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SIAM Journal on Discrete Mathematics
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
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FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Efficiently Testing Sparse GF(2) Polynomials
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Testing juntas nearly optimally
Proceedings of the forty-first annual ACM symposium on Theory of computing
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ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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Nearly tight bounds for testing function isomorphism
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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We develop a query-efficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a k-junta f : {0, 1}n → {0, 1} given oracle access to a function f′ : {0, 1}n → {0, 1} that is only close to f. After a preprocessing step, which takes Õ(k) queries, generating each sample to the core of f takes only one query to f′. We then plug in our sample extractor in the "testing by implicit learning" framework of Diakonikolas et al. [DLM+07], improving the query complexity of testers for various Boolean function classes. In particular, for some of the classes considered in [DLM+07], such as s-term DNF formulas, size-s decision trees, size-s Boolean formulas, s-sparse polynomials over F2, and size-s branching programs, the query complexity is reduced from Õ(s4/ε2) to Õ(s/ε2). This shows that using the new sample extractor, testing by implicit learning can lead to testers having better query complexity than those tailored to a specific problem, such as the tester of Parnas et al. [PRS02] for the class of monotone s-term DNF formulas. In terms of techniques, we extend the tools used in [CGM11] for testing function isomorphism to juntas. Specifically, while the original analysis in [CGM11] allowed query-efficient noisy sampling from the core of any k-junta f, the one presented here allows similar sampling from the core of the closest k-junta to f, even if f is not a k-junta but just close to being one. One of the observations leading to this extension is that the junta tester of Blais [Bla09], based on which the aforementioned sampling is achieved, enjoys a certain weak form of tolerance.