Abstract
LetR be a unidirectional asynchronous ring ofn identical processors each with a single input bit. Letf be any cyclic nonconstant function ofn boolean variables. Moran and Warmuth (1986) prove that anydeterministic algorithm that evaluatesf onR has communication complexity Ω (n logn) bits. They also construct a family of cyclic nonconstant boolean functions that can be evaluated inO(n logn) bits by a deterministic algorithm.
This contrasts with the following new results:
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1.
There exists a family of cyclic nonconstant boolean functions which can be evaluated with expected complexity\(O(n\sqrt {\log n} )\) bits by arandomized algorithm forR.
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2.
Anynondeterministic algorithm forR which evaluates any cyclic nonconstant function has communication complexity\(\Omega (n\sqrt {\log n} )\) bits.
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Karl Abrahamson is an assistant professor in the Computer Science Department at Washington State University. Prior to WSU he was at the University of British Columbia. His research interests are in distributed algorithms and analysis of algorithms.
Andrew Adler is a faculty member of the Department of Mathematics at the University of British Columbia, Canada. He completed a Ph.D. in Mathematical Logic at the University of Toronto in 1966.
Lisa higham became an assistant professor in the Computer Science department at the University of Calgary, Canada after completing her Ph.D. at the University of British Columbia in 1988. Her research interests include the design and analysis of distributed and parallel algorithms and computational complexity.
David G. Kirkpatrick received the B.Sc. degree in mathematics from the University of British Columbia in 1970 and the M.Sc. and Ph.D. degrees from the University of Toronto in 1972 and 1974 respectively. He has been on the faculty of the Computer Sciences Department of the University of British Columbia since September 1978 (as a Full Professor since 1986). Previous positions include Visiting Assitant Professor at Cornell University (1974–75), Assistant Professor at Simon Fraser University (1975–78), and Visiting Research Scientist at IBM San Jose Research Laboratory (1983–84). Dr. Kirkpatrick is a founding Fellow of the British Columbia Advanced Systems Institute. Dr. Kirkpatrick's research interests include computational complexity, algorithmic combinatorics, computational geometry, and parallel and distributed computing.
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Killam Foundation.
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Abrahamson, K., Adler, A., Higham, L. et al. Randomized function evaluation on a ring. Distrib Comput 3, 107–117 (1989). https://doi.org/10.1007/BF01784021
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DOI: https://doi.org/10.1007/BF01784021