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Tolerating Faults in Hypercubes Using Subcube Partitioning
May 1992 (vol. 41 no. 5)
pp. 599-605

The authors examine the issue of running algorithms on a hypercube which has both node and edge faults, and they assume a worst-case distribution of the faults. It is proven that for any constant c, an n-dimensional hypercube (n-cube) with n/sup c/ faulty components contains a fault-tree subgraph that can implement a large class of hypercube algorithms with only a constant factor slowdown. In addition, the approach yields practical implementations for small numbers of faults. For example, it is shown that any regular algorithm can be implemented on an n-cube that has at most n-1 faults with slowdowns of at most two for computation and at most four for communication. This is the first result showing that an n-cube can tolerate more than O(n) arbitrarily placed faults with a constant factor slowdown.

[1] B. Aiello and T. Leighton, "Coding theory, hypercube embeddings, and fault tolerance," inProc. 3rd Annu. ACM Symp. Parallel Algorithms Architectures, 1991, pp. 125-136.
[2] F. Annexstein, "Fault tolerance of hypercube-derivative networks," inProc. 1st Annu. ACM Symp. Parallel Algorithms Architectures, 1989, pp. 179-188.
[3] B. Becker and H. U. Simon, "How robust is then-cube?,"Inform. Computat., vol. 77, pp. 162-178, 1988.
[4] J. Bruck, "Optimal broadcasting in faulty hypercubes via edge-disjoint embeddings," IBM Res. Rep., RJ7147, 1989.
[5] J. Bruck, R. Cypher, and D. Soroker, "Running algorithms efficiently on faulty hypercubes," inProc. 2nd Annu. ACM Symp. Parallel Algorithms Architectures, 1990, pp. 37-44.
[6] M. Y. Chan and S. J. Lee, "Fault-tolerant permutation routing in hypercubes," Univ. Texas at Dallas Tech. Rep., UTDCS-5-90.
[7] D. Dolev, J. Y. Halpern, B. Simons, and R. Strong, "A new look at fault-tolerant network routing,"Inform. Computat., vol. 72, no. 3, pp. 180-196, Mar. 1987.
[8] J. Hastad, T. Leighton, and M. Newman, "Fast computation using faulty hypercubes," inProc. 21st Annu. ACM Symp. Theory Comput., May 1989, pp. 251-263.
[9] D. Kleitman, "On the problem by Yuzvinsky on separating then-cube,"Discrete Math., vol. 60, pp. 207-213, 1986.
[10] T. Leighton and B. Maggs, "Expanders might be practical: Fast algorithms for routing around faults on multibutterflies," inProc. 30th Annu. IEEE Symp. Foundations Comput. Sci., 1989, pp. 384-389.
[11] M. Livingston, Q. Stout, N. Graham, and F. Harary, "Subcube fault-tolerance in hypercubes," Tech. Rep. CRL-TR-12-87, Univ. Michigan Comput. Res. Lab., Sept. 1987.
[12] F. P. Preparata and J. Vuillemin, "The cube-connected cycle: A versatile network for parallel computation,"Commun. ACM, vol. 24, pp. 300-309, May 1981.
[13] M. O. Rabin, "Efficient dispersal of information for security, load balancing, and fault tolerance,"J. ACM, vol. 36, no. 2, Apr. 1989.

Index Terms:
fault tolerance; node faults; subcube partitioning; edge faults; worst-case distribution; faulty components; fault-tree subgraph; hypercube algorithms; computational complexity; fault tolerant computing; graph theory; hypercube networks; parallel algorithms.
J. Bruck, R. Cypher, D. Soroker, "Tolerating Faults in Hypercubes Using Subcube Partitioning," IEEE Transactions on Computers, vol. 41, no. 5, pp. 599-605, May 1992, doi:10.1109/12.142686
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