Document Open Access Logo

Almost Optimal Distribution-Free Sample-Based Testing of k-Modality

Authors Dana Ron , Asaf Rosin

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2020.27.pdf
  • Filesize: 0.59 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sample-based property testing
  • Distribution-free property testing
  • k-modality

Metrics

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

Abstract

For an integer k ≥ 0, a sequence σ = σ₁,… ,σ_n over a fully ordered set is k-modal, if there exist indices 1 = a₀ < a₁ < … < a_{k+1} = n such that for each i, the subsequence σ_{a_i},… ,σ_{a_{i+1}} is either monotonically non-decreasing or monotonically non-increasing. The property of k-modality is a natural extension of monotonicity, which has been studied extensively in the area of property testing. We study one-sided error property testing of k-modality in the distribution-free sample-based model. We prove an upper bound of O({√{kn}log k}/ε) on the sample complexity, and an almost matching lower bound of Ω(√{kn}/ε). When the underlying distribution is uniform, we obtain a completely tight bound of Θ(√{kn/ε}), which generalizes what is known for sample-based testing of monotonicity under the uniform distribution.

Cite As Get BibTex

Dana Ron and Asaf Rosin. Almost Optimal Distribution-Free Sample-Based Testing of k-Modality. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.27

Author Details

Dana Ron
  • Tel Aviv University, Israel
Asaf Rosin
  • Tel Aviv University, Israel

Funding

  • Ron, Dana: Supported by the Israel Science Foundation (grant number 1146/18) and the Kadar-family award.
  • Rosin, Asaf: This work is part of the author’s MSc research.

Acknowledgements

We would like to thank Ronitt Rubinfeld, since this work started as part of a project in her course, and Eden Kuperwasser, for being a great project partner. We also thank an anonymous reviewer for helpful comments.

References

  1. Nir Ailon and Bernard Chazelle. Information theory in property testing and monotonicity testing in higher dimensions. Information and Computation, 204(11):1704-1717, 2006. Google Scholar
  2. Nir Ailon, Bernard Chazelle, Ding Liu, and C. Seshadhri. Estimating the distance to a monotone function. Random Structures and Algorithms, 31(3):371-383, 2007. Google Scholar
  3. Maria Balcan, Eric Blais, Avrim Blum, and Liu Yang. Active property testing. In Proceedings of the Fifty-Third Annual Symposium on Foundations of Computer Science (FOCS), pages 21-30, 2012. Google Scholar
  4. Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan, S. Pallavoor, and Sofya Raskhodnikova. Optimal unateness testers for real-valued functions. In Automata, Languages and Programming: Forty-fourth International Colloquium (ICALP), pages 5:1-5:14, 2017. Google Scholar
  5. Tugkan Batu, Ravi Kumar, and Ronitt Rubinfeld. Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of the Thirty-Sixth Annual ACM Symposium on the Theory of Computing (STOC), pages 381-390, 2004. Google Scholar
  6. Tugkan Batu, Ronitt Rubinfeld, and Patrick White. Fast approximate PCPs for multidimensional bin-packing problems. Information and Computation, 196(1):42-56, 2005. Google Scholar
  7. Aleksandra Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In Proceedings of the Fourty-Eighth Annual ACM Symposium on the Theory of Computing (STOC), pages 1021-1032, 2016. Google Scholar
  8. Omri Ben-Eliezer. Testing local properties of arrays. In Proceedings of the Tenth Innovations in Theoretical Computer Science conference (ITCS), pages 11:1-11:20, 2019. Google Scholar
  9. Omri Ben-Eliezer and Clément L. Canonne. Improved bounds for testing forbidden order patterns. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2093-2112, 2018. Google Scholar
  10. Omri Ben-Eliezer, Clément L. Canonne, Shoham Letzter, and Erik Waingarten. Finding monotone patterns in sublinear time. In Proceedings of the Sixteeth Annual Symposium on Foundations of Computer Science (FOCS), pages 1469-1494, 2019. Google Scholar
  11. Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing convexity of figures under the uniform distribution. In Proceedings of the Thirty-Second International Symposium on Computational Geometry (SoCG), pages 17:1-17:15, 2016. Google Scholar
  12. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. SIAM Journal on Computing, 41(6):1380-1425, 2012. Google Scholar
  13. Hadley Black, Deeparnab Chakrabarty, and C. Seshadhri. A o(d)⋅ polylog n monotonicity tester for Boolean functions over the hypergrid [n]^d. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2133-2151, 2018. Google Scholar
  14. Hadley Black, Deeparnab Chakrabarty, and C. Seshadhri. Domain reduction for monotonicity testing: A o(d) tester for Boolean functions in d-dimensions. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1975-1994, 2020. Google Scholar
  15. Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In Proceedings of the Twenty-Ninth IEEE Annual Conference on Computational Complexity (CCC), pages 309-320, 2014. Google Scholar
  16. Jop Briet, Sourav Chakraborty, David García-Soriano, and Arie Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012. Google Scholar
  17. Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-monotonicity. In Proceedings of the Eighth Innovations in Theoretical Computer Science conference (ITCS), pages 29:1-29:21, 2017. Google Scholar
  18. Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri. Property testing on product distributions: Optimal testers for bounded derivative properties. ACM Transactions on Algorithms, 13(2):20:1-20:30, 2017. Google Scholar
  19. Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In Proceedings of the Fourty-Fifth Annual ACM Symposium on the Theory of Computing (STOC), pages 419-428, 2013. Google Scholar
  20. Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014. Google Scholar
  21. Deeparnab Chakrabarty and C. Seshadhri. An o(n) monotonicity tester for Boolean functions over the hypercube. SIAM Journal on Computing, 45(2):461-472, 2016. Google Scholar
  22. Deeparnab Chakrabarty and C. Seshadhri. Adaptive Boolean monotonicity testing in totla influence time. In Proceedings of the Tenth Innovations in Theoretical Computer Science conference (ITCS), pages 20:1-20:7, 2019. Google Scholar
  23. Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean function monotonicity testing requires (almost) n^1/2 non-adaptive queries. In Proceedings of the Fourty-Seventh Annual ACM Symposium on the Theory of Computing (STOC), pages 519-528, 2015. Google Scholar
  24. Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In Proceedings of the Fifty-Fifth Annual Symposium on Foundations of Computer Science (FOCS), pages 286-295, 2014. Google Scholar
  25. Xi Chen and Erik Waingarten. Testing unateness nearly optimally. In Proceedings of the Fifteeth-First Annual ACM Symposium on the Theory of Computing (STOC), pages 547-558, 2019. Google Scholar
  26. Xi Chen, Erik Waingarten, and Jinyu Xie. Beyond Talagrand functions: New lower bounds for testing monotonicity and unateness. In Proceedings of the Fourty-Ninth Annual ACM Symposium on the Theory of Computing (STOC), pages 523-536, 2017. Google Scholar
  27. Xi Chen, Erik Waingarten, and Jinyu Xie. Boolean unateness testing with Õ(n^3/4) adaptive queries. In Proceedings of the Fifty-Eighth Annual Symposium on Foundations of Computer Science (FOCS), pages 868-879, 2017. Google Scholar
  28. Ilias Diakonikolas and Daniel Kane. A new approach for testing properties of discrete distributions. In Proceedings of the Fifty-Seventh Annual Symposium on Foundations of Computer Science (FOCS), pages 685-694, 2016. URL: https://doi.org/10.1109/FOCS.2016.78.
  29. Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved bounds for testing monotonicity. In Proceedings of the Third International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), pages 97-108, 1999. Google Scholar
  30. Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing (STOC), pages 259-268, January 1998. Google Scholar
  31. Shahar Fattal and Dana Ron. Approximating the distance to monotonicity in high dimensions. ACM Transactions on Algorithms, 6(3):52:1-52:37, 2010. Google Scholar
  32. Eldar Fischer. On the strength of comparisons in property testing. Information and Computation, 189(1):107-116, 2004. Google Scholar
  33. Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the Thirty-Fourth Annual ACM Symposium on the Theory of Computing (STOC), pages 474-483, 2002. Google Scholar
  34. Oded Goldreich. The uniform distribution is complete with respect to testing identity to a fixed distribution. ECCC TR16-015, 2016. Google Scholar
  35. Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301-337, 2000. Google Scholar
  36. Oded Goldreich and Dana Ron. On sample-based testers. ACM Transactions on Computing Theory, 8(2):7:1-7:54, 2016. Google Scholar
  37. Shirley Halevy and Eyal Kushilevitz. Distribution-free property testing. SIAM Journal on Computing, 37(4):1107-1138, 2007. Google Scholar
  38. Shirley Halevy and Eyal Kushilevitz. Testing monotonicity over graph products. Random Structures and Algorithms, 33(1):44-67, 2008. Google Scholar
  39. Michael Kearns and Dana Ron. Testing problems with sub-learning sample complexity. Journal of Computer and System Sciences, 61(3):428-456, 2000. Google Scholar
  40. Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and Boolean isoperimentric-type theorems. SIAM Journal on Computing, 47(6):2238-2276, 2018. Google Scholar
  41. Subhash Khot and Igor Shinkar. An Õ(n) queries adaptive tester for unateness. In Proceedings of the Twentieth International Workshop on Randomization and Computation (RANDOM), pages 37:1-37:7, 2016. Google Scholar
  42. Ramesh Krishnan, S. Pallavoor, Sofya Raskhodnikova, and Nithin M. Varma. Paramaterized property testing of functions. ACM Transactions on Computing Theory, 9(4):17:1-17:19, 2018. Google Scholar
  43. Ramesh Krishnan, S. Pallavoor, Sofya Raskhodnikova, and Erik Waingarten. Approximating the distance to monotonicity of Boolean functions. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1995-2009, 2020. Google Scholar
  44. Ilan Newman, Yuri Rabinovich, Deepak Rajendraprasad, and Christian Sohler. Testing for forbidden order patterns in an array. Random Structures and Algorithms, 55(2):402-426, 2019. Google Scholar
  45. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. Journal of Computer and System Sciences, 72(6):1012-1042, 2006. Google Scholar
  46. Sofya Raskhodnikova. Monotonicity testing. Master’s thesis, MIT, 1999. Master’s thesis. Google Scholar
  47. Sofya Raskhodnikova, Dana Ron, Amir Shpilka, and Adam Smith. Strong lower bounds for approximating distribution support size and the distinct elements problem. SIAM Journal on Computing, 39(3):813-842, 2009. Google Scholar
  48. Wojciech Szpankowski. Average Case Analysis of Algorithms on Sequences. Wiley-Interscience, New York, 2001. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact