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Stabbing Circles for Sets of Segments in the Plane

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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.

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Notes

  1. Any vertex of \({\textsf {HVD}}(S)\) has degree three by the general position assumption that no four endpoints of segments in S are cocircular.

  2. Note that it is not always possible to shrink uv infinitesimally so that \(type(u)\) and \(type(v)\) consist of one element each. In particular, this is not possible when uv lies on an edge of \({\textsf {FCVD}}(S)\), and this situation does not contradict our general position assumption.

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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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Authors and Affiliations

  1. Universitat Politècnica de Catalunya, Barcelona, Spain

    Mercè Claverol & Carlos Seara

  2. Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland

    Elena Khramtcova & Evanthia Papadopoulou

  3. Department of Mathematics and European Centre of Excellence NTIS, University of West Bohemia, Plzeň, Czech Republic

    Maria Saumell

Authors
  1. Mercè Claverol
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  2. Elena Khramtcova
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  3. Evanthia Papadopoulou
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  4. Maria Saumell
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  5. Carlos Seara
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Corresponding authors

Correspondence to Evanthia Papadopoulou or Maria Saumell.

Additional information

A preliminary version of this paper appeared in Proc. 12th Latin American Theoretical Informatics Symposium (LATIN’16), pp. 290–305.

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Claverol, M., Khramtcova, E., Papadopoulou, E. et al. Stabbing Circles for Sets of Segments in the Plane. Algorithmica 80, 849–884 (2018). https://doi.org/10.1007/s00453-017-0299-z

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  • DOI: https://doi.org/10.1007/s00453-017-0299-z

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