Testing for forbidden order patterns in an array

TL;DR: In this paper, the authors studied the problem of testing the π-freeness of a sequence f : {1,..., n} → ℝ of length n contains a pattern π ∈ 𝔖k, k constant, be a (forbidden) pattern.

Abstract: In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → ℝ of length n contains a pattern π ∈ 𝔖k (𝔖k is the group of permutations of k elements), iff there are indices i1 f(iy) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ 𝔖k, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than ϵn places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error ϵ-tests of (ϵ−1 log n)O(k2) query complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error ϵ-test requires at least [EQUATION] queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n1−2/(k+1)).On the other hand, there always exists a non-adaptive one-sided error ϵ-test for π ∈ 𝔖k with O(ϵ−1/kn1−1/k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = (1, 3, 2), we describe an ϵ-test with (almost tight) query complexity of [EQUATION].Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ 𝔖3, tests π-freeness by making (ϵ−1 log n)O(1) queries.For all algorithms presented here, the running times are linear in their query complexity.