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Stabbing Circles for Sets of Segments in the Plane

  • Mercè Claverol
  • Elena Khramtcova
  • Evanthia PapadopoulouEmail author
  • Maria Saumell
  • Carlos Seara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

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 cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute 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 algorithm. We also observe that the stabbing circle problem for S can be solved in optimal \(O(n^2)\) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D.

Notes

Acknowledgments

M. C. and C. S. were supported by projects MTM2012-30951 and Gen.Cat. DGR2014SGR46. E. K. and E. P. were supported by SNF project 20GG21-134355, under the ESF EUROCORES, EuroGIGA/VORONOI program. 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-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Mercè Claverol
    • 1
  • Elena Khramtcova
    • 2
  • Evanthia Papadopoulou
    • 2
    Email author
  • Maria Saumell
    • 3
  • Carlos Seara
    • 1
  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 BohemiaPilsenCzech Republic

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