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

## 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(x^{i}), where x^{i} 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## Preview

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