The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.01 - January (2009 vol.20)
pp: 25-33
Iain A. Stewart , University of Durham, Durham
Yonghong Xiang , University of Durham, Durham
ABSTRACT
In this paper we give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Qnk is bipanconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Qnk is m-panconnected, for m=(n(k-1)+2k-6)/2, and (k-1)-pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Qnk, even in the presence of a faulty processor.
INDEX TERMS
Interconnection architectures, Path and circuit problems
CITATION
Iain A. Stewart, Yonghong Xiang, "Bipanconnectivity and Bipancyclicity in k-ary n-cubes", IEEE Transactions on Parallel & Distributed Systems, vol.20, no. 1, pp. 25-33, January 2009, doi:10.1109/TPDS.2008.45
REFERENCES
[1] E. Anderson, J. Brooks, C. Grassl, and S. Scott, “Performance of the Cray T3E Multiprocessor,” Proc. ACM/IEEE Conf. Supercomputing (SC '97), pp. 1-17, 1997.
[2] W.C. Athas and C.L. Seitz, “Multicomputers: Message-Passing Concurrent Computers,” Computer, vol. 21, pp. 9-24, 1988.
[3] Y.A. Ashir and I.A. Stewart, “On Embedding Cycles in $k$ -ary $n$ -cubes,” Parallel Processing Letters, vol. 7, pp. 49-55, 1997.
[4] S. Bettayeb, “On the $k$ -ary Hypercube,” Theoretical Computer Science, vol. 140, pp. 333-339, 1995.
[5] S. Borkar, R. Cohen, 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. ACM/IEEE Conf. Supercomputing (SC '88), pp. 330-339, 1988.
[6] B. Bose, B. Broeg, Y. Kwon, and Y. Ashir, “Lee Distance and Topological Properties of $k$ -ary $n$ -cubes,” IEEE Trans. Computers, vol. 44, pp. 1021-1030, 1995.
[7] Y. Bruck, R. Cypher, and C.T. Ho, “Efficient Fault-Tolerant Mesh and Hypercube Architectures,” Proc. 22nd Int'l Symp. Fault-Tolerant Computing (FTCS '92), pp. 162-169, 1992.
[8] J.P. Brunet and S.L. Johnsson, “All-to-All Broadcast and Applications on the Connection Machine,” Int'l J. Supercomputer Applications, vol. 6, pp. 241-256, 1992.
[9] J.M. Chang, J.S. Yang, J.S. Yang, Y.L. Wang, and Y. Cheng, “Panconnectivity, Fault-Tolerant Hamiltonicity and Hamiltonian-Connectivity in Alternating Group Graphs,” Networks, vol. 44, pp.302-310, 2004.
[10] R. Duncan, “A Survey of Parallel Computer Architectures,” Computer, vol. 23, pp. 5-16, 1990.
[11] T.H. Duncan, “Performance of the Intel iPSC/860 and Ncube 6400 Hypercubes,” Parallel Computing, vol. 17, pp. 1285-1302, 1991.
[12] J. Fan, X. Lin, and X. Jia, “Node-Pancyclicity and Edge-Pancyclicity of Crossed Cubes,” Information Processing Letters, vol. 93, pp. 133-138, 2005.
[13] J.F. Fang, “The Bipanconnectivity and m-Panconnectivity of the Folded Hypercube,” Theoretical Computer Science, vol. 385, pp. 286-300, 2007.
[14] S.Y. Hsieh and G.H. Chen, “Pancyclicity of Möbius Cubes with Maximal Edge Faults,” Parallel Computing, vol. 30, pp. 407-421, 2004.
[15] S.Y. Hsieh, T.J. Lin, and H.L. Huang, “Panconnectivity and Edge-Pancyclicity of 3-Ary $n$ -cubes,” J. Supercomputing, vol. 42, pp. 225-233, 2007.
[16] R.E. Kessler and J.L. Schwarzmeier, “CRAY T3D: A New Dimension for Cray Research,” Proc. 38th IEEE Int'l Computer Conf. (COMPCON '93), pp. 176-182, 1993.
[17] Y. Kikuchi and T. Araki, “Edge-Bipancyclicity and Edge-Fault-Tolerant Bipancyclicity of Bubble-Sort Graphs,” Information Processing Letters, vol. 100, pp. 52-59, 2006.
[18] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, 1992.
[19] T.K. Li, C.H. Tsai, J.J.M. Tan, and L.H. Hsu, “Bipanconnectivity and Edge-Fault-Tolerant Bipancyclicity of Hypercubes,” Information Processing Letters, vol. 87, pp. 107-110, 2003.
[20] M. Ma, G. Liu, and J.M. Xu, “Panconnectivity and Edge-Fault-Tolerant Pancyclicity of Augmented Cubes,” Parallel Computing, vol. 33, pp. 36-42, 2007.
[21] M.D. Noakes, D.A. Wallach, and W.J. Dally, “The J-Machine Multicomputer: An Architectural Evaluation,” Proc. 20th Ann. Int'l Symp. Computer Architecture (ISCA '93), pp. 224-235, 1993.
[22] J.H. Park, H.C. Kim, and H.S. Lim, “Panconnectivity and Pancyclicity of Hypercube-Like Interconnection Networks with Faulty Elements,” Theoretical Computer Science, vol. 377, pp.170-180, 2007.
[23] C.L. Seitz, “The Cosmic Cube,” Comm. ACM, vol. 28, pp. 22-33, 1985.
[24] C.L. Seitz, W.C. Athas, C.M. Flaig, A.J. Martin, J. Scizovic, C.S. Steele, and W.-K. Su, “Submicron Systems Architecture Project Semiannual Technical Report,” Technical Report Caltec-CS-TR-88-18, California Inst. of Tech nology, 1988.
[25] C.H. Tsai, “Linear Array and Ring Embeddings in Conditional Faulty Hypercubes,” Theoretical Computer Science, vol. 314, pp.431-443, 2004.
[26] C.H. Tsai and S.Y. Jang, “Path Bipancyclicity of Hypercubes,” Information Processing Letters, vol. 101, pp. 93-97, 2007.
[27] D. Wang, T. An, M. Pan, K. Wang, and S. Qu, “Hamiltonian-Like Properties of $k$ -ary $n$ -cubes,” Proc. Sixth Int'l Conf. Parallel and Distributed Computing, Applications and Technologies (PDCAT '05), pp. 1002-1007, 2005.
[28] J.M. Xu, Z.Z. Du, and M. Xu, “Edge-Fault-Tolerant Edge-Bipancyclicity of Hypercubes,” Information Processing Letters, vol. 96, pp. 146-150, 2005.
[29] M. Xu, X.D. Hu, and Q. Zhu, “Edge-Bipancyclicity of Star Graphs under Edge-Fault Tolerant,” Applied Math. and Computation, vol. 183, pp. 972-979, 2006.
[30] J.M. Xu and M.J. Ma, “Cycles in Folded Hypercubes,” Applied Math. Letters, vol. 19, pp. 140-145, 2006.
[31] M.C. Yang, T.K. Li, J.J.M. Tan, and L.H. Hsu, “Fault-Tolerant Cycle-Embedding of Crossed Cubes,” Information Processing Letters, vol. 88, pp. 149-154, 2003.
18 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool