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)


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.


Input Vector Boolean Function Undirected Graph Random Access Memory Single Processor 
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  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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