient Processor Allocation Algorithm for Torus-Based Chip Multiprocessors," Computers and Electrical Eng., vol. 37, pp. 91-105, Jan. 2011.
[7] E. Salminen, A. Kulmala, and T.D. Hamalainen, "Survey of Network-on-Chip Proposals," white paper, OCP-IP, pp. 1-13, 2008.
[8] C. Guo, G. Lu, D. Li, H. Wu, X. Zhang, Y. Shi, C. Tian, Y. Zhang, and S. Lu, "BCube: A High Performance, Server-Centric Network Architecture for Modular Data Centers," Proc. SIGCOMM, pp. 63-74, 2009.
[9] C. Guo, H. Wu, K. Tan, L. Shi, Y. Zhang, and S. Lu, "Dcell: A Scalable and Fault-Tolerant Network Structure for Data Centers," Proc. SIGCOMM, pp. 75-86, 2008.
[10] A. Youssef, "Cartesian Product Networks," Proc. Int'l Conf. Parallel Processing, vol. 1, pp. 684-685, 1991.
[11] A. Youssef, "Design and Analysis of Product Networks," Proc. Fifth Symp. Frontiers of Massively Parallel Computation (Frontiers '95), pp. 521-528, 1995.
[12] W.J. Dally, "Performance Analysis of K-Ary N-Cube Interconnection Networks," IEEE Trans. Computers, vol. 39, no. 6, pp. 775-785, June 1990.
[13] J.D.P. Rolim, O. Sýkora, and I. Vrto, "Optimal Cutwidths and Bisection Widths of 2- and 3-Dimensional Meshes," Proc. 21st Int'l Workshop Graph-Theoretic Concepts in Computer Science (WG), pp. 252-264, 1995.
[14] K. Efe and A. Fernández, "Products of Networks with Logarithmic Diameter and Fixed Degree," IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 9, pp. 963-975, Sept. 1995.
[15] K. Nakano, "Linear Layout of Generalized Hypercubes," Int'l J. Foundations Computer Science, vol. 14, no. 1, pp. 137-156, 2003.
[16] K. Efe and G.-L. Feng, "A Proof for Bisection Width of Grids," Proc. World Academy of Science, Eng. and Technology, vol. 27, no. 31, pp. 172-177, 2007.
[17] M.C. Azizoğlu and Ö. Eğecioğlu, "The Isoperimetric Number and the Bisection Width of Generalized Cylinders," Electronic Notes in Discrete Math., vol. 11, pp. 53-62, 2002.
[18] M.C. Azizoğlu and Ö. Eğecioğlu, "The Bisection Width and the Isoperimetric Number of Arrays," Discrete Applied Math., vol. 138, nos. 1/2, pp. 3-12, 2004.
[19] K. Efe and A. Fernández, "Mesh-Connected Trees: A Bridge between Grids and Meshes of Trees," IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 12, pp. 1281-1291, Dec. 1996.
[20] M.C. Azizoğlu and Ö. Eğecioğlu, "Extremal Sets Minimizing Dimension-Normalized Boundary in Hamming Graphs," SIAM J. Discrete Math., vol. 17, no. 2, pp. 219-236, 2003.
[21] Y. Pan, S.Q. Zheng, K. Li, and H. Shen, "An Improved Generalization of Mesh-Connected Computers with Multiple Buses," IEEE Trans. Parallel and Distributed Systems, vol. 12, no. 3, pp. 293-305, Mar. 2001.
31 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool