Testing for forbidden order patterns in an array

A sequence f ∶ [n] → R contains a pattern 𝜋 π∈S_k, that is, a permutations of [k], iff there are indices i₁ < … < i_k, such that f(i_x) > f(i_y) whenever π(x) > π(y). Otherwise, f is π-free. We study the property testing problem of distinguishing, for a fixed π, between π-free sequences and the sequences which differ from any π-free sequence in more than ϵ n places. Our main findings are as follows: (1) For monotone patterns, that is, π = (k,k − 1,…,1) and π = (1,2,…,k), there exists a nonadaptive one-sided error ϵ-test of (ϵ𝜖−1 log n)O(k²) query complexity. For any other π, any nonadaptive one-sided error test requires Ω(√n) queries. The latter lower-bound is tight for π = (1,3,2). For specific π∈S_k it can be strengthened to Ω(n^{1 − 2/(k + 1)}). The general case upper-bound is O(ϵ^−1/kn^{1 − 1/k}). (2) For adaptive testing the situation is quite different. In particular, for any π∈S₃ there exists an adaptive ϵ-tester of (ϵ𝜖−1 log n)O⁽¹⁾ query complexity.