Abstract
An optimal tree contraction algorithm for the boolean hypercube and the constant degree hypercubic networks, such as the shuffle exchange or the butterfly network, is presented. The algorithm is based on novel routing techniques and, for certain small subtrees, simulates optimal PRAM algorithms. For trees of size n, stored on a p processor hypercube in in-order, the running time of the algorithm is O([n/p] log p). The resulting speed-up of O(p/ log p) is optimal due to logarithmic communication overhead, as shown by a corresponding lower bound.
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References
H. Abelson. Lower bounds on information transfer in distributed computations. Journal of the ACM, 27:384–392, 1980.
K. Abrahamson, N. Dadoun, D.G. Kirkpatrick and T. Przytycka. A simple parallel tree contraction algorithm. Journal of Algorithms, 10:287–302, 1989.
R. Brent. The parallel evaluation of general arithmetical expressions. Journal of the ACM, 21:201–206, 1974.
S. Buss, S. Cook, A. Gupta and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21:755–780, 1992.
H. Gazit. and G.L. Miller and S.-H. Teng. Optimal tree contraction in the EREW model. In: Tewksbury, Stuart K. and Bradley W. Dickinson and Stuart C. Schwartz (eds.): Concurrent Computations: Algorithms, Architecture, and Technology. Plenum Press: New York-London (1988), 139–156.
A. Gibbons and W. Rytter. An optimal parallel algorithms for dynamic expression evaluation and its applications. Proceedings of the 6th Conference on Foundations of Software Technology and Theoretical Computer Science, Springer Verlag, LNCS-241:453–469, 1986.
S.R. Kosaraju and A.L. Delcher. Optimal parallel evaluation of tree-structured computations by raking. Proceedings of the 3rd Aegean Workshop on Computing: VLSI Algorithms and Architectures, 101–110, 1988.
F.T. Leighton. Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann Publishers, 1992.
E. W. Mayr and R. Werchner. Optimal routing of parentheses on the hypercube. Proceedings of the 4th Annual ACM Symposium on Parallel Algorithms and Architectures, 109–117, 1992.
G. L. Miller and J. H. Reif. Parallel tree contraction and its applications. Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 478–489, 1985.
D. Nassimi and S. Sahni. Data broadcasting in SIMD computers. IEEE Transactions on Computers, C-30:101–107, 1981.
D. Nassimi and S. Sahni. Parallel permutation and sorting algorithms and a new generalized connection network. JACM, 29:642–667, 1982.
G. Pietsch and E. Schömer. Optimal parallel recognition of bracket languages on hypercubes. 8th Annual Symposium on Theoretical Aspects of Computer Science, Springer Verlag, LNCS-480:434–443.
E. J. Schwabe. On the computational equivalence of hypercube-derived networks. Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, 388–397, 1990.
J.T. Schwartz. Ultracomputers. ACM Transactions on Programming Languages and Systems, 2:484–521, 1980.
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© 1993 Springer-Verlag Berlin Heidelberg
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Mayr, E.W., Werchner, R. (1993). Optimal tree contraction on the hypercube and related networks. In: Lengauer, T. (eds) Algorithms—ESA '93. ESA 1993. Lecture Notes in Computer Science, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57273-2_64
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DOI: https://doi.org/10.1007/3-540-57273-2_64
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