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41st Annual Symposium on Foundations of Computer Science
Super-linear time-space tradeoff lower bounds for randomized computation
Redondo Beach, California
November 12-November 14
ISBN: 0-7695-0850-2
P. Beame, Dept. of Comput. Sci. & Eng., Washington Univ., Seattle, WA, USA
M. Saks, Dept. of Comput. Sci. & Eng., Washington Univ., Seattle, WA, USA
Xiaodong Sun, Dept. of Comput. Sci. & Eng., Washington Univ., Seattle, WA, USA
E. Vee, Dept. of Comput. Sci. & Eng., Washington Univ., Seattle, WA, USA
We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by M. Ajtai (1999) in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for deterministic Boolean branching programs computing an explicit function based on quadratic forms over GF(2). Our results also give a quantitative improvement over those given by Ajtai. Ajtai shows, for certain specific functions, that any branching program using space S=o(n) requires time T that is superlinear. The functional form of the superlinear bound is not given in his paper, but optimizing the parameters in his arguments gives T= /spl Omega/(n log log n/log log log n) for S=0(n/sup 1-/spl epsiv//). For the same functions considered by Ajtai, we prove a time-space tradeoff of the form T=/spl Omega/(n/spl radic/(log(n/S)/log log(n/S))). In particular for space 0(n/sup 1-/spl epsiv//), this improves the lower bound on time to /spl Omega/(n/spl radic/(log n/log log n)).
Index Terms:
randomised algorithms; computational complexity; probability; super-linear time-space tradeoff lower bounds; randomized computation; decision problems; deterministic RAM algorithms; deterministic Boolean branching programs; branching program; time-space tradeoff; lower bound
Citation:
P. Beame, M. Saks, Xiaodong Sun, E. Vee, "Super-linear time-space tradeoff lower bounds for randomized computation," focs, pp.169, 41st Annual Symposium on Foundations of Computer Science, 2000
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