Improved Algorithm for Permutation Testing

Abstract

For a permutation $\pi :[K]\to [K]$, a sequence $f:\{1,2,\cdots ,n\}\to \mathbb{R}$ 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.