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Bounds For Dispersers, Extractors, And DepthTwo Superconcentrators
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
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Extractors for a constant number of polynomially small minentropy independent sources
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded ex ..."
Abstract

Cited by 38 (10 self)
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We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded extractors in simple ways. We introduce a new technique to condense independent somewhererandom sources which looks like a useful way to manipulate independent sources. Our techniques are different from those used in recent work [BIW04, BKS + 05, Raz05, Bou05] for this problem in the sense that they do not rely on any results from additive number theory. Using Bourgain’s extractor [Bou05] as a black box, we obtain a new extractor for 2 independent blocksources with few blocks, even when the minentropy is as small as polylog(n). We also show how to modify the 2 source disperser for linear minentropy of Barak et al. [BKS + 05] and the 3 source extractor of Raz [Raz05] to get dispersers/extractors with exponentially small error and linear output length where previously both were constant. In terms of Ramsey Hypergraphs, for every constant 1> γ> 0 our construction gives a family of explicit O(1/γ)uniform hypergraphs on N vertices that avoid cliques and independent sets of (log N)γ size 2.
Tight Bounds for DepthTwo Superconcentrators
 IN PROC. OF FOCS
, 1997
"... We show that the minimum size of a depthtwo Nsuperconcentrator is \Theta(N log² N= log log N ). Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are ..."
Abstract

Cited by 24 (2 self)
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We show that the minimum size of a depthtwo Nsuperconcentrator is \Theta(N log² N= log log N ). Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. We present two different methods for showing lower bounds. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs due to Kovari, S'os and Tur'an, this gives an almost optimal lower bound of \Omega\Gamma N(log N= log log N )²) on the size of N  superconcentrators. The second method, based on the work of Hansel, gives the optimal lower bound. The method of the Kovari, S'os and Tur'an can be extended to give tight lower bounds for extractors, both in terms of the number of truly random bits needed to extract one additional bit and in terms of the unavoidable entr...
Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers
, 2009
"... We extend the “method of multiplicities ” to get the following results, of interest in combinatorics and randomness extraction. 1. We show that every Kakeya set in F n q, the ndimensional vector space over the finite field on q elements, must be of size at least q n /2 n. This bound is tight to wit ..."
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Cited by 14 (5 self)
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We extend the “method of multiplicities ” to get the following results, of interest in combinatorics and randomness extraction. 1. We show that every Kakeya set in F n q, the ndimensional vector space over the finite field on q elements, must be of size at least q n /2 n. This bound is tight to within a 2 + o(1) factor for every n as q → ∞. 2. We give improved “randomness mergers”, i.e., seeded functions that take as input k (possibly correlated) random variables in {0, 1} N and a short random seed and output a single random variable in {0, 1} N that is statistically close to having entropy (1−δ)·N when one of the k input variables is distributed uniformly. The seed we require is only (1/δ)·log kbits long, which significantly improves upon previous construction of mergers. The “method of multiplicities”, as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset with high multiplicity. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple
KAKEYA SETS, NEW MERGERS AND OLD EXTRACTORS
"... Abstract. A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random variables on n bits each, u ..."
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Abstract. A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random variables on n bits each, uses a O(log(nk)) seed, and whose error is 1/nk. Our merger can be viewed as a derandomized version of the merger of Lu, Reingold, Vadhan and Wigderson (2003). Its analysis generalizes the recent resolution of the Kakeya problem in finite fields of Dvir (2008). Following the plan set forth by TaShma (1996), who defined mergers as part of this plan, our merger provides the last “missing link ” to a simple and modular construction of extractors for all entropies, which is optimal to constant factors in all parameters. This complements the elegant construction of such extractors given by Guruswami, Umans and Vadhan (2007). We also give simple extensions of our merger in two directions. First, we generalize it to handle the case where no source is uniform – in that case the merger will extract the entropy present in the most random of the given sources. Second, we observe that the merger works just as well in the computational setting, when the sources are efficiently samplable, and computational notions of entropy replace the information theoretic ones.