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On the Performance of Randomized Embedding of Reproduction Trees in Static Networks

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Abstract

High performance computing requires high quality load distribution of processes of a parallel application over processors in a parallel computer at runtime such that both the maximum load and dilation are minimized. The performance of a simple randomized tree embedding algorithm that dynamically supports tree-structured parallel computations on arbitrary static networks is analyzed in this paper. The algorithm spreads newly created tree nodes to neighboring processors, which actually provides randomized dilation-1 tree embedding in static networks. We develop a linear system of equations that characterizes expected loads on all processors under the reproduction tree model, which can generate trees of arbitrary size and shape. It is shown that as the tree size becomes large, the asymptotic performance ratio of the randomized tree embedding algorithm is the ratio of the maximum processor degree to the average processor degree. This implies that the simple randomized tree embedding algorithm is able to generate high quality load distributions on virtually all static networks commonly employed in parallel and distributed computing.

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References

  1. S. Bhatt and J.-Y. Cai, Take a Walk, Grow a Tree, Proc. of IEEE Symp. on Foundations of Computer Science, pp. 469-478 (1988).

  2. S. Bhatt and J.-Y. Cai, Taking Random Walks to Grow Trees in Hypercubes, J. ACM, 40(3):741-764 (1993).

    Google Scholar 

  3. F. T. Leighton, M. J. Newman, A. G. Ranade, and E. J. Schwabe, Dynamic Tree Embeddings in Butterflies and Hypercubes, SIAM J. Comp., 21(4):639-654 (1992).

    Google Scholar 

  4. W. Aiello and F. T. Leighton, Coding Theory, Hypercube Embeddings, and Fault Tolerance, Proc. of ACM Symp. on Parallel Algorithms and Architectures (1991).

  5. ö. Eğgecioğglu and M. Ibel, Asymptotic Hypercube Embeddings of Dynamic k-ary Trees, Congressus Numerantium, 126:21-32 (1997).

    Google Scholar 

  6. V. Heun and E. W. Mayr, Efficient Dynamic Embedding of Arbitrary Binary Trees into Hypercubes, Lecture Notes in Computer Science, Vol. 1117, Springer-Verlag, pp. 287-298 (1996).

  7. K. Li, A Method for Evaluating the Expected Load of Dynamic Tree Embeddings in Hypercubes, International Journal on Foundations of Computer Science, 11(2):207-230 (2000).

    Google Scholar 

  8. A. G. Ranade, Optimal Speedup for Backtrack Search on a Butterfly Network, Proc. of 3rd ACM Symp. on Parallel Algorithms and Architectures, pp. 40-48 (1991).

  9. R. M. Karp and Y. Zhang, Randomized Parallel Algorithms for Backtrack Search and Branch-and-Bound Computation, J. ACM, 40(3):765-789 (1993).

    Google Scholar 

  10. K. Li, Efficient Randomized Load Distribution for Tree Structured Computations on Parallel and Distributed Computer Systems, Int. J. Comput. Math., 71:21-34 (1999).

    Google Scholar 

  11. C. Kaklamanis and G. Persiano, Branch-and-Bound and Backtrack Search on Mesh-Connected Arrays of Processors, Proc. of ACM Symp. on Parallel Algorithms and Architectures, pp. 118-126 (1992).

  12. M. A. Palis and D. S. L. Wei, Backtracking and Branch-and-Bound on Mesh-Connected Computers with Reconfigurable Buses, Proc. of 7th International Conference on Parallel and Distributed Computing and Systems, pp. 243-247 (1995).

  13. K. Li, Comparative Performance Evaluation of a Random-Walk-Based Algorithm for Embedding Dynamically Evolving Trees in Hypercubic Networks, to appear in Computation and Concurrency: Practice and Experience, Vol. 15 (2003).

  14. S. Bhatt, D. Greenberg, T. Leighton, and P. Liu, Tight Bounds for On-Line Tree Embeddings, SIAM J. Comp., 29(2):474-491 (1999).

    Google Scholar 

  15. J. Gaber and B. Toursel, Randomized Load Distribution of Arbitrary Trees on a Distributed Network, Proc. of 13th Annual ACM Symp. on Applied Computing, Atlanta, Georgia, pp. 564-568 (February 1998).

  16. K. Li, Y. Pan, H. Shen, G. H. Young, and S.-Q. Zheng, Lower Bounds for Dynamic Tree Embedding in Bipartite Graphs, J. Parallel Distr. Com., 53(2):119-143 (1998).

    Google Scholar 

  17. H. Shen, K. Li, Y. Pan, G. H. Young, and S.-Q. Zheng, Performance Analysis for Dynamic Tree Embedding in k-Partite Networks by Random Walk, J. Parallel Distr. Com., 50(1):144-156 (1998).

    Google Scholar 

  18. T. Harris, The Theory of Branching Processes, Springer, Berlin (1963).

    Google Scholar 

  19. J. Pearl, Heuristics: Intelligent Search Strategies for Computer Problem Solving, Addison–Wesley, Reading, Massachusetts (1984).

    Google Scholar 

  20. I. Stojmenovićc, Direct Interconnection Networks, in Parallel and Distributed Computing Handbook, A. Y. Zomaya (ed.), McGraw–Hill, pp. 537-567 (1996).

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Authors and Affiliations

  1. Department of Computer Science, State University of New York, New Paltz, New York, 12561

    Keqin Li

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  1. Keqin Li
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Li, K. On the Performance of Randomized Embedding of Reproduction Trees in Static Networks. International Journal of Parallel Programming 31, 393–406 (2003). https://doi.org/10.1023/A:1027336712552

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