Optimal Adaptive Detection of Monotone Patterns
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We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed 2 ≤ k ∈N and ε > 0, consider the problem of finding a length-k increasing subsequence in an array f [n] →R, provided that f is ε-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is Θ((log n)^log_2 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 [2004] for monotonicity testing (which corresponds to the case k=2), and implying in particular that the query complexity of testing whether the longest increasing subsequence (LIS) has constant length is Θ(log n).
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