For a permutation $\pi:[k] \to [k]$, a function $f:[n] \to \mathbb{R}$ contains a $\pi$-appearance if there exists $1 \leq i_1 < i_2 < \dots < i_k \leq n$ such that for all $s,t \in [k]$, $f(i_s) < f(i_t)$ if and only if $\pi(s) < \pi(t)$. The function is $\pi$-free if it has no $\pi$-appearances. In this paper, we investigate the problem of testing whether an input function $f$ is $\pi$-free or whether $f$ differs on at least $\varepsilon n$ values from every $\pi$-free function. This is a generalization of the well-studied monotonicity testing and was first studied by Newman, Rabinovich, Rajendraprasad and Sohler (Random Structures and Algorithms 2019). We show that for all constants $k \in \mathbb{N}$, $\varepsilon \in (0,1)$, and permutation $\pi:[k] \to [k]$, there is a one-sided error $\varepsilon$-testing algorithm for $\pi$-freeness of functions $f:[n] \to \mathbb{R}$ that makes $\tilde{O}(n^{o(1)})$ queries. We improve significantly upon the previous best upper bound $O(n^{1 - 1/(k-1)})$ by Ben-Eliezer and Canonne (SODA 2018). Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.

翻译：对于排列$\pi:[k] \to [k]$，若存在$1 \leq i_1 < i_2 < \dots < i_k \leq n$使得对所有$s,t \in [k]$，有$f(i_s) < f(i_t)$当且仅当$\pi(s) < \pi(t)$，则函数$f:[n] \to \mathbb{R}$包含一个$\pi$出现。若函数没有$\pi$出现，则称其为$\pi$自由。本文研究测试输入函数$f$是否为$\pi$自由，或者$f$在至少$\varepsilon n$个值上与每个$\pi$自由函数不同的判定问题。这是经典单调性测试的推广，最早由Newman、Rabinovich、Rajendraprasad和Sohler（Random Structures and Algorithms 2019）研究。我们证明：对所有常数$k \in \mathbb{N}$、$\varepsilon \in (0,1)$及排列$\pi:[k] \to [k]$，存在针对函数$f:[n] \to \mathbb{R}$的$\pi$自由性的单边错误$\varepsilon$测试算法，其查询复杂度为$\tilde{O}(n^{o(1)})$。这显著改进了Ben-Eliezer与Canonne（SODA 2018）此前的最佳上界$O(n^{1 - 1/(k-1)})$。我们的算法是自适应的，而此前的最佳上界已知对于非自适应算法是紧的。