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

Authors Dana Ron , Asaf Rosin

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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.

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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

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.

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

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