Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
Abstract.
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 \(O(m^3)\), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient KDSs for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector and Voronoi center KDSs handle each event in \(O(\log^2 m)\) 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.
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Acknowledgments.
We would like to thank Man-Kwun Chiu and Yago Diez for interesting discussions during the initial stage of this research. We would also like to thank the anonymous reviewers, who helped us significantly improve the presentation of this paper and make it more accessible to a wider audience.
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Information & Authors
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Published In
SIAM Journal on Discrete Mathematics
Pages: 2276 - 2311
DOI: 10.1137/20M1384804
ISSN (online): 1095-7146
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 8 December 2020
Accepted: 5 July 2023
Published online: 12 October 2023
Keywords
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Funding Information
National Science Foundation (NSF): CCF-1422311
Funding: The first author was partially supported by MEXT KAKENHI 17K12635 and the NSF award CCF-1422311. The second and third authors were supported by JST ERATO Grant JPMJER1201, Japan. The fourth author was supported by The Netherlands Organisation for Scientific Research under project 612.001.651.
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