Testing for forbidden order patterns in an array

Abstract

A sequence contains a pattern , that is, a permutations of [k], iff there are indices i1 < … < ik, such that f(ix) > f(iy) 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 query complexity. For any other π, any nonadaptive one‐sided error test requires queries. The latter lower‐bound is tight for π = (1,3,2). For specific it can be strengthened to Ω(n1 − 2/(k + 1)). The general case upper‐bound is O(ϵ−1/kn1 − 1/k). (2) For adaptive testing the situation is quite different. In particular, for any there exists an adaptive ϵ‐tester of query complexity.