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Theory of Computing Systems

, Volume 63, Issue 3, pp 418–449 | Cite as

Property Testing Lower Bounds via a Generalization of Randomized Parity Decision Trees

  • Roei TellEmail author
Article
  • 37 Downloads

Abstract

A few years ago, Blais, Brody, and Matulef: Comput. Complex. 21(2), 311–358 (2012) presented a methodology for proving lower bounds for property testing problems by reducing them from problems in communication complexity. Recently, Bhrushundi, Chakraborty, and Kulkarni (2014) showed that some reductions of this type can be deconstructed to two separate reductions, from communication complexity to randomized parity decision trees and from the latter to property testing. This work follows up on these ideas. We introduce a model called linear-access algorithms, which is a generalization of randomized parity decision trees, and show several methods to reduce communication complexity problems to problems for linear-access algorithms and problems for linear-access algorithms to property testing problems. This approach yields a new interpretation for several well-known reductions, since we present these reductions as a composition of two steps with fundamentally different functionalities. Furthermore, we demonstrate the potential of proving lower bounds on property testing problems by reducing them directly from problems for linear-access algorithms. In particular, we provide an alternative and simple proof for a known lower bound of Ω(k) queries on testing “k-linearity”; that is, the property of k-sparse linear functions over \(\mathbb {F}_{2}\). This alternative proof relies on a theorem by Linial and Samorodnitsky: Combinatorica 22(4), 497–522 (2002). We then extend this result to a new lower bound of Ω(s) queries for testing s-sparse degree-d polynomials over \(\mathbb {F}_{2}\), for any \(d\in \mathbb {N}\). In addition we provide a simple proof for the hardness of testing some families of linear subcodes.

Keywords

Property testing Communication complexity Parity decision trees Linear-access algorithms Affine subspaces Linear codes 

Notes

Acknowledgments

The author thanks Tom Gur for suggesting the initial observation motivating the study and for several helpful discussions during the research process. The author is grateful to Avishay Tal for pointing him to the work of Linial and Samorodnitsky. The author also thanks his advisor, Oded Goldreich, for his guidance and support in the research and writing process. This research was partially supported by the Israel Science Foundation (grant No. 671/13).

References

  1. 1.
    Bhattacharyya, A., Fischer, E., Hatami, H., Hatami, P., Lovett, S.: Every Locally Characterized Affine-Invariant Property is Testable. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC ’13, pp 429–436. ACM, New York (2013)Google Scholar
  2. 2.
    Bhrushundi, A., Chakraborty, S., Kulkarni, R.: Property testing bounds for linear and quadratic functions via parity decision trees. In: CSR, pp. 97–110 (2014)Google Scholar
  3. 3.
    Blais, E., Brody, J., Matulef, K.: Property testing lower bounds via communication complexity. Comput. Complex. 21(2), 311–358 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blais, E., Kane, D.M.: Tight bounds for testing k-linearity. In: APPROX-RANDOM, pp. 435–446 (2012)Google Scholar
  5. 5.
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with Applications to Numerical Problems. In: Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, STOC ’90, pp 73–83. ACM, New York (1990)Google Scholar
  6. 6.
    Buhrman, H., García-soriano, D., Matsliah, A., de Wolf, R.: The non-adaptive query complexity of testing k-parities. Chicago Journal of Theoretical Computer Science (2013)Google Scholar
  7. 7.
    Chor, B., Goldreich, O.: Unbiased bits from sources of weak randomness and probabilistic communication complexity (extended abstract). In: FOCS, pp. 429–442 (1985)Google Scholar
  8. 8.
    Cohen, G., Shinkar, I.: The complexity of dnf of parities. Electron. Colloq. Comput. Complex. (ECCC) 21, 99 (2014)zbMATHGoogle Scholar
  9. 9.
    Goldreich, O.: On testing computability by small width obdds. In: APPROX-RANDOM, pp. 574–587 (2010)Google Scholar
  10. 10.
    Goldreich, O.: On the communication complexity methodology for proving lower bounds on the query complexity of property testing. Electron. Colloq. Comput. Complex. (ECCC) 20, 73 (2013)Google Scholar
  11. 11.
    Goldreich, O.: Introduction to Property Testing (working draft), April 26, 2017. Accessed at http://www.wisdom.weizmann.ac.il/oded/pt-intro.html, May 15, 2017
  12. 12.
    Håstad, J., Wigderson, A.: The randomized communication complexity of set disjointness. Theory Comput. 3(11), 211–219 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Klauck, H.: Rectangle size bounds and threshold covers in communication complexity. In: IEEE Conference on Computational Complexity, pp. 118–134 (2003)Google Scholar
  14. 14.
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  15. 15.
    Linial, N., Samorodnitsky, A.: Linear codes and character sums. Combinatorica 22(4), 497–522 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Montanaro, A., Osborne, T.: On the communication complexity of xor functions. CoRR, arXiv:0909.3392 (2009)
  17. 17.
    Razborov, A.A.: on the distributional complexity of disjointness. In: ICALP, pp. 249–253 (1990)Google Scholar
  18. 18.
    Shaltiel, R.: An introduction to randomness extractors. In: ICALP (2), pp. 21–41 (2011)Google Scholar
  19. 19.
    Shpilka, A., Tal, A., Volk, B.L.: On the structure of boolean functions with small spectral norm. In: ITCS, pp. 37–48 (2014)Google Scholar
  20. 20.
    Tell, R.: An alternative proof of an Ω(k) lower bound for testing k-linear boolean functions. Electron. Colloq. Comput. Complex. (ECCC) 21, 73 (2014)Google Scholar
  21. 21.
    Zhang, Z., Shi, Y.: On the parity complexity measures of boolean functions. Theor. Comput. Sci. 411(26-28), 2612–2618 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceWeizmann Institute of ScienceRehovotIsrael

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