Optimal Distribution-Free Sample-Based Testing Of subsequence-freeness

Dana Ron*, Asaf Rosin

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In this work, we study the problem of testing subsequence-freeness. For a given subsequence (word) w = w1... wk, a sequence (text) T = t1... tn is said to contain w if there exist indices 1 ≤ i1 < · · · < ik ≤ n such that tij = wj 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, ti) 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(k2 log k/ε) samples suffice for this testing task. We obtain the following results. • The number of samples sufficient for 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, ti) 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 under the uniform distribution and with queries (Canonne et al., Theory of Computing, 2019).

Original languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Pages337-356
Number of pages20
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: 10 Jan 202113 Jan 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual
Period10/01/2113/01/21

Funding

FundersFunder number
Israel Science Foundation1146/18

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