Improved algorithm for permutation testing

We study the problem of testing forbidden patterns. The patterns that are of significant interest include monotone pattern and $(1,3,2)$-pattern. For the problem of testing monotone patterns, \cite{newman2019testing} propose a non-adaptive algorithm with query complexity $(\log n)^{O(k^2)}$. \cite{ben2019finding} then improve the query complexity of non-adaptive algorithm to $\Omega((\log n)^{\lfloor\log k\rfloor})$. Further, \cite{ben2019optimal} propose an adaptive algorithm for testing monotone pattern with optimal query complexity $O(\log n)$. However, the adaptive algorithm and the analysis are rather complicated. We provide a simple adaptive algorithm with one-sided error for testing monotone permutation. We also present an algorithm with improved query complexity for testing $(1,3,2)$-pattern.Comment: There were some mistakes for the proposal of a simpler algorithm for testing monotone patter

Optimal Adaptive Detection of Monotone Patterns

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((logn)⌊log2k⌋). In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes O(logn) queries. This is optimal, matching the classical Ω(logn) 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(logn)