Letzter, Shoham; Ben-Eliezer, Omri; Waingarten, Erik; (2022) Finding Monotone Patterns in Sublinear Time, Adaptively. In: Leibniz International Proceedings in Informatics. (pp. pp. 1-19). Leibniz-Zentrum für Informatik, Dagstuhl Publishing: Leibniz, Germany.

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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.

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