Deterministic extractors for small-space sources


Kamp, Jesse, Anup Rao, Salil Vadhan, and David Zuckerman. “Deterministic extractors for small-space sources.” Journal of Computer and System Sciences 77, no. 1 (2011): 191-220.
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STOC-05-2006.pdf258 KB


Version History: Special issue to celebrate Richard Karp's Kyoto Prize. Extended abstract in STOC '06.

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space \(s\) sources on\(\{0,1\}^n\) as sources generated by width \(2^s\) branching programs. Specifically, there is a constant \(η > 0\) such that for any \(ζ > n^{−η}\), our algorithm extracts \(m = (δ − ζ)n\) bits that are exponentially close to uniform (in variation distance) from space \(s\) sources with min-entropy \(δn\), where \(s = Ω(ζ^ 3n)\). Previously, nothing was known for \(δ \ll 1/2,\), even for space \(0\). Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of \(r\) independent smaller sources over \(\{0, 1\}^\ell\), where the total min-entropy over all the smaller sources is \(k\). We give deterministic extractors for such sources when \(k\) is as small as \(\mathrm{polylog}(r)\), for small enough \(\ell\).


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Last updated on 07/17/2020