This Article 
 Bibliographic References 
 Add to: 
Subcube Determination in Faulty Hypercubes
August 1997 (vol. 46 no. 8)
pp. 871-879

Abstract—A hypercube may operate in a gracefully degraded manner, after faults arise, by supporting the execution of parallel algorithms in smaller fault-free subcubes. In order to reduce execution slowdown in a hypercube with given faults, it is essential to identify the maximum healthy subcubes in the faulty hypercube because the time for executing a parallel algorithm tends to depend on the dimension of the assigned subcube. This paper describes an efficient procedure capable of determining all maximum fault-free subcubes in a faulty hypercube. The procedure is a distributed one, since every healthy node next to a failed component performs the same procedure independently and concurrently. Based on interesting properties of faulty hypercubes, this procedure exhibits empirically polynomial time complexity with respect to the system dimension and the number of faults, for a practical range of dimensions. It compares favorably with prior methods when the number of faults is in the order of the system dimension. This procedure can deal with node failures and link failures uniformly and equally efficiently.

[1] C. L. Seitz,“The cosmic cube,”CACM, vol. 28, pp. 22–33, Jan. 1985.
[2] J.C. Peterson et al., "The Mark III Hypercube-Ensemble Concurrent Computer," Proc. 1985 Int'l Conf. Parallel Processing, pp. 71-73, Aug. 1985.
[3] R. Arlauskas, "iPSC/2 System: A Second Generation Hypercube," Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 38-42, Jan. 1988.
[4] NCUBE Corporation, n-Cube-2 Processor Manual. NCUBE Corporation, 1990.
[5] W.D. Hillis, The Connection Machine, MIT Press, Cambridge, Mass., 1985.
[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] D.A. Rennels, "On Implementing Fault-Tolerance in Binary Hypercubes," Proc. 16th Int'l Symp. Fault-Tolerant Computing, pp. 344-349, July 1986.
[8] S.-C. Chau and A.L. Liestman, "A Proposal for a Fault-Tolerant Binary Hypercube Architecture," Proc. 19th Int'l Symp. Fault-Tolerant Computing, pp. 323-330, June 1989.
[9] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[10] F. Ozguner and C. Aykanat, "A Reconfiguration Algorithm for Fault Tolerance in a Hypercube Multiprocessor," Information Processing Letters, vol. 29, pp. 247-254, Nov. 988.
[11] M.A. Sridhar and C.S. Raghavendra, "On Finding Maximal Subcubes in Residual Hypercubes," Proc. Second IEEE Symp. Parallel and Distributed Processing, pp. 870-873, Dec. 1990.
[12] B. Becker and H.-U. Simon, "How Robust Is the n-Cube?" Proc. IEEE 27th Symp. Foundatons of Computer Science, pp. 283-291, Oct. 1986.
[13] S. Latifi, "Distributed Subcube Identification Algorithms for Reliable Hypercubes," Information Processing Letters, vol. 38, pp. 315-321, June 1991.
[14] J.R. Armstrong and F.G. Gray, "Fault Diagnosis in a Boolean n Cube Array of Microprocessors," IEEE Trans. Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[15] 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.

Index Terms:
Faulty hypercubes, prime subcubes, product-of-sums, reconfiguration, sum-of-products, time complexity.
Hsing-Lung Chen, Nian-Feng Tzeng, "Subcube Determination in Faulty Hypercubes," IEEE Transactions on Computers, vol. 46, no. 8, pp. 871-879, Aug. 1997, doi:10.1109/12.609276
Usage of this product signifies your acceptance of the Terms of Use.