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On the average-case complexity of property testing. (English) Zbl 1343.68296
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, 124-135 (2011).
Summary: Motivated by a study of M. Zimand [“On derandomizing probabilistic sublinear-time algorithms”, in: Proceedings of the 22th annual IEEE conference on computational complexity, CCC 2007. Washington DC: IEEE Computer Society. 1–9 (2007; doi:10.1109/CCC.2007.19)], we consider the average-case complexity of property testing (focusing, for clarity, on testing properties of Boolean strings). We make two observations:
1.
In the context of average-case analysis with respect to the uniform distribution (on all strings of a fixed length), property testing is trivial. Specifically, either the yes-instances (i.e., instances having the property) or the no-instances (i.e., instances that are far from having the property) are exponentially rare, and thus the tester may just reject (resp., accept) obliviously of the input.
2.
Turning to average-case derandomization with respect to distributions that assigns noticeable probability mass to both yes-instances and no-instances, we identify a natural class of distributions and testers for which average-case derandomization results can be obtained directly (i.e., without using randomness extractors). Furthermore, the resulting deterministic algorithm may preserve the non-adaptivity of the original tester. (In contrast, Zimand’s argument utilizes a strong type of randomness extractors and introduces adaptivity into the testing process.)

For the entire collection see [Zbl 1220.68005].
MSC:
68W20 Randomized algorithms
68Q25 Analysis of algorithms and problem complexity
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