Testing for forbidden order patterns in an array: NEWMAN et al.

A sequence f:[n]-> R contains a pattern pi is an element of Sk, that is, a permutations of [k], iff there are indices i(1) < horizontal ellipsis < i(k), such that f(i(x)) > f(i(y)) whenever pi(x) > pi(y). Otherwise, f is pi-free. We study the property testing problem of distinguishing, for a fixed pi, between pi-free sequences and the sequences which differ from any pi-free sequence in more than epsilon n places. Our main findings are as follows: (1) For monotone patterns, that is, pi = (k,k - 1, horizontal ellipsis ,1) and pi = (1,2, horizontal ellipsis ,k), there exists a nonadaptiveone-sided error epsilon-test of (epsilon-1logn)O(k2) query complexity. For any other pi, any nonadaptive one-sided error test requires omega(n) queries. The latter lower-bound is tight for pi = (1,3,2). For specific pi is an element of Sk it can be strengthened to omega(n(1 - 2/(k + 1))). The general case upper-bound is O(epsilon(-1/k)n(1 - 1/k)). (2) For adaptive testing the situation is quite different. In particular, for any pi is an element of S3 there exists an adaptive epsilon-tester of (epsilon-1logn)O(1) query complexity.