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A tight Ω(loglog n)-bound on the time for parallel Ram's to compute nondegenerated boolean functions

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Foundations of Computation Theory (FCT 1983)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 158))

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Abstract

A function f: {0, 1}n → {0, 1} is said to depend on dimension i iff there exists an input vector x such that f(x) differs from f(xi), where xi agrees with x in every dimension except i. In this case x is said to be critical for f with respect to i. f is called nondegenerated iff it depends on all n dimensions.

The main result of this paper is that for each nondegenerated function f: {0, 1}n → {0, 1} there exists an input vector x which is critical with respect to at least Ω(log n) dimensions. A function achieving this bound is presented.

Together with earlier results from Cook,Dwork [2] and Reischuk [3] we can conclude that a parallel RAM requires at least Ω(loglog n) steps to compute f.

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References

  1. A. Borodin,J. Hopcroft. Routing and Merging on Parallel Models of Computation. Proc. 14'th annual ACM, 5/1982. pp.338–344.

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  2. S. Cook,C. Dwork. Bounds on the Time for Parallel RAM's to Compute Simple Functions. Proc. 14'th annual ACM, 5/1982. pp.231–233.

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  3. R.Reischuk. A Lower Time Bound for Parallel RAM's without Simultaneous Writes. IBM Research Report RJ3431 (40917), 3/1982.

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Marek Karpinski

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© 1983 Springer-Verlag Berlin Heidelberg

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Simon, HU. (1983). A tight Ω(loglog n)-bound on the time for parallel Ram's to compute nondegenerated boolean functions. In: Karpinski, M. (eds) Foundations of Computation Theory. FCT 1983. Lecture Notes in Computer Science, vol 158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12689-9_124

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  • DOI: https://doi.org/10.1007/3-540-12689-9_124

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12689-8

  • Online ISBN: 978-3-540-38682-7

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