Strongly Sublinear Algorithms for Testing Pattern Freeness

For a permutation $π:[k] \to [k]$, a function $f:[n] \to \mathbb{R}$ contains a $π$-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 $π(s) < π(t)$. The function is $π$-free if it has no $π$-appearances. In this paper, we investigate the problem of testing whether an input function $f$ is $π$-free or whether $f$ differs on at least $\varepsilon n$ values from every $π$-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 $π:[k] \to [k]$, there is a one-sided error $\varepsilon$-testing algorithm for $π$-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.