Loss-less condensers, unbalanced expanders, and extractors
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Extracting Randomness Using Few Independent Sources
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Extracting Randomness via Repeated Condensing
SIAM Journal on Computing
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
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Condensers are functions which receive two inputs--a random string of bits chosen according to some unknown distribution and an independent uniform (short) seed--and output a string of bits which somehow preserves the randomness of the input. The parameters of interest here are the seed length, output length and how much randomness is preserved. Here we present explicit algorithms for condensers which have constant seed size. Our constructions improve on previous constant-seed condensers of Barak et al (2005). When the input distribution has high min-entropy, we provide a condenser having optimal rate and seed chosen from {1, 2, 3}. The analysis of this construction is considerably simpler than those of previous constructions. For the low min-entropy regime, we provide a different construction which can be viewed as a pseudorandom coloring of hypergraphs. The analysis of this condenser involves a generalization of the celebrated Balog-Szemerédi-Gowers Theorem. As an example of the simplicity of the ideas behind this generalization, we improve Bourgain-Katz-Tao sum-product estimates in just a few lines.