TR98-072 Authors: Ziv Bar-Yossef, Oded Goldreich, Avi Wigderson

Publication: 14th December 1998 09:20

Downloads: 2024

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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 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.