Document Open Access Logo

Sublinear-Time Computation in the Presence of Online Erasures

Authors Iden Kalemaj , Sofya Raskhodnikova , Nithin Varma

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2022.90.pdf
  • Filesize: 0.88 MB
  • 25 pages

Document Identifiers

Author Details

Iden Kalemaj
  • Department of Computer Science, Boston University, MA, USA
Sofya Raskhodnikova
  • Department of Computer Science, Boston University, MA, USA
Nithin Varma
  • Chennai Mathematical Institute, India

Funding

  • Kalemaj, Iden: This work was supported by NSF award CCF-1909612 and Boston University’s Dean’s Fellowship.
  • Raskhodnikova, Sofya: This work was supported by NSF award CCF-1909612.
  • Varma, Nithin: This work was done in part while the author was a postdoctoral researcher at the University of Haifa, Israel, where he was supported by the Israel Science Foundation, grant number 497/17 and the PBC Fellowship for Postdoctoral Fellows by the Israeli Council of Higher Education.

Acknowledgements

We are thankful to Kobbi Nissim for suggesting investigating settings with online adversarial erasures. We thank Noga Ron-Zewi for pointing out relevant references and the anonymous ITCS reviewers for helpful comments.

Cite As Get BibTex

Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-Time Computation in the Presence of Online Erasures. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 90:1-90:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.90

Abstract

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase t input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm’s access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant t with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of t, showing that the query complexity is Θ(log t). In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for t = 1. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Randomized algorithms
  • property testing
  • Fourier analysis
  • linear functions
  • quadratic functions
  • Lipschitz and monotone functions
  • sorted sequences

Metrics

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

Related Versions

References

  1. Nir Ailon, Bernard Chazelle, Seshadhri Comandur, and Ding Liu. Estimating the distance to a monotone function. Random Structures and Algorithms, 31(3):371-383, 2007. URL: https://doi.org/10.1002/rsa.20167.
  2. Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing Reed-Muller codes. IEEE Transactions on Information Theory, 51(11):4032-4039, 2005. URL: https://doi.org/10.1109/TIT.2005.856958.
  3. Sanjeev Arora and Madhu Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365-426, 2003. URL: https://doi.org/10.1007/s00493-003-0025-0.
  4. Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing Lipschitz functions on hypergrid domains. Algorithmica, 74(3):1055-1081, 2016. URL: https://doi.org/10.1007/s00453-015-9984-y.
  5. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 21-31, 1991. URL: https://doi.org/10.1145/103418.103428.
  6. László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. URL: https://doi.org/10.1007/BF01200056.
  7. Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, and Madhu Sudan. Linearity testing in characteristic two. IEEE Transactions on Information Theory, 42(6):1781-1795, 1996. URL: https://doi.org/10.1109/18.556674.
  8. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCPs, and nonapproximability-towards tight results. SIAM Journal on Computing (SICOMP), 27(3):804-915, 1998. URL: https://doi.org/10.1137/S0097539796302531.
  9. Mihir Bellare, Shafi Goldwasser, Carsten Lund, and Alexander Russell. Efficient probabilistically checkable proofs and applications to approximations. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 294-304, 1993. URL: https://doi.org/10.1145/167088.167174.
  10. Mihir Bellare and Madhu Sudan. Improved non-approximability results. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 184-193, 1994. URL: https://doi.org/10.1145/195058.195129.
  11. Aleksandrs Belovs. Adaptive lower bound for testing monotonicity on the line. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 31:1-31:10, 2018. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.31.
  12. Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Ron D. Rothblum. Hard properties with (very) short PCPPs and their applications. In Proceedings, Innovations in Theoretical Computer Science (ITCS), pages 9:1-9:27, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.9.
  13. Eli Ben-Sasson, Madhu Sudan, Salil P. Vadhan, and Avi Wigderson. Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 612-621, 2003. URL: https://doi.org/10.1145/780542.780631.
  14. Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 164-173, 2014. URL: https://doi.org/10.1145/2591796.2591887.
  15. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. SIAM Journal on Computing (SICOMP), 41(6):1380-1425, 2012. URL: https://doi.org/10.1137/110826655.
  16. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of Reed-Muller codes. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 488-497, 2010. URL: https://doi.org/10.1109/FOCS.2010.54.
  17. Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In Proceedings, IEEE Conference on Computational Complexity (CCC), pages 309-320, 2014. URL: https://doi.org/10.1109/CCC.2014.38.
  18. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993. URL: https://doi.org/10.1016/0022-0000(93)90044-W.
  19. Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri. Property testing on product distributions: Optimal testers for bounded derivative properties. ACM Transactions on Algorithms (TALG), 13(2):20:1-20:30, 2017. URL: https://doi.org/10.1145/3039241.
  20. Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 419-428, 2013. URL: https://doi.org/10.1145/2488608.2488661.
  21. Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014. URL: https://doi.org/10.4086/toc.2014.v010a017.
  22. Irit Dinur and Venkatesan Guruswami. PCPs via low-degree long code and hardness for constrained hypergraph coloring. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 340-349, 2013. URL: https://doi.org/10.1109/FOCS.2013.44.
  23. Kashyap Dixit, Madhav Jha, Sofya Raskhodnikova, and Abhradeep Thakurta. Testing the Lipschitz property over product distributions with applications to data privacy. In Theory of Cryptography Conference (TCC), pages 418-436, 2013. URL: https://doi.org/10.1007/978-3-642-36594-2_24.
  24. Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma. Erasure-resilient property testing. SIAM Journal on Computing (SICOMP), 47(2):295-329, 2018. URL: https://doi.org/10.1137/16M1075661.
  25. Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 97-108, 1999. URL: https://doi.org/10.1007/978-3-540-48413-4_10.
  26. Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. Journal of Computer and System Sciences, 60(3):717-751, 2000. URL: https://doi.org/10.1006/jcss.1999.1692.
  27. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292, 1996. URL: https://doi.org/10.1145/226643.226652.
  28. Eldar Fischer. On the strength of comparisons in property testing. Information and Computation, 189(1):107-116, 2004. URL: https://doi.org/10.1016/j.ic.2003.09.003.
  29. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 474-483, 2002. URL: https://doi.org/10.1145/509907.509977.
  30. Katalin Friedl and Madhu Sudan. Some improvements to total degree tests. In Third Israel Symposium on Theory of Computing and Systems (ISTCS), pages 190-198, 1995. URL: https://doi.org/10.1109/ISTCS.1995.377032.
  31. Peter Gemmell, Richard J. Lipton, Ronitt Rubinfeld, Madhu Sudan, and Avi Wigderson. Self-testing/correcting for polynomials and for approximate functions. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 32-42, 1991. URL: https://doi.org/10.1145/103418.103429.
  32. Oded Goldreich. On multiple input problems in property testing. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 704-720, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.704.
  33. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998. URL: https://doi.org/10.1145/285055.285060.
  34. Venkatesan Guruswami and Atri Rudra. Tolerant locally testable codes. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 306-317, 2005. URL: https://doi.org/10.1007/11538462_26.
  35. Elad Haramaty, Amir Shpilka, and Madhu Sudan. Optimal testing of multivariate polynomials over small prime fields. SIAM Journal on Computing (SICOMP), 42(2):536-562, 2013. URL: https://doi.org/10.1137/120879257.
  36. Johan Håstad and Avi Wigderson. Simple analysis of graph tests for linearity and PCP. Random Structures and Algorithms, 22(2):139-160, 2003. URL: https://doi.org/10.1002/rsa.10068.
  37. Madhav Jha and Sofya Raskhodnikova. Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM Journal on Computing (SICOMP), 42(2):700-731, 2013. URL: https://doi.org/10.1137/110840741.
  38. Charanjit S. Jutla, Anindya C. Patthak, Atri Rudra, and David Zuckerman. Testing low-degree polynomials over prime fields. Random Structures and Algorithms, 35(2):163-193, 2009. URL: https://doi.org/10.1002/rsa.20262.
  39. Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-time computation in the presence of online erasures. CoRR, abs/2109.08745, 2021. Google Scholar
  40. Tali Kaufman, Simon Litsyn, and Ning Xie. Breaking the epsilon-soundness bound of the linearity test over GF(2). SIAM Journal on Computing (SICOMP), 39(5):1988-2003, 2010. URL: https://doi.org/10.1137/080715548.
  41. Tali Kaufman and Dana Ron. Testing polynomials over general fields. SIAM Journal on Computing (SICOMP), 36(3):779-802, 2006. URL: https://doi.org/10.1137/S0097539704445615.
  42. Swastik Kopparty and Shubhangi Saraf. Tolerant linearity testing and locally testable codes. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 601-614, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_45.
  43. Amit Levi, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Erasure-resilient sublinear-time graph algorithms. In Proceedings, Innovations in Theoretical Computer Science (ITCS), pages 80:1-80:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.80.
  44. Alessandro Mantelero. The EU proposal for a general data protection regulation and the roots of the `right to be forgotten'. Computer Law and Security Review, 29(3):229-235, 2013. URL: https://doi.org/10.1016/j.clsr.2013.03.010.
  45. Dana Moshkovitz. Low-degree test with polynomially small error. Computational Complexity, 26(3):531-582, 2017. URL: https://doi.org/10.1007/s00037-016-0149-4.
  46. Dana Moshkovitz and Ran Raz. Sub-constant error low degree test of almost-linear size. SIAM Journal on Computing (SICOMP), 38(1):140-180, 2008. URL: https://doi.org/10.1137/060656838.
  47. Ilan Newman and Nithin Varma. New sublinear algorithms and lower bounds for LIS estimation. In Proceedings, International Colloquium on Automata, Languages and Programming (ICALP), pages 100:1-100:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.100.
  48. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
  49. Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Parameterized property testing of functions. ACM Transactions on Computation Theory, 9(4):17:1-17:19, 2018. URL: https://doi.org/10.1145/3155296.
  50. Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Erik Waingarten. Approximating the distance to monotonicity of Boolean functions. Random Structures and Algorithms, 2021. Preliminary version appeared in SODA 2020. URL: https://doi.org/10.1137/1.9781611975994.123.
  51. Ramesh Krishnan Pallavoor Suresh. Improved Algorithms and New Models in Property Testing. PhD thesis, Boston University, 2020. Google Scholar
  52. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. Journal of Computer and System Sciences, 6(72):1012-1042, 2006. URL: https://doi.org/10.1016/j.jcss.2006.03.002.
  53. Sofya Raskhodnikova. Testing if an array is sorted. Encyclopedia of Algorithms, pages 2219-2222, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_700.
  54. Sofya Raskhodnikova, Noga Ron-Zewi, and Nithin Varma. Erasures versus errors in local decoding and property testing. Random Structures and Algorithms, 59(4):640-670, 2021. URL: https://doi.org/10.1002/rsa.21031.
  55. Sofya Raskhodnikova and Ronitt Rubinfeld. Linearity testing/testing Hadamard codes. In Encyclopedia of Algorithms, pages 1107-1110. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_202.
  56. Sofya Raskhodnikova and Nithin Varma. Brief announcement: Erasure-resilience versus tolerance to errors. In Proceedings, International Colloquium on Automata, Languages and Programming (ICALP), pages 111:1-111:3, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.111.
  57. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 475-484, 1997. URL: https://doi.org/10.1145/258533.258641.
  58. Noga Ron-Zewi and Madhu Sudan. A new upper bound on the query complexity of testing generalized Reed-Muller codes. Theory of Computing, 9:783-807, 2013. URL: https://doi.org/10.4086/toc.2013.v009a025.
  59. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing (SICOMP), 25(2):252-271, 1996. URL: https://doi.org/10.1137/S0097539793255151.
  60. Michael E. Saks and C. Seshadhri. Estimating the longest increasing sequence in polylogarithmic time. SIAM Journal on Computing (SICOMP), 46(2):774-823, 2017. URL: https://doi.org/10.1137/130942152.
  61. Alex Samorodnitsky. Low-degree tests at large distances. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 506-515, 2007. URL: https://doi.org/10.1145/1250790.1250864.
  62. Alex Samorodnitsky and Luca Trevisan. A PCP characterization of NP with optimal amortized query complexity. In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 191-199, 2000. URL: https://doi.org/10.1145/335305.335329.
  63. Alex Samorodnitsky and Luca Trevisan. Gowers uniformity, influence of variables, and PCPs. SIAM Journal on Computing (SICOMP), 39(1):323-360, 2009. URL: https://doi.org/10.1137/070681612.
  64. Amir Shpilka and Avi Wigderson. Derandomizing homomorphism testing in general groups. SIAM Journal on Computing (SICOMP), 36(4):1215-1230, 2006. URL: https://doi.org/10.1137/S009753970444658X.
  65. Madhu Sudan and Luca Trevisan. Probabilistically checkable proofs with low amortized query complexity. In Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pages 18-27, 1998. URL: https://doi.org/10.1109/SFCS.1998.743425.
  66. Luca Trevisan. Recycling queries in PCPs and in linearity tests (extended abstract). In Proceedings, ACM Symposium on Theory of Computing (STOC), pages 299-308, 1998. URL: https://doi.org/10.1145/276698.276769.
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 About DROPS Imprint Privacy Contact