This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Embedding Cube-Connected Cycles Graphs into Faulty Hypercubes
October 1994 (vol. 43 no. 10)
pp. 1210-1219

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults.

[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?" inProc. 27th Annu. IEEE Symp. on Foundations of Comput. Sci., 1986, pp. 283-291.
[4] J. Bruck, "On optimal broadcasting in faulty hypercubes," IBM Res. Rep., RJ7147, 1989. To appear inDisc. Appl. Math., vol. 53, Sept. 1994.
[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, "On the existence of hamiltonian circuits in faulty hypercubes,"SIAM J. Discrete Math., vol. 4, no. 4, pp. 511-527, 1991.
[7] M. Y. Chan and S. J. Lee, "Fault-tolerant embedding of complete binary trees in hypercubes,"IEEE Trans. Parallel Distrib. Syst., vol. 4, no. 3, pp. 277-288, 1993.
[8] M. Y. Chan and S. J. Lee, "Fault-tolerant permutation routing in hypercubes," Tech. Rep., UTDCS-5-90, Univ. of Texas at Dallas, 1990.
[9] 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.
[10] J. Hastad, T. Leighton, and M. Newman, "Fast computation using faulty hypercubes," inProc. 21st Annu. ACM Symp. Theory Comput., May 1989, pp. 251-263.
[11] C. T. Ho and S. L. Johnsson, "Embedding meshes in boolean cubes by graph decomposition,"J. Parallel and Distrib. Computing, Apr. 1990, pp. 325-339.
[12] M. Livingston, Q. Stout, N. Graham, and F. Harary, "Subcube faulttolerance in hypercubes," Tech. Rep. CRL-TR-12-87, Univ. of Michigan Computing Res. Lab., Sept. 1987.
[13] M. Livingston and Q. Stout, "Embeddings in hypercubes," InMath.l Comput. Modeling, vol. 11, pp. 222-227,1988.
[14] 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.
[15] M. O. Rabin, "Efficient dispersal of information for security, load balancing, and fault tolerance,"J. ACM, vol. 36, no. 2, Apr. 1989.
[16] R. E. Tarjan and U. Vishkin, "An efficient parallel biconnectivity algorithm,"SIAM J. Comput., vol. 14, 4, pp. 862-874, Nov. 1985.
[17] C. D. Thompson, "Area-time complexity for VLSI," inProc. Eleventh Annu. ACM Symp. Theory Comput., 1979, pp. 81-88.

Index Terms:
hypercube networks; fault tolerant computing; reliability; cube-connected cycles graphs embedding; faulty hypercubes; ascend-descend algorithms; bitonic sort; spanning torus.
Citation:
J. Bruck, R. Cypher, D. Soroker, "Embedding Cube-Connected Cycles Graphs into Faulty Hypercubes," IEEE Transactions on Computers, vol. 43, no. 10, pp. 1210-1219, Oct. 1994, doi:10.1109/12.324546
Usage of this product signifies your acceptance of the Terms of Use.