Abstract
We consider the problem of distribution-free property testing of functions. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, but no distribution-free property testing algorithm was known for any (non-trivial) property. By extending known results (from “standard”, uniform distribution property testing), we present the first such distribution-free algorithms for two of the central problems in this field:
-
A distribution-free testing algorithm for low-degree multivariate polynomials with query complexity O(d 2 + d · ε − 1), where d is the total degree of the polynomial.
-
A distribution-free monotonicity testing algorithm for functions f:[n]d →A for low-dimensions (e.g., when d is a constant) with query complexity \(O(\frac{\log^d n \cdot 2^d}{\epsilon})\).
The same approach that is taken for the distribution-free testing of low-degree polynomials is shown to apply also to several other problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Dar, S., Parnas, M., Ron, D.: Testing of clustering. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 240–251 (2000)
Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient testing of large graphs. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 656–666 (1999)
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing low-degree polynomials over GF(2). To appear in Proceedings of Random 2003 (2003)
Alon, N., Krivelevich, M., Newman, I., Szegedy, M.: Regular languages are testable with a constant number of queries. SIAM Journal on Computing 30, 1842–1862 (2001); also appeared in Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 645–655 (1999)
Alon, N., Shapira, A.: Testing satisfiablity. In: Proccedings of 13th SODA (2001)
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. JACM 45(1), 501–555 (1998); preliminary version appeared in Proc. 33th FOCS (1992)
Arora, S., Safra, S.: Probabilistic checkable proofs: A new characterization of NP. JACM 45(1), 70–122 (1998); a preliminary version appeared in Proc. 33rd FOCS (1992)
Arora, S., Sudan, M.: Improved low-degree testing and its applications. In: Proceedings of the 29th ACM STOC, pp. 485–495 (1997)
Batu, T., Rubinfeld, R., White, P.: Fast approximation PCPs for multidimensional bin-packing problems. In: Proceedings of the 3rd International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 246–256 (1999)
Blum, M., Luby, M., Rubinfeld, R.: Self testing/correcting with applications to numerical problems. Journal of Computer and System Sceince 47, 549–595 (1993)
Bshouty, N.: Private communication
Czumaj, A., Sohler, C.: Testing hypergraph coloring. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 493–505. Springer, Heidelberg (2001)
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonicity. In: Proceedings of the 3rd International Workshop on Randomized and Approximation Techniques in Computer Science, pp. 97–108 (1999)
Ergün, E., Kannan, S., Kumar, R., Rubinfeld, R., Viswanathan, M.: Spotcheckers. Journal of Computing and System Science 60, 717–751 (2000); a preliminary version appeared in Proc. 30th STOC (1998)
Fischer, E.: On the strength of comparisons in property testing (manuscript) (available at ECCC TR00-083)
Fischer, E.: The art of uninformed decisions: A primer to property testing. The Computational Complexity Column of The bulletin of the European Association for Theoretical Computer Science 75, 97–126 (2001)
Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing Juntas. In: Proceedings of the 43rd FOCS, pp. 103–112 (2002)
Fischer, E., Lehman, E., Newman, I., Raskhodnikova, S., Rubinfeld, R., Samorodnitsky, A.: Monotonicity testing over general poset domains. In: Proceedings of the 34th ACM STOC, pp. 474–483 (2002)
Fischer, E., Newman, I.: Testing of matrix properties. In: Proceedings of the 33rd ACM STOC, pp. 286–295 (2001)
Gemmell, P., Lipton, R., Rubinfeld, R., Sudan, M., Wigderson, A.: Self testing/ correcting for polynomials and for approximate functions. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 32–42 (1991)
Goldreich, O.: Combinatorical property testing – a survey. In: Pardalos, P., Rajasekaran, S., Rolim, J. (eds.) Randomized Methods in Algorithms Design, AMSDIMACS, pp. 45–61 (1998)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samorodnitsky, A.: Testing Monotonicity. Combinatorica 20(3), 301–337 (2000) (a preliminary version appeared in Proc. 39th FOCS, 1998)
Goldreich, O., Goldwasser, S., Ron, D.: Propert testing and its connection to learning and approximation. Journal of the ACM 45(4), 653–750 (1998) (a preliminary version appeared in Proc. 37th FOCS, 1996)
Goldreich, O., Ron, D.: Property testing in bounded degree graphs. In: Proccedings of the 31st Annual ACM Symposium on the Theory of Computing, pp. 406–415 (1997)
Goldreich, O., Trevisan, L.: Three theorems regarding testing graph properties. In: Proceedings of FOCS 2001, pp. 460–469 (2001)
Kearns, M.J., Vzirani, U.V.: An introduction to Computational Learning Theory. MIT Press, Cambridge (1994)
Kohayakawa, Y., Nagle, B., Rodl, V.: Efficient testing of hypergraphs. In: Proceedings of ICALP 2002 (2002)
Newman, I.: Testing of functions that have small width branching programs. In: Proceedings of the 41st Annual Symposium of Foundations of Computer Science, pp. 251–258 (2000)
Parnas, M., Ron, D.: Testing metric properties. In: Proceedings of the 33rd ACM STOC, pp. 276–285 (2001)
Parnas, M., Ron, D.: Testing the diameter of graphs. In: RANDOM APPROX, pp. 85–96 (1999)
Parnas, M., ron, D., Rubinfeld, R.: Testing parenthesis languages. In: Proceedings of the 5th International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 261–272 (2001)
Parnas, M., Ron, D., Samorodnitsky, A.: Proclaiming dictators and juntas ot testing boolean formulae. In: RANDOM APPROX, pp. 273–284 (2001)
Ron, D.: Property testing (a tutorial). In: Rajasekaran, S., Pardalos, P.M., Reif, J.H., Rolin, J.D.P. (eds.) Handbook of Randomized Computing. Kluwer Press, Dordrecht (2001)
Rubinfeld, R.: Robust functional equations and their applications to program testing. SIAM Journal on Computing 28(6), 1972–1997 (1999) (appeared in Proceedings of the 35th Annual Symposium of Foundations of Computer Science, 1994)
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applocations to program testing. SIAM Journal of Computing 25(2), 252–271 (1996) (first appeared as a technical report, Cornell University, 1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Halevy, S., Kushilevitz, E. (2003). Distribution-Free Property Testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-45198-3_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40770-6
Online ISBN: 978-3-540-45198-3
eBook Packages: Springer Book Archive