Improved algorithm for permutation testing

A sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $k$, if there is a sequence of indices $(i_1, i_2, \cdots, i_k)$ ($i_1<i_2<\cdots<i_k$), satisfying that $f(i_a)<f(i_b)$ if $\pi(a)<\pi(b)$, for $a,b\in [k]$. Otherwise, $f$ is referred to as $\pi$-free. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two special cases, the monotone permutations and the $(1,3,2)$ permutation. \cite{ben2019finding} improved the $(\log n)^{O(k^2)}$ non-adaptive query complexity of \cite{newman2017testing} to $O((\log n)^{\lfloor \log_{2} k\rfloor})$. Further, \cite{ben2019optimal} proposed an adaptive algorithm with $O(\log n)$ query complexity. However, no progress has yet been made on the problem of testing $(1,3,2)$-pattern. In this work, we present an adaptive algorithm for testing $(1,3,2)$-pattern. The query complexity of our algorithm is $O(\epsilon^{-2}\log^4 n)$, which significantly improves over the $O(\epsilon^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.