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

  • Hans-Ulrich Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 158)

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.

Keywords

Input Vector Boolean Function Undirected Graph Random Access Memory Single Processor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. Borodin,J. Hopcroft. Routing and Merging on Parallel Models of Computation. Proc. 14'th annual ACM, 5/1982. pp.338–344.Google Scholar
  2. [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.Google Scholar
  3. [3]
    R.Reischuk. A Lower Time Bound for Parallel RAM's without Simultaneous Writes. IBM Research Report RJ3431 (40917), 3/1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Hans-Ulrich Simon
    • 1
  1. 1.Institut für angewandte Mathematik und Informatik der Universität des SaarlandesSaarbrückenW.-Germany

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