改进的排列测试算法
Improved Algorithm for Permutation Testing
AI总结：
本文提出一种测试$(1,3,2)$-freeness的自适应算法，查询复杂度为$O(ε^{-2}\log^4 n)$，通过引入嵌入模式的新结构，显著改进了此前$O(ε^{-7}\log^{26}n)$的查询复杂度。
摘要：
对于排列$π: [K]\rightarrow [K]$，序列$f: \{1,2,\cdots, n\}\rightarrow \mathbb R$包含大小为$K$的$π$-模式，如果存在一列索引$(i_1, i_2, \cdots, i_K)$（$i_1<i_2<\cdots<i_K$），满足对于$a,b\in [K]$，若$π(a)<π(b)$则$f(i_a)<f(i_b)$。否则，称$f$为$π$-free。当$π= (1,2,\cdots, K)$时，称为单调模式。\cite{newman2017testing}开创了带单边误差的$π$-freeness测试研究，聚焦于测试单调排列和$(1,3,2)$排列两个特定问题。针对单调排列$(1,2,\cdots,K)$测试，\cite{ben2019finding}将\cite{newman2017testing}的$(\log n)^{O(K^2)}$非自适应查询复杂度改进至$O((\log n)^{\lfloor \log_{2} K\rfloor})$。进一步，\cite{ben2019optimal}提出了查询复杂度为$O(\log n)$的自适应算法。然而，测试$(1,3,2)$-freeness问题此前无任何进展。本文提出一种测试$(1,3,2)$-freeness的自适应算法，其查询复杂度为$O(ε^{-2}\log^4 n)$，显著优于\cite{newman2017testing}的$O(ε^{-7}\log^{26}n)$查询自适应算法。此改进主要归功于提出一种嵌入模式中的新结构。
英文摘要：
For a permutation $π: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $π$-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 $π(a)<π(b)$, for $a,b\in [K]$. Otherwise, $f$ is referred to as $π$-free. For the special case where $π= (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $π$-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 $O(ε^{-2}\log^4 n)$, which significantly improves over the $O(ε^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
