Strongly Sublinear Algorithms for Testing Pattern Freeness

Abstract

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 $\varepsilon n$ valuesfrom every $\pi$-free function. This is a generalization of the well-studiedmonotonicity testing and was first studied by Newman, Rabinovich,Rajendraprasad and Sohler (Random Structures and Algorithms 2019). We show thatfor all constants $k \in \mathbb{N}$, $\varepsilon \in (0,1)$, and permutation$\pi:[k] \to [k]$, there is a one-sided error $\varepsilon$-testing algorithmfor $\pi$-freeness of functions $f:[n] \to \mathbb{R}$ that makes$\tilde{O}(n^{o(1)})$ queries. We improve significantly upon the previous bestupper bound $O(n^{1 - 1/(k-1)})$ by Ben-Eliezer and Canonne (SODA 2018). Ouralgorithm is adaptive, while the earlier best upper bound is known to be tightfor nonadaptive algorithms.Comment: 28 pages, 2 figures; We thank anonymous reviewers for comments that helped us significantly improve the presentatio