On Kinetic Delaunay Triangulations

Rubin, Natan

doi:10.1145/2746228

Cited by 1 publication

(1 citation statement)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is one of the long outstanding open problems if this bound can be improved [8,9]. Only recently, Rubin [24] showed that if all sites move linearly and with the same speed, the number of changes is at most O(n 2+ε ) for some arbitrarily small ε > 0. For arbitrary speeds, the best known bound is still O(n 3 β 4 (n)).…”

Section: Related Workmentioning

confidence: 99%

Kinetic Geodesic Voronoi Diagrams in a Simple Polygon

Korman,

van Renssen,

Roeloffzen

et al. 2023

SIAM J. Discrete Math.

View full text Add to dashboard Cite

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 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(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.

show abstract

Section: Related Workmentioning

confidence: 99%