Stabbing Circles for Sets of Segments in the Plane

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...

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Bibliographic Details
Published in:Algorithmica Vol. 80; no. 3; pp. 849 - 884
Main Authors: Claverol, Mercè, Khramtcova, Elena, Papadopoulou, Evanthia, Saumell, Maria, Seara, Carlos
Format: Journal Article Publication
Language:English
Published: New York Springer US 01.03.2018
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Cluster Voronoi diagrams
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Farthest-color Voronoi diagram
Geometria computacional
Hausdorff Voronoi diagram
Informàtica
Lower bounds
Matemàtiques i estadística
Mathematics of Computing
Segments
Special Issue on Theoretical Informatics
Stabbing circle
Stabbing line segments
Theory of Computation
Voronoi diagram
Voronoi graphs
Voronoi polygons
Àrees temàtiques de la UPC
Voronoi diagram
Hausdorff Voronoi diagram
Cluster Voronoi diagrams
Stabbing circle
Stabbing line segments
Farthest-color Voronoi diagram
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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 Ω ( n log n ) lower bound.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-017-0299-z