Deterministic Amplification of Space-Bounded Probabilistic Algorithms

This paper initiates the study of deterministic amplification of space-bounded probabilistic algorithms. The straightforward implementations of known amplification methods cannot be used for such algorithms, since they consume too much space. We present a new implementation of the Ajtai-Komlós-Szemerédi method, that enables to amplify an S-space algorithm that uses r random bits and errs with probability to an O(kS)-space algorithm that uses r +O(k) random bits and errs with probability (k). This method can be used to reduce the error probability of BPL algorithms below any constant, with only a constant addition of new random bits. This is weaker than the exponential reduction that can be achieved for BPP algorithms by methods that use only O(r) random bits. However, we prove that any black-box amplification method that uses O(r) random bits and makes at most p parallel simulations reduces the error to at most O(p). Hence, in BPL, where p should be a constant, the error cannot be reduced to less than a constant. This means that our method is optimal with respect to black-box amplification methods, that use O(r) random bits. The new implementation of the AKS method is based on explicit constructions of constant-space online extractors and online expanders. These are extractors and expanders, for which neighborhoods can be computed in a constant space by a Turing machine with a one-way input tape.