Algorithmica

, Volume 80, Issue 3, pp 849–884 | Cite as

Stabbing Circles for Sets of Segments in the Plane

Article
Part of the following topical collections:
  1. Special Issue on Theoretical Informatics

Abstract

Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to two cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute a representation of all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a \(O(n \log ^2{n})\) time and O(n) space algorithm. We also observe that the stabbing circle problem for S can be solved in worst-case optimal \(O(n^2)\) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. Finally we show that the problem of computing the stabbing circle of minimum radius for a set of n parallel segments of equal length has an \(\varOmega (n \log n)\) lower bound.

Keywords

Stabbing circle Stabbing line segments Voronoi diagram Cluster Voronoi diagrams Hausdorff Voronoi diagram Farthest-color Voronoi diagram 

Notes

Acknowledgements

M. C. and C. S. were supported by Projects MTM2015-63791-R (MINECO/FEDER) and Gen.Cat. DGR2014SGR46. E. K. and E. P. were supported by Projects SNF 20GG21-134355, under the ESF EUROCORES Program EuroGIGA/VORONOI, and SNF 200021E-154387. M. S. was supported by Project LO1506 of the Czech Ministry of Education, Youth and Sports, and by Project NEXLIZ CZ.1.07/2.3.00/30.0038, co-financed by the European Social Fund and the state budget of the Czech Republic.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Faculty of InformaticsUniversità della Svizzera italiana (USI)LuganoSwitzerland
  3. 3.Department of Mathematics and European Centre of Excellence NTISUniversity of West BohemiaPlzeňCzech Republic

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