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A sample of samplers: a computational perspective on sampling. (English) Zbl 1343.68297
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 302-332 (2011).
Summary: We consider the problem of estimating the average of a huge set of values. That is, given oracle access to an arbitrary function $$f:\{0,1\}^n \rightarrow [0,1]$$, we wish to estimate $$2^{-n} \sum_{x\in\{0,1\}^n} f(x)$$ upto an additive error of $$\epsilon$$. We are allowed to employ a randomized algorithm that may err with probability at most $$\delta$$.
We survey known algorithms for this problem and focus on the ideas underlying their construction. In particular, we present an algorithm that makes $$O(\epsilon ^{-2}\cdot \log (1/\delta ))$$ queries and uses $$n + O(\log (1/\epsilon )) + O(\log (1/\delta ))$$ coin tosses, both complexities being very close to the corresponding lower bounds.
For the entire collection see [Zbl 1220.68005].

##### MSC:
 68W20 Randomized algorithms 05C81 Random walks on graphs 68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) 94A20 Sampling theory in information and communication theory
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