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29
Extractors with weak random seeds
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... We show how to extract random bits from two or more independent weak random sources in cases where only one source is of linear minentropy and all other sources are of logarithmic minentropy. Our main results are as follows: 1. A long line of research, starting by Nisan and Zuckerman [15], gives e ..."
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Cited by 67 (6 self)
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We show how to extract random bits from two or more independent weak random sources in cases where only one source is of linear minentropy and all other sources are of logarithmic minentropy. Our main results are as follows: 1. A long line of research, starting by Nisan and Zuckerman [15], gives explicit constructions of seededextractors, that is, extractors that use a short seed of truly random bits to extract randomness from a weak random source. For every such extractor E, with seed of length d, we construct an extractor E ′ , with seed of length d ′ = O(d), that achieves the same parameters as E but only requires the seed to be of minentropy larger than (1/2 + δ) · d ′ (rather than fully random), where δ is an arbitrary small constant. 2. Fundamental results of Chor and Goldreich and Vazirani [6, 22] show how to extract Ω(n) random bits from two (independent) sources of length n and minentropy larger than (1/2 + δ) · n, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits (with optimal probability of error) when only one source is of minentropy (1/2 + δ) · n and the other source is of logarithmic minentropy. 1 3. A recent breakthrough of Barak, Impagliazzo and Wigderson [4] shows how to extract Ω(n) random bits from a constant number of (independent) sources of length n and minentropy larger than δn, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits (with optimal probability of error) when only one source is of minentropy δn and all other (constant number of) sources are of logarithmic minentropy. 4. A very recent result of Barak, Kindler, Shaltiel, Sudakov and Wigderson [5] shows how to extract a constant number of random bits from three (independent) sources of length n and minentropy larger than δn, where δ is an arbitrary small constant. We show how to extract Ω(n) random bits, with subconstant probability of error, from one source of minentropy δn and two sources of logarithmic minentropy.
Szemerédi’s regularity lemma for sparse graphs
 Foundations of Computational Mathematics
, 1997
"... A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for ..."
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Cited by 50 (17 self)
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A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.
Superpolynomial lower bounds for monotone span programs
, 1996
"... In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2) by Beimel, Gál, Paterson [BGP]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are ba ..."
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Cited by 44 (6 self)
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In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2) by Beimel, Gál, Paterson [BGP]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paleytype bipartite graphs via Weil’s character sum estimates. We prove an n Ω(log n / log log n) lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (nonmonotone) span programs over arbitrary fields.
Improved Randomness Extraction from Two Independent Sources
 In Proc. of 8th RANDOM
, 2004
"... Given two independent weak random sources X, Y , with the same length \ell and minentropies bX,bY whose sum is greater than \ell \Omega(polylog(\ell/\epsilon)), we construct a deterministic twosource extractor (aka "blender") that extracts max(bX,bY) (bX bY\ell4log(1/\epsilon)) bits which are \e ..."
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Cited by 23 (6 self)
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Given two independent weak random sources X, Y , with the same length \ell and minentropies bX,bY whose sum is greater than \ell \Omega(polylog(\ell/\epsilon)), we construct a deterministic twosource extractor (aka "blender") that extracts max(bX,bY) (bX bY\ell4log(1/\epsilon)) bits which are \epsilonclose to uniform. In contrast, best previously published construction [4] extracted at most 2(bX bY\ell2log(1/\epsilon)) bits. Our main technical tool is a construction of a strong twosource extractor that extracts (bX bY\ell2log(1/\epsilon)) bits which are \epsilonclose to being uniform and independent of one of the sources(aka "strong blender"), so that they can later be reused as a seed to a seeded extractor. Our strong twosource extractor construction improves the best previously published construction of such strong blenders [7] by a factor of 2, applies to more sources X and Y , and is considerably simpler than the latter. Our methodology also unifies several of the previous twosource extractor constructions from the literature.
A disproof of a conjecture of Erdős in Ramsey theory
 J. London Math. Soc
, 1989
"... Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of monochrom ..."
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Cited by 19 (0 self)
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Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of monochromatic Kt's in a random colouring of Kn. We present counterexamples to this conjecture and discuss properties of the extremal graphs. 1.
Some Combinatorial Aspects of TimeStamp Systems
 Europ. J. Combinatorics
, 1990
"... The aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2 k \Gamma 1 and an effective ..."
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Cited by 9 (4 self)
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The aim of this paper is to outline a combinatorial structure appearing in distributed computing, namely a directed graph in which a certain family of subsets with k vertices have a successor. It has been proved that the number of vertices of such a graph is at least 2 k \Gamma 1 and an effective construction has been given which needs k2 k\Gamma1 vertices. This problem is issued from some questions related to the labeling of processes in a system for determining the order in which they were created. By modifying some requirements on the distributed system, we show that there arise other combinatorial structures leading to the construction of solutions whose size becomes a linear function of the input. 0 Introduction Let us first describe in detail the problem of timestamping. In a system, we consider two kinds of events, namely the creation and the death of processes. We assume that two such events cannot occur simultaneously. A global "scheduler" assigns a timestamp to any proc...
ONEDIMENSIONAL ASYMPTOTIC CLASSES OF FINITE STRUCTURES
"... Abstract. A collection C of finite Lstructures is a 1dimensional asymptotic class if for every m ∈ N and every formula ϕ(x, ¯y), where ¯y =(y1,...,ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈Cand ā ∈ Mm,eitherϕ(M,ā)  ≤C,orforsomeµ ∈ E, ..."
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Cited by 9 (0 self)
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Abstract. A collection C of finite Lstructures is a 1dimensional asymptotic class if for every m ∈ N and every formula ϕ(x, ¯y), where ¯y =(y1,...,ym): (i) There is a positive constant C and a finite set E ⊂ R>0 such that for every M ∈Cand ā ∈ Mm,eitherϕ(M,ā)  ≤C,orforsomeµ ∈ E,
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This r ..."
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Cited by 6 (0 self)
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Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...