Testing for Forbidden Order Patterns in an Array

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 is the group of permutations of k elements), iff there are indices i1 < i2 < · · · < ik, such that f (ix) > f (iy) whenever π(χ) > 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 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 en 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 (∊−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 Ω(γ/η) 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 with O(e−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 Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any tests π-freeness by making (∊−1 logn)O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.