TY - GEN

T1 - Kakeya sets, new mergers and old extractors

AU - Dvir, Zeev

AU - Wigderson, Avi

PY - 2008

Y1 - 2008

N2 - 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 resolution of 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 factors in 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.

AB - 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 resolution of 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 factors in 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.

UR - http://www.scopus.com/inward/record.url?scp=57949092809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57949092809&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.23

DO - 10.1109/FOCS.2008.23

M3 - Conference contribution

AN - SCOPUS:57949092809

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 625

EP - 633

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -