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Covering and Packing with Spheres by Diagonal Distortion in ℝn

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Rainbow of Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6570))

Abstract

We address the problem of covering ℝn with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in ℝn, for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3.

This research is partially supported by DARPA under grant HR0011-09-0065 and NSF under grant DBI-0820624.

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Author information

Authors and Affiliations

  1. IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

    Herbert Edelsbrunner & Michael Kerber

  2. Departments of Computer Science and of Mathematics, Duke University, Durham, North Carolina, USA

    Herbert Edelsbrunner

  3. Geomagic, Research Triangle Park, North Carolina, USA

    Herbert Edelsbrunner

Authors
  1. Herbert Edelsbrunner
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  2. Michael Kerber
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Science Centre, The University of Auckland, 38 Princes Street, 1142, Auckland, New Zealand

    Cristian S. Calude

  2. Leiden Institute of Center Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

  3. Turku Centre for Computer Science (TUCS), University of Turku, Leminkäisenkatu 14, 20014, Turku, Finland

    Arto Salomaa

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Edelsbrunner, H., Kerber, M. (2011). Covering and Packing with Spheres by Diagonal Distortion in ℝn . In: Calude, C.S., Rozenberg, G., Salomaa, A. (eds) Rainbow of Computer Science. Lecture Notes in Computer Science, vol 6570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19391-0_2

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  • DOI: https://doi.org/10.1007/978-3-642-19391-0_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-19390-3

  • Online ISBN: 978-3-642-19391-0

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