Elsevier

Journal of Algorithms

Volume 42, Issue 1, January 2002, Pages 109-134
Journal of Algorithms

Regular Article
The 2-Center Problem with Obstacles

https://doi.org/10.1006/jagm.2001.1194Get rights and content

Abstract

Given a set S of n points in the plane and a set O of pairwise disjoint simple polygons with a total of m edges, we wish to find two congruent disks of smallest radius whose union covers S and whose centers lie outside the polygons in O (referred to as locational constraints in facility location theory). We present an algorithm to solve this problem in randomized expected time O(m log2(mn) + mn log2n log(mn)). We also present an efficient approximation scheme that constructs, for a given ε > 0, two disks as above of radius at most (1 + ε)r*, where r* is the optimal radius, in time O(1/ε log(1/ε)(m log2m + n log2n)) or in randomized expected time O(1/ε log(1/ε)((m + n log n) log(mn))).

References (17)

  • P.K. Agarwal et al.

    Computing a segment center for a planar point set

    J. Algorithms

    (1993)
  • T.Y. Chan

    More planar two-center algorithms

    Comput. Geom. Theory Appl.

    (1999)
  • M.H. Overmars et al.

    Maintenance of configurations in the plane

    J. Comput. Syst. Sci.

    (1981)
  • S.-W. Cheng, O. Cheong, H. Everett and R. van Oostrum, Hierarchical vertical decomposition, ray shooting, and circular...
  • R. Cole

    Slowing down sorting networks to obtain faster sorting algorithms

    J. ACM

    (1987)
  • R. Cole et al.

    An optimal-time algorithm for slope selection

    SIAM J. Comput.

    (1989)
  • J. Craig

    Geometric algorithms in Adept RAPID

  • D. Eppstein, Dynamic three-dimensional linear programming, inProc. 32th Annu. IEEE Sympos. Found. Comput. Sci., 1991,...
There are more references available in the full text version of this article.

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Work on this paper by Ken Goldberg and Dan Halperin has been supported by a grant from U.S.–Israeli Binational Science Foundation. Work by Dan Halperin and Micha Sharir has been supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), by a Franco–Israeli research grant “factory of the future” (monitored by AFIRST/France and The Israeli Ministry of Science), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. A preliminay version of this paper appeared in Proceedings of the 16th ACM Symposium on Computational Geometry, Hong Kong, 2000, pp. 80–90.

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