Bipartite diameter and other measures under translation

Boris Aronov, Omrit Filtser, Matthew J. Katz, Khadijeh Sheikhan

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Let A and B be two sets of points in Rd, where |A| = |B| = n and the distance between them is defined by some bipartite measure dist(A, B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A, B) = max{d(a, b) | a ∈ A, b ∈ B}, where d(a, b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A, B) = diam(A, B) − d(A, B), where d(A, B) = min{d(a, b) | a ∈ A, b ∈ B}, and (iii) the union width in two and three dimensions, that is union_width(A, B) = width(A ∪ B). For each of these measures we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R2 and R3, for uniformity we describe a roughly O(n9/4)-time algorithm, and for union width we offer a near-linear-time algorithm in R2 and a quadratic-time one in R3

    Original languageEnglish (US)
    Title of host publication36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019
    EditorsRolf Niedermeier, Christophe Paul
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    ISBN (Electronic)9783959771009
    DOIs
    StatePublished - Mar 1 2019
    Event36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019 - Berlin, Germany
    Duration: Mar 13 2019Mar 16 2019

    Publication series

    NameLeibniz International Proceedings in Informatics, LIPIcs
    Volume126
    ISSN (Print)1868-8969

    Conference

    Conference36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019
    CountryGermany
    CityBerlin
    Period3/13/193/16/19

    Keywords

    • Geometric optimization
    • Minimum-width annulus
    • Translation-invariant similarity measures

    ASJC Scopus subject areas

    • Software

    Cite this

    Aronov, B., Filtser, O., Katz, M. J., & Sheikhan, K. (2019). Bipartite diameter and other measures under translation. In R. Niedermeier, & C. Paul (Eds.), 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019 [8] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 126). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2019.8