A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

• Published in: Algorithmica Vol. 82; no. 4; pp. 915 - 937
• Main Author: Liu, Chih-Hung
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Lower bounds; Mathematics of Computing; Polygons; Theory of Computation; Upper bounds; Voronoi graphs; Computational geometry; Voronoi diagrams; Simple polygons; Geodesic distance
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-019-00624-2

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Ω ( n + m log m ) , and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O ( ( n + m ) log ( n + m ) ) and O ( n + m log m log 2 n ) time, which are optimal for m = Ω ( n ) and m = O ( n log 3 n ) , respectively. In this paper, we give a construction algorithm with O ( n + m ( log m + log 2 n ) ) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O ( log n ) time, then the construction time will become the optimal O ( n + m log m ) . In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O ( log n ) time.