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TR08-058 | 1st June 2008 00:00

Kakeya sets, new mergers and old extractors


Authors: Zeev Dvir, Avi Wigderson
Publication: 4th June 2008 15:51
Downloads: 1370


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

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