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Finding Monotone Patterns in Sublinear Time, Adaptively

Authors Omri Ben-Eliezer , Shoham Letzter , Erik Waingarten

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Author Details

Omri Ben-Eliezer
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Shoham Letzter
  • University College London, UK
Erik Waingarten
  • Stanford University, CA, USA

Funding

  • Letzter, Shoham: Research supported by Dr. Max Rössler, the Walter Haefner Foundation and by the ETH Zurich Foundation.
  • Waingarten, Erik: This work is supported by the National Science Foundation under Award No. 2002201 and Moses Charikar’s Simons Investigator award.

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Omri Ben-Eliezer, Shoham Letzter, and Erik Waingarten. Finding Monotone Patterns in Sublinear Time, Adaptively. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 17:1-17:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.17

Abstract

We investigate adaptive sublinear algorithms for finding monotone patterns in sequential data. Given fixed 2 ≤ k ∈ m N and ε > 0, consider the problem of finding a length-k increasing subsequence in a sequence f : [n] → ℝ, provided that f is ε-far from free of such subsequences. It was shown by Ben-Eliezer et al. [FOCS 2019] that the non-adaptive query complexity of the above task is Θ((log n)^⌊log₂ k⌋). In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes O(log n) queries. This is optimal, matching the classical Ω(log n) adaptive lower bound by Fischer [Inf. Comp. 2004] for monotonicity testing (which corresponds to the case k = 2). Equivalently, our result implies that testing whether a sequence decomposes into k monotone subsequences can be done with O(log n) queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • property testing
  • monotone patterns
  • monotone decomposition
  • adaptivity

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