# Increasing the output length of zero-error dispersers

Ariel Gabizon, Ronen Shaltiel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

Let be a class of probability distributions over a finite set Ω. A function D : Ω → {0,1}m is a disperser for with entropy threshold k and error if for any distribution X in such that X gives positive probability to at least 2 k elements we have that the distribution D(X) gives positive probability to at least (1 - ε)2m elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called "extractors") for various classes of distributions while trying to maximize m as a function of k. In this paper we are interested in explicitly constructing zero-error dispersers (that is dispersers with error ε = 0). For several interesting classes of distributions there are explicit constructions in the literature of zero-error dispersers with "small" output length m and we give improved constructions that achieve "large" output length, namely m = Ω(k). We achieve this by developing a general technique to improve the output length of zero-error dispersers (namely, to transform a disperser with short output length into one with large output length). This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given in [29] building on earlier work by [15]). Nevertheless, we stress that our techniques are different than those of [29] and in particular give non-trivial results in the errorless case. Using our approach we construct improved zero-error dispersers for the class of 2-sources. More precisely, we show that for any constant δ > 0 there is a constant η > 0 such that for sufficiently large n there is a poly-time computable function D : {0,1}n × {0,1}n → {0,1}ηnsuch that for any two independent distributions X 1, X2 over {0,1}n such that both of them support at least 2δn elements we get that the output distribution D(X1, X2) has full support. This improves the output length of previous constructions by [2] and has applications in Ramsey Theory and in constructing certain data structures [13]. We also use our techniques to give explicit constructions of zero-error dispersers for bit-fixing sources and affine sources over polynomially large fields. These constructions improve the best known explicit constructions due to [26,14] and achieve m = Ω(k) for bit-fixing sources and m = k - o(k) for affine sources.

Original language English Approximation, Randomization and Combinatorial Optimization Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings 430-443 14 https://doi.org/10.1007/978-3-540-85363-3_34 Published - 2008 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 - Boston, MA, United StatesDuration: 25 Aug 2008 → 27 Aug 2008

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 5171 LNCS 0302-9743 1611-3349

### Conference

Conference 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 United States Boston, MA 25/08/08 → 27/08/08

## ASJC Scopus subject areas

• Theoretical Computer Science
• General Computer Science

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