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Dispersion in Disks

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Abstract

We present three new approximation algorithms with improved constant ratios for selecting n points in n disks such that the minimum pairwise distance among the points is maximized.

  1. (1)

    A very simple O(nlog n)-time algorithm with ratio 0.511 for disjoint unit disks.

  2. (2)

    An LP-based algorithm with ratio 0.707 for disjoint disks of arbitrary radii that uses a linear number of variables and constraints, and runs in polynomial time.

  3. (3)

    A hybrid algorithm with ratio either 0.4487 or 0.4674 for (not necessarily disjoint) unit disks that uses an algorithm of Cabello in combination with either the simple O(nlog n)-time algorithm or the LP-based algorithm.

The LP algorithm can be extended for disjoint balls of arbitrary radii in ℝd, for any (fixed) dimension d, while preserving the features of the planar algorithm. The algorithm introduces a novel technique which combines linear programming and projections for approximating Euclidean distances. The previous best approximation ratio for dispersion in disjoint disks, even when all disks have the same radius, was 1/2. Our results give a positive answer to an open question raised by Cabello, who asked whether the ratio 1/2 could be improved.

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI, 53201-0784, USA

    Adrian Dumitrescu

  2. Department of Computer Science, Utah State University, Logan, UT, 84322-4205, USA

    Minghui Jiang

Authors
  1. Adrian Dumitrescu
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  2. Minghui Jiang
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Correspondence to Adrian Dumitrescu.

Additional information

A preliminary version of this paper [5] appeared in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, Nancy, France, March 2010.

The work of A.D. was supported in part by NSF CAREER grant CCF-0444188. Part of the research by this author was done at Ecole Polytechnique Fédérale de Lausanne.

The work of M.J. was supported in part by NSF grant DBI-0743670.

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Dumitrescu, A., Jiang, M. Dispersion in Disks. Theory Comput Syst 51, 125–142 (2012). https://doi.org/10.1007/s00224-011-9331-x

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  • DOI: https://doi.org/10.1007/s00224-011-9331-x

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