Testing local properties of arrays

Omri Ben-Eliezer*

*Corresponding author for this work

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


We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f : [n]d → Σ is k-local if it can be defined by a family of k x . . . x k forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are 2-local; convexity is (usually) 3-local; and many typical problems in computational biology and computer vision involve o(n)local properties. In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any k-local property of ddimensional arrays is testable by a simple canonical one-sided error non-adaptive ϵ-test, whose query complexity is O(ϵ−1k log ϵn/k ) for d = 1 and O(cdϵ−1/dk · nd−1) for d > 1. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet Σ. The query complexity is optimal for a wide range of parameters: For d = 1, this matches the query complexity of many previously investigated local properties, while for d > 1 we design and analyze new constructions of k-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity.

Original languageEnglish
Title of host publication10th Innovations in Theoretical Computer Science, ITCS 2019
EditorsAvrim Blum
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770958
StatePublished - 1 Jan 2019
Event10th Innovations in Theoretical Computer Science, ITCS 2019 - San Diego, United States
Duration: 10 Jan 201912 Jan 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference10th Innovations in Theoretical Computer Science, ITCS 2019
Country/TerritoryUnited States
CitySan Diego


  • Hypergrid
  • Local properties
  • Monotonicity testing
  • Pattern matching
  • Property testing


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