Kakeya Sets, New Mergers and Old Extractors

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 gener...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Zeev Dvir, Avi Wigderson
• Keywords: Constant Factors, FOCS 2008, Random, Random Variables, Theoretical Computer Science