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An adaptivity hierarchy theorem for property testing

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Abstract

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries.

In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of “rounds of adaptivity” it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in \({k+1}\) rounds, where the queries in the \({i}\) ’th round may depend on the answers obtained in the previous \({i-1}\) rounds. Then, we ask the following question:

Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity?

We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every \({n \in \mathbb{N}}\) and \({0 \le k \le n^{0.33}}\) there exists a property \({\mathcal{P}_{n,k}}\) of functions for which (1) there exists a \({k}\)-adaptive tester for \({\mathcal{P}_{n,k}}\) with query complexity \({{\tilde O}{(k)}}\), yet (2) any \({(k-1)}\)-adaptive tester for \({\mathcal{P}_{n,k}}\) must make \({{\tilde \Omega}{(n/k^2)}}\) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.

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References

  • Scott Aaronson & Avi Wigderson (2008). Algebrization: a new barrier in complexity theory. In Proceedings of STOC, 731–740. http://doi.acm.org/10.1145/1374376.1374481

  • Alon, Noga, Fischer, Eldar, Krivelevich, Michael, Szegedy, Mario: Efficient Testing of Large Graphs. Combinatorica 20(4), 451–476 (2000). https://doi.org/10.1007/s004930070001

    Article  MathSciNet  MATH  Google Scholar 

  • Ben-Sasson, Eli, Goldreich, Oded, Harsha, Prahladh, Sudan, Madhu, Vadhan, Salil P.: Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding. SIAM Journal on Computing 36(4), 889–974 (2006). https://doi.org/10.1137/S0097539705446810

    Article  MathSciNet  MATH  Google Scholar 

  • Ben-Sasson, Eli, Harsha, Prahladh, Raskhodnikova, Sofya: Some 3CNF Properties Are Hard to Test. SIAM J. Comput. 35(1), 1–21 (2005). https://doi.org/10.1137/S0097539704445445

    Article  MathSciNet  MATH  Google Scholar 

  • Arnab Bhattacharyya & Yuichi Yoshida (2017). Property Testing. Forthcoming. https://propertytestingbook.wordpress.com/

  • Abhishek Bhrushundi, Sourav Chakraborty & Raghav Kulkarni (2014). Property Testing Bounds for Linear and Quadratic Functions via Parity Decision Trees. In CSR, volume 8476 of Lecture Notes in Computer Science, 97–110. Springer

  • Eric Blais (2008). Improved bounds for testing juntas. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 317–330. Springer

  • Eric Blais (2009). Testing juntas nearly optimally. In Proceedings of STOC, 151–158. ACM

  • Blais, Eric, Brody, Joshua, Matulef, Kevin: Property Testing Lower Bounds via Communication Complexity. Computational Complexity 21(2), 311–358 (2012). https://doi.org/10.1007/s00037-012-0040-x

    Article  MathSciNet  MATH  Google Scholar 

  • Eric Blais & Daniel M. Kane (2012). Tight Bounds for Testing \(k\)-Linearity. In Proceedings of APPROX-RANDOM, volume 7408 of Lecture Notes in Computer Science, 435–446. Springer

  • Blum, Manuel, Luby, Michael, Rubinfeld, Ronitt: Self-Testing/Correcting with Applications to Numerical Problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)

    Article  MathSciNet  Google Scholar 

  • Joshua Brody, Kevin Matulef & Chenggang Wu (2011). Lower Bounds for Testing Computability by Small Width OBDDs. In TAMC, volume 6648 of Lecture Notes in Computer Science, 320–331. Springer

  • Buhrman, Harry, García-Soriano, David, Matsliah, Arie, de Wolf, Ronald: The non-adaptive query complexity of testing \(k\)-parities, p. 2013. Chicago J. Theor. Comput, Sci (2013)

    MATH  Google Scholar 

  • Harry Buhrman & Ronald de Wolf: Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci. 288(1), 21–43 (2002). https://doi.org/10.1016/S0304-3975(01)00144-X

    Article  MathSciNet  MATH  Google Scholar 

  • Clément L. Canonne (2015). A Survey on Distribution Testing: your Data is Big. But is it Blue? Electronic Colloquium on Computational Complexity (ECCC) 22, 63

  • Clément L. Canonne & Tom Gur (2017). An Adaptivity Hierarchy Theorem for Property Testing. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, Ryan O'Donnell, editor, volume 79 of LIPIcs, 27:1–27:25. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.CCC.2017.27

  • Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten & Jinyu Xie (2017). Settling the query complexity of non-adaptive junta testing. In Computational Complexity Conference (CCC), volume 79 of LIPIcs, 26:1–26:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik

  • Dingzhu Du & Frank K. Hwang (2000). Combinatorial Group Testing and Its Applications. Applied Mathematics. World Scientific. ISBN 9789810241070. https://books.google.com/books?id=KW5-CyUUOggC

  • Oded Goldreich (editor) (2010). Property Testing - Current Research and Surveys [outgrow of a workshop at the Institute for Computer Science (ITCS) at Tsinghua University, January 2010], volume 6390 of Lecture Notes in Computer Science. Springer. ISBN 978-3-642-16366-1. http://dx.doi.org/10.1007/978-3-642-16367-8

  • Goldreich, Oded: On the Communication Complexity Methodology for Proving Lower Bounds on the Query Complexity of Property Testing. Electronic Colloquium on Computational Complexity (ECCC) 20, 73 (2013)

    Google Scholar 

  • Oded Goldreich (2017). Introduction to Property Testing. Forthcoming. http://www.wisdom.weizmann.ac.il/~oded/pt-intro.html

  • Goldreich, Oded, Goldwasser, Shafi, Ron, Dana: Property Testing and Its Connection to Learning and Approximation. Journal of the ACM 45(4), 653–750 (1998)

    Article  MathSciNet  Google Scholar 

  • Oded Goldreich, Tom Gur & Ilan Komargodski (2015). Strong Locally Testable Codes with Relaxed Local Decoders. In Conference on Computational Complexity, volume 33 of LIPIcs, 1–41. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik

  • Oded Goldreich & Dana Ron: On Testing Expansion in Bounded-Degree Graphs. Electronic Colloquium on Computational Complexity (ECCC) 7, 20 (2000)

    MATH  Google Scholar 

  • Oded Goldreich & Dana Ron: Algorithmic Aspects of Property Testing in the Dense Graphs Model. SIAM J. Comput. 40(2), 376–445 (2011). https://doi.org/10.1137/090749621

    Article  MathSciNet  MATH  Google Scholar 

  • Oded Goldreich & Luca Trevisan: Three theorems regarding testing graph properties. Random Struct. Algorithms 23(1), 23–57 (2003). https://doi.org/10.1002/rsa.10078

    Article  MathSciNet  MATH  Google Scholar 

  • Tom Gur & Ron D. Rothblum (2017). A Hierarchy Theorem for Interactive Proofs of Proximity. In 8th Innovations in Theoretical Computer Science (ITCS), volume 67 of LIPIcs, 39:1–39:43. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik

  • Indyk, Piotr, Price, Eric, Woodruff, David P.: On the Power of Adaptivity in Sparse Recovery. In Proceedings of FOCS 285–294, (2011). https://doi.org/10.1109/FOCS.2011.83

  • Eyal Kushilevitz & Noam Nisan (1997). Communication complexity. Cambridge University Press. ISBN 978-0-521-56067-2

  • Kevin Matulef, Ryan O'Donnell, Ronitt Rubinfeld & Rocco A. Servedio (2009). Testing \(\pm \)1-weight halfspace. In Proceedings of APPROX-RANDOM, volume 5687 of Lecture Notes in Computer Science, 646–657. Springer

  • Noam Nisan & Avi Wigderson (1993). Rounds in Communication Complexity Revisited. SIAM Journal on Computing 22(1), 211–219. ISSN 0097-5397. http://dx.doi.org/10.1137/0222016

    Article  MathSciNet  Google Scholar 

  • Christos H. Papadimitriou & Michael Sipser (1982). Communication Complexity. In Proceedings of STOC, Proceedings of STOC, 196–200. ACM, New York, NY, USA. ISBN 0-89791-070-2. http://doi.acm.org/10.1145/800070.802192

  • Sofya Raskhodnikova & Adam D. Smith (2006). A Note on Adaptivity in Testing Properties of Bounded Degree Graphs. Electronic Colloquium on Computational Complexity (ECCC) 13(089). http://eccc.hpi-web.de/eccc-reports/2006/TR06-089/index.html

  • Ron, Dana: Property Testing: A Learning Theory Perspective. Foundations and Trends in Machine Learning 1(3), 307–402 (2008). https://doi.org/10.1561/2200000004

    Article  MATH  Google Scholar 

  • Ron, Dana: Algorithmic and Analysis Techniques in Property Testing. Foundations and Trends in Theoretical Computer Science 5(2), 73–205 (2009). https://doi.org/10.1561/0400000029

    Article  MathSciNet  MATH  Google Scholar 

  • Dana Ron & Rocco A. Servedio (2013). Exponentially Improved Algorithms and Lower Bounds for Testing Signed Majorities. In Proceedings of SODA, 1319–1336. SIAM

  • Dana Ron & Gilad Tsur: Testing computability by width-two OBDDs. Theor. Comput. Sci. 420, 64–79 (2012)

    Article  MathSciNet  Google Scholar 

  • Ronitt Rubinfeld & Madhu Sudan: Robust Characterization of Polynomials with Applications to Program Testing. SIAM Journal on Computing 25(2), 252–271 (1996)

    Article  MathSciNet  Google Scholar 

  • Mert Sağlam & Gábor Tardos (2013). On the Communication Complexity of Sparse Set Disjointness and Exists-Equal Problems. In Proceedings of FOCS, 678–687. IEEE Computer Society

  • Rocco A. Servedio, Li-Yang Tan & John Wright (2015). Adaptivity Helps for Testing Juntas. In Conference on Computational Complexity, volume 33 of LIPIcs, 264–279. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik

  • Tell, Roei: Deconstructions of Reductions from Communication Complexity to Property Testing using Generalized Parity Decision Trees. Electronic Colloquium on Computational Complexity (ECCC) 21, 115 (2014)

    Google Scholar 

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Canonne, C.L., Gur, T. An adaptivity hierarchy theorem for property testing. comput. complex. 27, 671–716 (2018). https://doi.org/10.1007/s00037-018-0168-4

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