Time-space trade-off lower bounds for randomized computation of decision problems

Abstract

We prove the first time-space lower bound trade-offs 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 extension of those used by Ajtai and by Beame, Jayram, and Saks that applied to deterministic branching programs. Our results also give a quantitative improvement over the previous results.Previous time-space trade-off results for decision problems can be divided naturally into results for functions withBoolean domain, that is, each input variable is {0,1}-valued, and the case oflarge domain, where each input variable takes on values from a set whose size grows with the number of variables.In the case of Boolean domain, Ajtai exhibited an explicit class of functions, and proved that any deterministic Boolean branching program or RAM using spaceS=o(n) requires superlinear timeTto compute them. The functional form of the superlinear bound is not given in his paper, but optimizing the parameters in his arguments givesT= Ω(nlog logn/log log logn) forS=O(n1-ϵ). For the same functions considered by Ajtai, we prove a time-space trade-off (for randomized branching programs with error) of the formT= Ω(n√ log(n/S)/log log (n/S)). In particular, for spaceO(n1-ϵ), this improves the lower bound on time to Ω(n√ logn/log logn).In the large domain case, we prove lower bounds of the formT= Ω(n√ log(n/S)/log log (n/S)) for randomized computation of the element distinctness function and lower bounds of the formT= Ω(nlog (n/S)) for randomized computation of Ajtai's Hamming closeness problem and of certain functions associated with quadratic forms over large fields.