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Embedding and Reconfiguration of Binary Trees in Faulty Hypercubes
March 1996 (vol. 7 no. 3)
pp. 237-245

Abstract—We consider the problem of embedding and reconfiguring binary tree structures in faulty hypercubes. We assume that the number of faulty nodes is at most (n2), where n is the number of dimensions of the hypercube; we further assume that the location of faulty nodes are known. Our embedding techniques are based on a key concept called free dimension, which can be used to partition a cube into subcubes such that each subcube contains at most one faulty node. Using this approach, two distributed schemes are provided for embedding and reconfiguration in faulty hypercubes. We extend the free dimension concept to degree of occupancy and use this to develop a distributed scheme for reconfiguration of binary tree in faulty hypercubes with up to $\lceil\, 3n/2\, \rceil$ node faults.

[1] P. Banerjee, "Strategies for Reconfiguring Hypercubes Under Faults," Proc. 20th Int'l Symp. Fault Tolerant Computing, pp. 210-217, June 1990.
[2] S.N. Bhatt and I.C.F. Ipsen, "How to Embed Trees in Hypercubes," Yale Unive. Research Report RR-443, 1985.
[3] J. Bruck, R. Cypher, and D. Soroker, "Running Algorithms Efficiently on Faulty Hypercubes," Computer Architecture News, vol. 19, no. 1, pp. 89-96, Mar. 1991.
[4] J. Bruck, R. Cypher, and C.-T. Ho, "Wildcard Dimensions, Coding Theory and Fault-Tolerant Meshes and Hypercubes," Proc. 23rd Int'l Symp. Fault-Tolerant Computing, pp. 260-267, June 1993.
[5] 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.
[6] M.Y. Chan and S.J. Lee, "Fault Tolerant Embedding of Complete Binary Trees and Rings in Hypercubes," Univ. of Texas at Dallas, Technical Report UTDCS-17-89, Aug. 1898.
[7] S.R. Deshpande and R.M. Jenevein, "Scalability of a Binary Tree on a Hypercube," Int'l Conf. Parallel Processing, pp. 661-668, 1986.
[8] S. Dutt and J.P. Hayes, "An Automorphic Approach to the Design of Fault-Tolerant Multiprocessors," Proc. 19th Fault Tolerant Computing Symp., pp. 496-503,Chicago, June 1989.
[9] J. Hastad, T. Leighton, and M. Newman, "Reconfiguring a Hypercube in the Presence of Faults," ACM Theory of Computing, pp. 274-284, 1987.
[10] T.H. Horowitz and S. Sahni, Fundamentals of Computer Algorithms. Computer Science Press, 1978.
[11] S.L. Johnsson, "Communication Efficient Basic Linear Algebra Computations on Hypercube Architectures," J. Parallel and Distributed Computing, vol. 4, pp. 133-172, 1987.
[12] T.C. Lee, "Quick Recovery of Embedded Structures in Hypercube Computers," Proc. Fifth Distributed Memory Computing Conf., pp. 1,426-1,435, Apr. 1990.
[13] D. Nassimi and S. Sahni, "Optimal BPC Permutations on a Cube Connected SIMD Computer," IEEE Trans. Computers, pp. 338-341, Apr. 1982.
[14] S. Nugent, "The iPSC/2 Direct-Connect Communications Technology," Proc. Third Conf. Hypercube Concurrent Computers and Applications, pp. 51-60, Jan. 1998.
[15] F.J. Provost and R. Melhem, "Distributed Fault Tolerant Embedding of Binary Trees and Rings in Hypercubes," Proc. Int'l Workshop Defect and Fault Tolerance in VLSI Systems, 1988.
[16] 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.
[17] C.S. Raghavendra,P.-J. Yang,, and S.-B. Tien,“Free dimensions—an effective approach to achieving fault tolerance in hypercubes,” 22nd Ann. Int’l Symp. Fault-Tolerant Computing, pp. 170-177, 1992.
[18] 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.
[19] D.A. Rennels, "On Implementing Fault-Tolerance in Binary Hypercubes," Proc.16th Int'l Symp. Fault-Tolerant Computing, pp. 344-349, 1986.
[20] Y. Saad and M. Schultz, "Topological Properties of Hypercubes," IEEE Trans. Computers, vol. 37, no. 7, pp. 867-872, July 1988.
[21] S.B. Tien, C.S. Raghavendra, and M.A. Sridhar, "Reconfiguration of Embedded Task Graphs in Faulty Hypercubes by Automorphisms," Hawaii Int'l Conf. System Science, pp. 91-100, 1990.
[22] S.B. Tien and C.S. Raghavendra, "Optimal Global Operations and Broadcasting on Faulty Hypercubes," Technical Report, Dept. of Electrical Eng., Southern Illinois Univ., 1991.
[23] J.D. Ullman, Computational Aspects of VLSI. Computer Science Press, 1984.
[24] A.Y. Wu, "Embedding of Tree Networks into Hypercubes," J. Parallel and Distributed Computing, pp. 238-249, 1985.
[25] P.J. Yang, S.B. Tien, and C.S. Raghavendra, "Embedding of Multidimensional Meshes on to Faulty Hypercubes," Int'l Conf. Parallel Processing, pp. 571-574, 1991.
[26] P.J. Yang and C.S. Raghavendra, "Embedding and Reconfiguration of Binary Trees in Faulty Hypercubes," Washington State Univ. Technical Report, 1994.

Index Terms:
Fault tolerance, hypercube, binary tree, embedding, free dimension.
Citation:
Pei-Ji Yang, C.s. Raghavendra, "Embedding and Reconfiguration of Binary Trees in Faulty Hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 3, pp. 237-245, March 1996, doi:10.1109/71.491577
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