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European Symposium on Algorithms

ESA 2002: Algorithms — ESA 2002 pp 42–53Cite as

Translating a Planar Object to Maximize Point Containment

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2461))

Abstract

Let C be a compact set in ℝ2 and let S be a set of n points in ℝ2. We consider the problem of computing a translate of C that contains the maximum number, κ*, of points of S. It is known that this problem can be solved in a time that is roughly quadratic in n. We show how random-sampling and bucketing techniques can be used to develop a near-linear-time Monte Carlo algorithm that computes a placement of C containing at least (1 - ɛ)κ*. points of S, for given ɛ > 0, with high probability. We also present a deterministic algorithm that solves the ?-approximate version of the optimal-placement problem and runs in O((n 1+δ+ n/ɛ) logm) time, for arbitrary constant δ > 0, if C is a convex m-gon.

Agarwal was supported by NSF grants ITR-333-1050, EIA-9870724, EIA-9972879, CCR-00-86013, and CCR-9732787, and by a grant from the U.S.-Israeli Binational Science Foundation. Sharir was supported by NSF Grants CCR-97-32101 and CCR- 00-98246, by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University, and by a grant from the U.S.-Israeli Binational Science Foundation. Smid was supported by NSERC.

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Author information

Authors and Affiliations

  1. Center for Geometric Computing and Department of Computer Science, Duke University, Durham, NC 27708-0129, USA

    Pankaj K. Agarwal

  2. Institut für Informatik, Universität Frankfurt, D-60054 Frankfurt am Main, Germany

    Torben Hagerup

  3. Max-Planck-Institute for Computer Science, 66123, Saarbrücken, Germany

    Rahul Ray

  4. School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Micha Sharir

  5. School of Computer Science, Carleton University, K1S 5B6, Ottawa, Canada

    Michiel Smid

  6. Institut für Theoretische Informatik, ETH Zürich, CH-8092, Zürich, Switzerland

    Emo Welzl

Authors
  1. Pankaj K. Agarwal
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  2. Torben Hagerup
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  3. Rahul Ray
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  4. Micha Sharir
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  5. Michiel Smid
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  6. Emo Welzl
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Editor information

Editors and Affiliations

  1. Fakultät II: Mathematik und Naturwissenschaften, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Rolf Möhring

  2. Department of Mathematics and Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Rajeev Raman

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© 2002 Springer-Verlag Berlin Heidelberg

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Agarwal, P.K., Hagerup, T., Ray, R., Sharir, M., Smid, M., Welzl, E. (2002). Translating a Planar Object to Maximize Point Containment. In: Möhring, R., Raman, R. (eds) Algorithms — ESA 2002. ESA 2002. Lecture Notes in Computer Science, vol 2461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45749-6_8

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  • DOI: https://doi.org/10.1007/3-540-45749-6_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44180-9

  • Online ISBN: 978-3-540-45749-7

  • eBook Packages: Springer Book Archive

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