Publication 1995 Issue No. 3 - March Abstract - Optimal Simulation of Full Binary Trees on Faulty Hypercubes
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Optimal Simulation of Full Binary Trees on Faulty Hypercubes
March 1995 (vol. 6 no. 3)
pp. 269-286
 ASCII Text x Bethany M. Y. Chan, Francis Y. L. Chin, Chung-Keung Poon, "Optimal Simulation of Full Binary Trees on Faulty Hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 3, pp. 269-286, March, 1995.
 BibTex x @article{ 10.1109/71.372776,author = {Bethany M. Y. Chan and Francis Y. L. Chin and Chung-Keung Poon},title = {Optimal Simulation of Full Binary Trees on Faulty Hypercubes},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {6},number = {3},issn = {1045-9219},year = {1995},pages = {269-286},doi = {http://doi.ieeecomputersociety.org/10.1109/71.372776},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - Optimal Simulation of Full Binary Trees on Faulty HypercubesIS - 3SN - 1045-9219SP269EP286EPD - 269-286A1 - Bethany M. Y. Chan, A1 - Francis Y. L. Chin, A1 - Chung-Keung Poon, PY - 1995VL - 6JA - IEEE Transactions on Parallel and Distributed SystemsER -

Abstract—We study the problem of running full binary tree based algorithms on a hypercube with faulty nodes. The key to this problem is to devise a method for embedding a full binary tree into the faulty hypercube. Based on a novel embedding strategy, we present two results for embedding an $\left(n - 1\right)$-tree (a full binary tree with $2^\left\{n-1\right\} - 1$ nodes) into an $n$-cube (a hypercube with $2^n$ nodes) with unit dilation and load. For the problem where the root of the tree must be mapped to a specified hypercube node (specified root embedding problem), we show that up to $n - 2$ (node or edge) faults can be tolerated. This result is optimal in the following sense: 1) it is time-optimal, 2) $\left(n - 1\right)$-tree is the largest full binary tree that can be embedded in an $n$-cube, and 3) $n - 2$ faults is the maximum number of worst-case faults that can be tolerated in the specified root problem. Furthermore, we also show that any algorithm for this problem cannot be totally recursive in nature. For the problem where the root can be mapped to any nonfaulty hypercube node (variable root embedding problem), we show that up to $2n - 3 - \lceil\log n\rceil$ faults can be tolerated. Thus we have improved upon the previous result of $n - 1 - \lceil\log n \rceil.$ In addition, we show that the algorithm for the variable root embedding problem is optimal within a class of algorithms called recursive embedding algorithms as far as the number of tolerable faults is concerned. Finally, we show that when an $O\left(1/\sqrt\left\{n\right\}\right)$ fraction of nodes in the hypercube are faulty, it is not always possible to have an $O\left(1\right)$-load variable root embedding no matter how large the dilation is.

Index Terms—Embedding, hypercubes, full binary trees, dilation, simulation, faulty architecture.

[1] F. Annextein, "Fault-Tolerance of Hypercube-Derivative Networks," Proc. First ACM Symp. Parallel Algorithms and Architectures, pp. 179-188, 1989.
[2] B. Becker and H. Simon,“How robust is the n-cube?” Information and Computation, pp. 162-178, 1988.
[3] S. Bhatt, F. Chung, T. Leighton, and A. Rosenberg,“Optimal simulations of tree machines,”inProc. 27th Annual Symp. Foundations Comput. Sci., 1986, pp. 274–282.
[4] S. N. Bhatt and I. C. F. Ipsen,“How to embed trees in hypercubes,”Yale University, Res. Rep.YALEU/DCS/RR-443, Dec. 1985.
[5] J. Bruck, R. Cypher, and D. Soroker,“Running algorithms efficiently on faulty hypercubes,”inProc.2nd Annual ACM Symp. on Parallel Algorithms and Architectures, 1990, pp. 37–44.
[6] M.Y. Chan,“Embedding of grids into optimal hypercubes,” SIAM J. Computing, vol. 20, pp. 834-864, Oct. 1991.
[7] M. Y. Chan and F. Chin,“On embedding rectangular grids in hypercubes,”IEEE Trans. Comput. 37, pp. 1285–1288, 1988.
[8] ——,“A parallel algorithm for an efficient mapping of grids into hypercubes,”IEEE Trans. Parallel Distrib. Syst., vol. 4, pp. 933-946, Aug. 1993.
[9] M. Y. Chan and S. J. Lee,“Fault-tolerant embeddings of complete binary trees in hypercubes,”IEEE Trans. Parallel Distrib. Syst., vol. 4, pp. 277–288, Mar. 1993.
[10] M.Y. Chan and S.J. Lee, “On the Existence of Hamiltonian Circuits in Faulty Hypercubes,” SIAM J. Discrete Mathematics, vol. 4, no. 4, pp. 511-527, Nov. 1991.
[11] C.-T. Ho and S.L. Johnsson, “Embedding Meshes in Boolean Cubes by Graph Decomposition,” J. Parallel and Distributed Computing, vol. 8, pp. 325-339, 1990.
[12] J. Hastad, T. Leighton, and M. Newman, "Reconfiguring a Hypercube in the Presence of Faults," ACM Theory of Computing, pp. 274-284, 1987.
[13] J. Hastad,T. Leighton,, and M. Newman,“Fast computation using faulty hypercubes,” Proc. 21st ACM Symp. Theory of Computing, 1989.
[14] T. Leighton,M. Newman,A.G. Ranada, and E. Schwabe,"Dynamic tree embeddings in butterflies and hypercubes," Proc. ACM Symp. Parallel Algorithms and Architectures, pp. 224-234, 1989.
[15] M. Livingston and Q. Stout,“Embeddings in hypercubes,”Mathematical Computer Modelling, vol. 11, pp. 222–227.
[16] M. Livingston, Q. Stout, N. Graham, and F. Harary,“Subcube fault-tolerance in hypercubes,”Univ. of Michigan Computing Research Laboratory, Tech. Rep. CRL-TR-12-87, Sept. 1987.
[17] B. Monien and I. H. Sudborough,“Simulating binary trees on hypercubes,”Proc. 3rd Aegean Workshop on Comput., 1988, pp. 170–180.
[18] A. Wagner,“Embedding arbitrary binary trees in a hypercube,”J. Parallel Distrib. Comput. 7, no. 3, pp. 503–520, 1989.
[19] A. Wagner and D. Corneil,“Embedding trees in a hypercube is NP-complete,” SIAM J. Computing, vol. 19, no. 3, pp. 570-590, June 1990.
[20] A. Y. Wu,“Embedding of tree networks into hypercubes,”J. Parallel Distrib. Comput. 2, pp. 238–249, 1985.
[21] A. Wang, R. Cypher, and E. Mayr,“Embedding complete binary trees in faulty hypercubes,”Tech. Rep. RJ 7821 (72203), Nov. 1990.

Citation:
Bethany M. Y. Chan, Francis Y. L. Chin, Chung-Keung Poon, "Optimal Simulation of Full Binary Trees on Faulty Hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 3, pp. 269-286, March 1995, doi:10.1109/71.372776
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