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Issue No.08 - August (2008 vol.19)
pp: 1071-1085
ABSTRACT
Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Qkn in which the number of node faults fv and the number of link faults fe are such that fn+fe ≤ 2n-2. We prove that given any two healthy nodes s and e of Qkn, there is a path from s to e of length at least kn-2fn-1 (resp. kn-2fn-2) if the nodes s and e have different (resp. the same) parities (the parity of a node in Qkn is the sum modulo 2 of the elements in the n-tuple over {0,1,...,k-1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n=2.
INDEX TERMS
Interconnection architectures, Fault tolerance
CITATION
Iain A. Stewart, Yonghong Xiang, "Embedding Long Paths in k-Ary n-Cubes with Faulty Nodes and Links", IEEE Transactions on Parallel & Distributed Systems, vol.19, no. 8, pp. 1071-1085, August 2008, doi:10.1109/TPDS.2007.70787
REFERENCES
[1] B. Aiello and T. Leighton, “Coding Theory, Hypercube Embeddings and Fault Tolerance,” Proc. Third Ann. ACM Symp. Parallel Algorithms and Architectures (SPAA '91), pp. 125-136, 1991.
[2] S.G. Akl, Parallel Computation: Models and Methods. Prentice Hall, 1997.
[3] E. Anderson, J. Brooks, C. Grassl, and S. Scott, “Performance of the Cray T3E Multiprocessor,” Proc. ACM/IEEE Conf. Supercomputing, pp. 1-17, 1997.
[4] W.C. Athas and C.L. Seitz, “Multicomputers: Message-Passing Concurrent Computers,” Computer, vol. 21, pp. 9-24, 1988.
[5] 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.
[6] 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.
[7] 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. Int'l Conf. Supercomputing, pp. 330-339, 1988.
[8] R.J. Cole, B.M. Maggs, and R.K. Sitaraman, “Reconfiguring Arrays with Faults I: Worst-Case Faults,” SIAM J. Computing, vol. 26, pp.1581-1611, 1997.
[9] R. Duncan, “A Survey of Parallel Computer Architectures,” Computer, vol. 23, pp. 5-16, 1990.
[10] T.H. Duncan, “Performance of the Intel iPSC/860 and Ncube 6400 Hypercubes,” Parallel Computing, vol. 17, pp. 1285-1302, 1991.
[11] J.-S. Fu, “Longest Fault-Free Paths in Hypercubes with Vertex Faults,” Information Sciences, vol. 176, pp. 759-771, 2006.
[12] J. Hastad, T. Leighton, and M. Newman, “Reconfiguring a Hypercube in the Presence of Faults,” Proc. 19th Ann. ACM Symp. Theory of Computing (STOC '97), pp. 274-284, 1997.
[13] S.-Y. Hsieh and N.-W. Chang, “Optimal Node-to-Node Path Embedding in Hypercubes with Conditional Faults,” IEEE Trans. Parallel and Distributed Systems, to be published.
[14] R.E. Kessler and J.L. Schwarzmeier, “CRAY T3D: A New Dimension for Cray Research,” Proc. 38th IEEE CS Int'l Conf. (Compcon '93), pp. 176-182, 1993.
[15] H.-C. Kim and J.-H. Park, “Fault Hamiltonicity of Two-Dimensional Torus Networks,” Proc. Fifth Japan-Korea Joint Workshop Algorithms and Computation (WAAC '00), pp. 110-117, 2000.
[16] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays. Trees. Hypercubes. Morgan Kaufmann, 1992.
[17] F.T. Leighton, B.M. Maggs, and R.K. Sitaraman, “On the Fault Tolerance of Some Popular Bounded-Degree Networks,” SIAM J.Computing, vol. 27, pp. 1303-1333, 1998.
[18] 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.
[19] C.L. Seitz, “The Cosmic Cube,” Comm. ACM, vol. 28, pp. 22-33, 1985.
[20] 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.
[21] I.A. Stewart, “Distributed Algorithms for Building Hamiltonian Cycles in $k\hbox{-}{\rm Ary}\;n\hbox{-}{\rm Cubes}$ and Hypercubes with Faulty Links,” Proc. 12th Int'l Conf. Parallel and Distributed Systems (ICPADS '06), pp.308-318, 2006.
[22] M.-C. Yang, J.J.M. Tan, and L.H. Hsu, “Hamiltonian Circuit and Linear Array Embeddings in Faulty $k\hbox{-}{\rm Ary}\;n\hbox{-}{\rm Cubes}$ ,” J. Parallel and Distributed Computing, vol. 67, no. 4, pp. 362-368, 2007.
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