## Abstract

Let C be a class of probability distributions over a finite set Ω. A function D Ω → {0,1}^{m} is a disperser for C with entropy threshold k and error ε if for any distribution X in C 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-ε2^{m} 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. For several interesting classes of distributions there are explicit constructions in the literature of zero-error dispersers with "small" output length m. In this paper we develop a general technique to improve the output length of zero-error dispersers. This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given by Shaltiel (CCC'06; Proceedings of the 21st Annual IEEE Conference on Computational Complexity, (2006) 46-60.) building on earlier work by Gabizon, Raz and Shaltiel (SIAM J Comput 36 (2006) 1072-1094). Our techniques are different than those of Shaltiel (CCC'06; Proceedings of the 21st Annual IEEE Conference on Computational Complexity (2006) 46-60) and in particular give non-trivial results in the errorless case. Using our approach we construct improved zero-error 2-source dispersers. 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}^{η n} such that for every two independent distributions X_{1}, X_{2} over {0,1}^{n} each with support size at least 2^{δ n}, the output distribution D(X_{1}, X_{2}) has full support. This improves the output length of previous constructions by Barak, Kindler, Shaltiel, Sudakov and Wigderson (Proceedings of the 37th Annual ACM Symposium on Theory of Computing (2005) 1-10) and has applications in Ramsey theory and in improved constructions of certain data structures from the work of Fiat and Naor [SIAM J Comput 22 (1993)]. 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 These constructions improve the best known explicit constructions due to Rao (unpublished data) and Gabizon and Raz [Combinatorica 28 (2008)] and achieve m=Ω(k) for bit-fixing sources and m=k-o(k) for affine sources over polynomial size fields.

Original language | English |
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Pages (from-to) | 74-104 |

Number of pages | 31 |

Journal | Random Structures and Algorithms |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

## Keywords

- Derandomization
- Extractors
- Pseudorandomness
- Ramsey graphs

## All Science Journal Classification (ASJC) codes

- Software
- Applied Mathematics
- General Mathematics
- Computer Graphics and Computer-Aided Design