Optimal Distribution-Free Sample-Based Testing of Subsequence-Freeness with One-Sided Error

Abstract

In this work, we study the problem of testing subsequence-freeness. For a given subsequence (word) w = w 1 … w k , a sequence (text) T = t 1 … t n is said to contain w if there exist indices 1 ≤ i 1 < … < i k ≤ n such that t ij = w j for every 1 ≤ j ≤ k . Otherwise, T is w -free. While a large majority of the research in property testing deals with algorithms that perform queries, here we consider sample-based testing (with one-sided error). In the “standard” sample-based model (i.e., under the uniform distribution), the algorithm is given samples ( i , t i ) where i is distributed uniformly independently at random. The algorithm should distinguish between the case that T is w -free, and the case that T is ε-far from being w -free (i.e., more than an ε-fraction of its symbols should be modified so as to make it w -free). Freitag, Price, and Swartworth (Proceedings of RANDOM, 2017) showed that O (( k 2 log k )ε) samples suffice for this testing task. We obtain the following results. – The number of samples sufficient for one-sided error sample-based testing (under the uniform distribution) is O ( k ε). This upper bound builds on a characterization that we present for the distance of a text T from w -freeness in terms of the maximum number of copies of w in T , where these copies should obey certain restrictions. – We prove a matching lower bound, which holds for every word w . This implies that the above upper bound is tight. – The same upper bound holds in the more general distribution-free sample-based model. In this model, the algorithm receives samples ( i , t i ) where i is distributed according to an arbitrary distribution p (and the distance from w -freeness is measured with respect to p ). We highlight the fact that while we require that the testing algorithm work for every distribution and when only provided with samples, the complexity we get matches a known lower bound for a special case of the seemingly easier problem of testing subsequence-freeness with one-sided error under the uniform distribution and with queries (Canonne et al., Theory of Computing , 2019).