Document Open Access Logo

Erasure-Resilient Property Testing

Authors Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, Nithin Varma

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2016.91.pdf
  • Filesize: 0.56 MB
  • 15 pages

Document Identifiers

Author Details

Kashyap Dixit
Sofya Raskhodnikova
Abhradeep Thakurta
Nithin Varma

Cite AsGet BibTex

Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma. Erasure-Resilient Property Testing. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 91:1-91:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.91

Abstract

Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless. In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P. We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.
Keywords
  • Randomized algorithms
  • property testing
  • error correction
  • monotoneand Lipschitz functions

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. N. Ailon and B. Chazelle. Information theory in property testing and monotonicity testing in higher dimension. Inform. and Comput., 204(11):1704-1717, 2006. Google Scholar
  2. N. Ailon, B. Chazelle, S. Comandur, and D. Liu. Estimating the distance to a monotone function. Random Structures Algorithms, 31(3):371-383, 2007. Google Scholar
  3. Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing Lipschitz functions on hypergrid domains. Algorithmica, 74(3):1055-1081, 2016. URL: http://dx.doi.org/10.1007/s00453-015-9984-y.
  4. Maria-Florina Balcan, Eric Blais, Avrim Blum, and Liu Yang. Active property testing. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 21-30, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.64.
  5. T. Batu, R. Rubinfeld, and P. White. Fast approximate PCPs for multidimensional bin-packing problems. Inform. and Comput., 196(1):42-56, 2005. Google Scholar
  6. Tugkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. J. ACM, 60(1):4, 2013. URL: http://dx.doi.org/10.1145/2432622.2432626.
  7. Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing Convexity of Figures Under the Uniform Distribution, 2015. To appear in SoCG 2016. Google Scholar
  8. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. SIAM J. Comput., 41(6):1380-1425, 2012. Google Scholar
  9. E. Blais, J. Brody, and K. Matulef. Property testing lower bounds via communication complexity. Comp. Complexity, 21(2):311-358, 2012. Google Scholar
  10. Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 309-320, 2014. Google Scholar
  11. J. Briët, S. Chakraborty, D. García-Soriano, and A. Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012. Google Scholar
  12. Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri. Property testing on product distributions: Optimal testers for bounded derivative properties. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1809-1828, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.121.
  13. Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 419-428, 2013. URL: http://dx.doi.org/10.1145/2488608.2488661.
  14. Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a017.
  15. Kashyap Dixit, Madhav Jha, Sofya Raskhodnikova, and Abhradeep Thakurta. Testing the lipschitz property over product distributions with applications to data privacy. In TCC, pages 418-436, 2013. URL: http://dx.doi.org/10.1007/978-3-642-36594-2_24.
  16. Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Randomization, Approximation, and Combinatorial Algorithms and Techniques, Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems RANDOM-APPROX'99, Berkeley, CA, USA, August 8-11, 1999, Proceedings, pages 97-108, 1999. Google Scholar
  17. Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. J. Comput. Syst. Sci., 60(3):717-751, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1692.
  18. Shahar Fattal and Dana Ron. Approximating the distance to convexity. Unpublished manuscript. Uploaded at URL: http://www.eng.tau.ac.il/~danar/Public-pdf/app-conv.pdf.
  19. Shahar Fattal and Dana Ron. Approximating the distance to monotonicity in high dimensions. ACM Transactions on Algorithms, 6(3), 2010. URL: http://dx.doi.org/10.1145/1798596.1798605.
  20. E. Fischer. On the strength of comparisons in property testing. Inform. and Comput., 189(1):107-116, 2004. Google Scholar
  21. Eldar Fischer and Lance Fortnow. Tolerant versus intolerant testing for boolean properties. Theory of Computing, 2(9):173-183, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a009.
  22. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, STOC'02, pages 474-483, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.509977.
  23. O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samorodnitsky. Testing monotonicity. Combinatorica, 20:301-337, 2000. Google Scholar
  24. O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. Google Scholar
  25. Oded Goldreich and Dana Ron. On proximity-oblivious testing. SIAM J. Comput., 40(2):534-566, 2011. URL: http://dx.doi.org/10.1137/100789646.
  26. Oded Goldreich and Dana Ron. On sample-based testers. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 337-345, 2015. URL: http://dx.doi.org/10.1145/2688073.2688080.
  27. Oded Goldreich and Igor Shinkar. Two-sided error proximity oblivious testing. Random Struct. Algorithms, 48(2):341-383, 2016. URL: http://dx.doi.org/10.1002/rsa.20582.
  28. S. Halevy and E. Kushilevitz. Testing monotonicity over graph products. Random Structures Algorithms, 33(1):44-67, 2008. Google Scholar
  29. Madhav Jha and Sofya Raskhodnikova. Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM J. Comput., 42(2):700-731, 2013. Google Scholar
  30. Michael J. Kearns and Dana Ron. Testing problems with sublearning sample complexity. J. Comput. Syst. Sci., 61(3):428-456, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1656.
  31. E. Lehman and D. Ron. On disjoint chains of subsets. J. Combin. Theory Ser. A, 94(2):399-404, 2001. Google Scholar
  32. M. Parnas, D. Ron, and R. Rubinfeld. Tolerant property testing and distance approximation. J. Comput. System Sci., 6(72):1012-1042, 2006. Google Scholar
  33. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. On testing convexity and submodularity. SIAM J. Comput., 32(5):1158-1184, 2003. URL: http://dx.doi.org/10.1137/S0097539702414026.
  34. Bruce A. Reed. The height of a random binary search tree. J. ACM, 50(3):306-332, 2003. URL: http://dx.doi.org/10.1145/765568.765571.
  35. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996. URL: 10.1137/S0097539793255151, URL: http://dx.doi.org/10.1137/S0097539793255151.
  36. Michael E. Saks and C. Seshadhri. Estimating the longest increasing sequence in polylogarithmic time. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 458-467, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.51.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact