For a permutation $\pi: [K]\rightarrow [K]$, 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. For the special case where $\pi = (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the $(1,3,2)$ permutation. For the problem of testing monotone permutation $(1,2,\cdots,K)$, \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 $(1,3,2)$-freeness. In this work, we present an adaptive algorithm for testing $(1,3,2)$-freeness. 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.

翻译：对于排列 $\pi: [K]\rightarrow [K]$，若存在索引序列 $(i_1, i_2, \cdots, i_K)$（$i_1<i_2<\cdots<i_K$），使得对所有 $a,b\in [K]$，有 $f(i_a)<f(i_b)$ 当且仅当 $\pi(a)<\pi(b)$，则序列 $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ 包含一个大小为 $K$ 的 $\pi$-模式。否则，称 $f$ 是 $\pi$-自由的。当 $\pi = (1,2,\cdots, K)$ 时，该模式称为单调模式。\cite{newman2017testing} 首次提出了具有单边错误的 $\pi$-自由性测试问题，并重点研究了两个特例：单调排列和 $(1,3,2)$ 排列的测试。对于单调排列 $(1,2,\cdots,K)$ 的测试问题，\cite{ben2019finding} 将 \cite{newman2017testing} 中 $(\log n)^{O(K^2)}$ 的非自适应查询复杂度改进为 $O((\log n)^{\lfloor \log_{2} K\rfloor})$。随后，\cite{ben2019optimal} 提出了查询复杂度为 $O(\log n)$ 的自适应算法。然而，在 $(1,3,2)$-自由性测试问题上尚未取得进展。本文提出了一种用于测试 $(1,3,2)$-自由性的自适应算法，其查询复杂度为 $O(\epsilon^{-2}\log^4 n)$，显著优于 \cite{newman2017testing} 中 $O(\epsilon^{-7}\log^{26}n)$ 查询的自适应算法。这一改进主要得益于在模式中嵌入一种新型结构的创新方法。