Deterministic Amplification of Space-Bounded Probabilistic Algorithms

• Authors: Ziv Bar-Yossef;Oded Goldreich;Avi Wigderson
• Venue: COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
• Year: 1999

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-Koml\'{o}s-Szemer\'{e}di method, that enables to amplify an $S$-space algorithm that uses $r$ random bits and errs with probability $\epsilon$ to an $O(kS)$-space algorithm that uses $r + O(k)$ random bits and errs with probability $\epsilon^{\Omega(k)}$.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 $\epsilon^{O(p)}$. 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 {\em online extractors} and {\em 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.