NDLI: Kakeya Sets, New Mergers and Old Extractors

Content Provider	IEEE Xplore Digital Library
Author	Dvir, Z. Wigderson, A.
Copyright Year	2008
Description	Author affiliation: Weizmann Inst. of Sci., Rehovot (Dvir, Z.) \|\| Inst. for Adv. Study, Princeton, NJ (Wigderson, A.)
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 vairables 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 resolutionof the Kakeya problem in finite fields of Dvir (2008). Following the plan set forth by Ta-Shma (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 factorsin all parameters. This complements the elegant construction of optimal extractor by Guruswami, Vadhan and Umans (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.
Starting Page	625
Ending Page	633
File Size	284828
Page Count	9
File Format	PDF
ISBN	9780769534367
ISSN	02725428
DOI	10.1109/FOCS.2008.23
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-10-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Corporate acquisitions Galois fields Computer science Entropy Mathematics Harmonic analysis Random variables Modular construction Data mining Needles kakeya extractors mergers
Content Type	Text
Resource Type	Article

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