This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Cyclic-Cubes: A New Family of Interconnection Networks of Even Fixed-Degrees
December 1998 (vol. 9 no. 12)
pp. 1253-1268

Abstract—We introduce a new family of interconnection networks that are Cayley graphs with fixed degrees of any even number greater than or equal to four. We call the proposed graphs cyclic-cubes because contracting some cycles in such a graph results in a generalized hypercube. These Cayley graphs have optimal fault tolerance and logarithmic diameters. For comparable number of nodes, a cyclic-cube can have a diameter smaller than previously known fixed-degree networks. The proposed graphs can adopt an optimum routing algorithm known for one of its subfamilies of Cayley graphs. We also show that a graph in the new family has a Hamiltonian cycle and, hence, there is an embedding of a ring. Embedding of meshes and hypercubes are also discussed.

[1] S.B. Akers and B. Krishnamurthy, "On Group Graphs and Their Fault Tolerance," IEEE Trans. Computers, vol. 36, no. 7, pp. 885-888, July 1987.
[2] 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.
[3] S. Lakshmivarahan, J.S. Jwo, and S.K. Dhall, "Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey," Parallel Computing, vol. 19, pp. 361-407, 1993.
[4] 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.
[5] S. Öhring, F. Sarkar, S.K. Das, and D.H. Hohndel, "Cayley Graph Connected Cycles: A New Class of Fixed-Degree Interconnection Networks," Proc. Int'l Conf. Systems Sciences, pp. 479-487, 1995.
[6] F.P. Preparata and J. Vuillemin, “The Cube-Connected Cycles: A Versatile Network for Parallel Computation,” Comm ACM, vol. 24, no. 5, pp. 300-309, 1981.
[7] P. Vadapalli and P.K. Srimani, “A New Family of Cayley Graph Interconnection Networks of Constant Degree Four” IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 1, pp. 26-32, Jan. 1996.
[8] C.L. Seitz et al., "The Architecture and Programming of the Ametak Series 2010," Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 33-37, Jan. 1988.
[9] C.L. Seitz et al., "Submicron Systems Architecture Project Semiannual Technical Report," Technical Report Caltec-CS-TR-88-18, Calif. Inst. of Tech nology, Nov. 1988.
[10] W.J. Dally et al., The J-Machine: A Fine-Grain Concurrent Computer. Elsevier Science Publishers B.V., 1989.
[11] W.J. Dally, "The Message-Driven Processor: A Multicomputer Processing Node with Efficient Mechanisms," IEEE Micro, pp. 23-39, Apr. 1992.
[12] S. Borkar, R. Cohn, G. Cox, S. Gleason, T. Gross, H.T. Kung, M. Lam, B. Moore, C. Peterson, J. Pieper, L. Rankin, P.S. Tseng, J. Sutton, J. Urbanski, and J. Webb iWarp: An Integrated Solution to High-Speed Parallel Computing, Proc. 1988 Int'l Conf. Supercomputing, pp. 330-339., IEEE CS and ACM SIGARCH, Orlando, Fla., Nov. 1988.
[13] B. Bose, B. Broeg, Y. Kwon, and Y. Ashir, "Lee Distance and Topological Properties of k-Ary n-Cubes," IEEE Trans. Computers, vol. 44, no. 8, pp. 1,021-1,030, Aug. 1995.
[14] N.-F. Tzeng and S. Wei, “Enhanced Hypercubes,” IEEE Trans. Computers, vol. 40, no. 3, pp. 284-294, Mar. 1991.
[15] A. El-Amawy and S. Latifi, "Properties and Performance of Folded Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 1, pp. 31-42, 1991.
[16] D.K. Pradhan, "Fault Tolerant VLSI Architectures Based on de Bruijn Graphs (Galileo in the Mid Nineties)," DIMACS Series Discrete Math., vol. 5, 1991.
[17] B. Alspach, "Cayley Graphs with Optimal Fault Tolerance," IEEE Trans. Computers, vol. 41, no. 10, pp. 1,337-1,339, Oct. 1992.
[18] 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.
[19] F. Harary, Graph Theory. Addison Wesley, 1971.
[20] G.A. Dirac, "Generalizations du Theoreme de Menger," C.R. Academie Science Paris, pp. 4,252-4,253, 1960.
[21] K. Menger, "Zur Allgemeinen Kurventheorie," Fundamentals Math., pp. 96-115, 1927.
[22] P.T. Gaugham and S. Yalamanchilli, "Adaptive Routing Protocols for Hypercube Interconnection Networks," Computer, May 1993.
[23] M. Marsic, "Hamiltonian Circuits in Cayley Graphs," Discrete Math., pp. 49-54, 1982.
[24] A. Tucker, Applied Combinatorics, second ed. John Wiley and Sons, 1984.
[25] P. Ramanathan and S. Chalasani, "Resource Placement with Multiple Adjacency Constraints in k-Ary n-Cubes," IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 5, pp. 511-519, May 1995.

Index Terms:
Cayley graphs, generalized hypercube, fixed degree, interconnection.
Citation:
Ada Wai-chee Fu, Siu-Cheung Chau, "Cyclic-Cubes: A New Family of Interconnection Networks of Even Fixed-Degrees," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 12, pp. 1253-1268, Dec. 1998, doi:10.1109/71.737700
Usage of this product signifies your acceptance of the Terms of Use.