Testing for Forbidden Order Patterns in an Array

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 ∈ S k (S k is the group of permutations of k elements), iff there are indices i 1 < i 2 < · · · < i k , such that f (i x ) > f (i y ) 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 ∈ S k , 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 n 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 -tests of (complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error -test requires at least Ω( √ n) queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n 1−2/(k+1) ).On the other hand, there always exists a nonadaptive one-sided error -test for π ∈ S k with O( −1/k n 1−1/k ) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = * Department of Computer Science, University of Haifa, Israel. (1, 3, 2), we describe an -test with (almost tight) query complexity of O( √ n).Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ S 3 , tests π-freeness by making ( −1 log n) O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.

Improved Bounds for Testing Forbidden Order Patterns

“…In Section 2 we present the results of Newman et al [22] on the problem of testing π-freeness. We then provide our results in Section 3.…”

“…Most of our results are in the nonadaptive case, and seem to yield a relatively good general understanding of this case. All results in [22] only consider one-sided testing, and we also mainly follow this paradigm.…”

“…A significant body of work has been dedicated to such properties, with the examples of monotonicity [12,2,9,13,7,24], the Lipschitz property [19,7], convexity [23], and k-monotonicity [5,17]. Very recently, Newman, Rabinovich, Rajendraprasad, and Sohler [22] initiated the study of a massively parameterized property, 1 order pattern freeness, which generalizes and subsumes some of the aforementioned properties as special cases. We refer the reader to [22] for a discussion of several motivations for testing order pattern freeness, stemming e.g.…”

“…Very recently, Newman, Rabinovich, Rajendraprasad, and Sohler [22] initiated the study of a massively parameterized property, 1 order pattern freeness, which generalizes and subsumes some of the aforementioned properties as special cases. We refer the reader to [22] for a discussion of several motivations for testing order pattern freeness, stemming e.g. from combinatorics and time series analysis.…”

“…We describe here the previous state of knowledge on testing pattern-freeness; all results here are established in [22], and focus on one-sided tests.…”