Strongly Sublinear Algorithms for Testing Pattern Freeness

For a permutation π : [k] → [k], a function f : [n] → ℝ contains a π-appearance if there exists 1 ≤ i₁ < i₂ < · · · < i_k ≤ n such that for all s, t ∈ [k], f(is) < f(it) 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 εn 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 [28]. We show that for all constants k ∈ N, ε ∈ (0, 1), and permutation π : [k] → [k], there is a one-sided error ε-testing algorithm for π-freeness of functions f : [n] → R that makes Õ(n^o(1)) queries. We improve significantly upon the previous best upper bound O(n^{1−1/(k−1))} by Ben-Eliezer and Canonne [7]. Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.