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Free Dimensions-An Effective Approach to Achieving Fault Tolerance in Hypercubes
September 1995 (vol. 44 no. 9)
pp. 1152-1157

Abstract—Hypercube network is an attractive structure for parallel processing due to its symmetry and regularity. In this paper, we use the concept of free dimensions to achieve fault tolerance in hypercubes without requiring additional spare processing nodes; such additional redundancy requires modification of hypercube structure. A free dimension is defined to be a dimension across which both end nodes are not faulty.

Given an n-dimensional hypercube, Qn, and a set of fn faulty nodes, we present an efficient algorithm to find free dimensions, and show that at least nf+ 1 free dimensions exist. Free dimensions can be used to partition Qn into subcubes such that each subcube contains at most one fault. Such a partitioning helps in achieving fault tolerance via emulation, embedding, reconfiguration. It also helps in designing efficient routing and broadcasting algorithms in faulty hypercubes.

[1] B. Aiello and T. Leighton, "Coding Theory, Hypercube Embeddings, and Fault Tolerance," Proc. Third Ann. ACM Symp. Parallel Algorithms and Architectures, pp. 125-136, July 1991.
[2] B. Becker and H. Simon,“How robust is the n-cube?” Information and Computation, pp. 162-178, 1988.
[3] J. Bruck, R. Cypher, and D. Soroker, "Tolerating Faults in Hypercubes Using Subcube Partitioning," IEEE Trans. Computers, vol. 41, no. 5, pp. 599-605, May 1992.
[4] M.Y. Chan and S.J. Lee, "Fault-Tolerant Embedding of Complete Binary Trees in Hypercubes," IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 3, pp. 277-288, Mar. 1993.
[5] Y. Chang,“Fault tolerant broadcasting in SIMD hypercubes,” Proc. Fifth Symp. Parallel and Distributed Processing, pp. 348-351, Dec. 1993.
[6] S.K. Chen, C.T. Liang, and W.T. Tsai, "An Efficient Multi-Dimensional Grids Reconfiguration Algorithm on Hypercubes," Proc. 18th In'l Symp. Fault-Tolerant Computing, pp. 368-373, 1988.
[7] M.S. Chen and K.G. Shin, "Depth-First Search Approach for Fault-Tolerant Routing in Hypercube Multicomputers," IEEE Trans. Parallel and Distributed Systems, vol. 1, no. 2, pp. 152-159, Apr. 1990.
[8] B.S. Chlebus, K. Diks, and A. Pelc, "Optimal Broadcasting in Fault Hypercubes," Proc. IEEE 21st Int'l Symp. Fault-Tolerant Computing, pp. 266-273, June 1991.
[9] S. Dutt and J.P. Hayes, “Designing Fault-Tolerant Systems Using Auto-morphisms,” J. Parallel and Distributed Computing, vol. 12, no. 3, pp. 249–268, 1991.
[10] F. Harary,J.P. Hayes,, and H.J. Wu,“A survey of the theory of hypercube graphs,” Comput. Math. Applic., pp. 277-289, 1988.
[11] J. Hastad,T. Leighton,, and M. Newman,“Fast computation using faulty hypercubes,” Proc. 21st ACM Symp. Theory of Computing, 1989.
[12] W.D. Hillis, The Connection Machine, MIT Press, Cambridge, Mass., 1985.
[13] T.C. Lee and J.P. Hayes,“A fault-tolerant communication scheme for hypercube computers,” IEEE Trans. Computers, vol. 41, no. 10, pp. 1,242-1,256, Oct. 1992.
[14] T.C. Lee,“Quick recovery of embedded structures in hypercube computers,” Proc. Fifth Distributed Memory Computing Conf., pp. 1,426-1,435, Apr. 1990.
[15] S. Nugent, "The iPSC/2 Direct-Connect Communications Technology," Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 51-60, Jan. 1998.
[16] S. Park and B. Bose, "Broadcasting in Hypercubes with Link/Node Failures," Proc. Fourth Symp. Frontiers of Massively Parallel Computation, pp. 286-290, 1992.
[17] M. Peercy and P. Banerjee, "Distributed Algorithms for Shortest-Path, Deadlock-Free Routing and Broadcasting in Arbitrarily Faulty Hypercubes," Proc. 20th Int'l Symp. Fault Tolerant Computing, pp. 218-225, June 1990.
[18] F.J. Provost and R. Melhem,“Distributed fault tolerant embedding of binary trees and rings inhypercubes,” Proc. Int’l Workshop Defect and Fault Tolerance in VLSI Systems, 1988.
[19] C.S. Raghavendra and M.A. Sridhar, "Broadcasting Algorithms in Faulty SIMD Hypercubes," Proc. Fourth IEEE Symp. Parallel and Distributed Processing, pp. 4-11, Dec. 1992.
[20] C.S. Raghavendra and M.A. Sridhar,“Fault-tolerant routing algorithms in hypercube multiprocessors,” Proc. Fourth ISMM Int’l Conf. Parallel and Distributed ComputingSystems.
[21] P. Ramanathan and K.G. Shin, "Reliable Broadcast in Hypercube Multicomputers," IEEE Trans. Computers, vol. 37, no. 12, pp. 1,654-1,657, Dec. 1988.
[22] D.A. Rennels,“On implementing fault tolerance in binary hypercubes,” Proc. 16th Int’l Symp. Fault-Tolerant Computing, pp. 344-349, 1986.
[23] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[24] S.-B. Tien and C.S. Raghavendra,“Algorithms and bounds for shortest paths and diameter in faultyhypercubes,” IEEE Trans. Parallel and Distributed Systems, pp. 713-718, June 1993.
[25] P.J. Yang,S.B. Tien,, and C.S. Raghavendra,“Reconfiguration of rings and meshes in faulty hypercubes,” technical report, Dept. of Elec. Eng., Univ. of Southern Calif., 1991.
[26] P.J. Yang, S.B. Tien, and C.S. Raghavendra, “Embedding of Rings and Meshes onto Faulty Hypercubes Using Free Dimensions,” IEEE Trans. Computers, vol. 43, no. 5, pp. 608-613, May 1994.
[27] P.J. Yang,S.B. Tien,, and C.S. Raghavendra,“Embedding of multidimensional meshes on to faulty hypercubes,” Int’l Conf. Parallel Processing, pp. I-571-I-574, 1991.
[28] P.J. Yang and C.S. Raghavendra,“Embedding and reconfiguration of binary trees in faulty hypercubes” Int’l Parallel Processing Symp., Mar. 1992.

Index Terms:
Hypercubes, fault tolerance, embedding, reconfiguration, routing and broadcasting.
Pei-Ji Yang, C.s. Raghavendra, Sing-Ban Tien, "Free Dimensions-An Effective Approach to Achieving Fault Tolerance in Hypercubes," IEEE Transactions on Computers, vol. 44, no. 9, pp. 1152-1157, Sept. 1995, doi:10.1109/12.464395
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