eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-29
75:1
75:17
10.4230/LIPIcs.ICALP.2020.75
article
Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
Korman, Matias
1
van Renssen, André
2
Roeloffzen, Marcel
3
Staals, Frank
4
Department of Computer Science, Tufts University, Medford, MA, USA
School of Computer Science, University of Sydney, Australia
Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
Department of Information and Computing Sciences, Utrecht University, The Netherlands
We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m³), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log² m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.75/LIPIcs.ICALP.2020.75.pdf
kinetic data structure
simple polygon
geodesic voronoi diagram