Testing Local Properties of Arrays

Author Omri Ben-Eliezer

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Omri Ben-Eliezer
  • Tel Aviv University, Tel Aviv 69978, Israel

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Omri Ben-Eliezer. Testing Local Properties of Arrays. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f:[n]^d -> Sigma 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 d-dimensional arrays is testable by a simple canonical one-sided error non-adaptive epsilon-test, whose query complexity is O(epsilon^{-1}k log{(epsilon n)/k}) for d = 1 and O(c_d epsilon^{-1/d} k * n^{d-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 Sigma. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Property Testing
  • Local Properties
  • Monotonicity Testing
  • Hypergrid
  • Pattern Matching


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