Deleting and testing forbidden patterns in multi-dimensional arrays

Omri Ben-Eliezer, Simon Korman, Daniel Reichman

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

    Abstract

    Analyzing multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms to understand local properties of the data. We focus on the natural problem of testing pattern freeness: given a large d-dimensional array A and a fixed d-dimensional pattern P over a finite alphabet Γ, we say that A is P-free if it does not contain a copy of the forbidden pattern P as a consecutive subarray. The distance of A to P-freeness is the fraction of the entries of A that need to be modified to make it P-free. For any ϵ > 0 and any large enough pattern P over any alphabet - other than a very small set of exceptional patterns - we design a tolerant tester that distinguishes between the case that the distance is at least ϵ and the case that the distance is at most adϵ, with query complexity and running time cdϵ-1, where ad < 1 and cd depend only on the dimension d. These testers only need to access uniformly random blocks of samples from the input A. To analyze the testers we establish several combinatorial results, including the following ddimensional modification lemma, which might be of independent interest: For any large enough d-dimensional pattern P over any alphabet (excluding a small set of exceptional patterns for the binary case), and any d-dimensional array A containing a copy of P, one can delete this copy by modifying one of its locations without creating new P-copies in A. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data.

    Original languageEnglish
    Title of host publication44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
    EditorsAnca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    ISBN (Electronic)9783959770415
    DOIs
    StatePublished - 1 Jul 2017
    Event44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland
    Duration: 10 Jul 201714 Jul 2017

    Publication series

    NameLeibniz International Proceedings in Informatics, LIPIcs
    Volume80
    ISSN (Print)1868-8969

    Conference

    Conference44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
    Country/TerritoryPoland
    CityWarsaw
    Period10/07/1714/07/17

    Keywords

    • Pattern matching
    • Property testing
    • Sublinear algorithms

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