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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Veröffentlicht in:	Algorithmica Jg. 80; H. 3; S. 849 - 884
Hauptverfasser:	Claverol, Mercè, Khramtcova, Elena, Papadopoulou, Evanthia, Saumell, Maria, Seara, Carlos
Format:	Journal Article Verlag
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.03.2018 Springer Nature B.V
Schlagworte:	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
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Zusammenfassung:	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.
Bibliographie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0178-4617 1432-0541
DOI:	10.1007/s00453-017-0299-z