Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
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)).
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
M. Saks, X. Sun, P. Beame and E. Vee, "Super-linear time-space tradeoff lower bounds for randomized computation," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 169.