A nearly optimal algorithm for the geodesic voronoi diagram of points in a simple polygon
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Journal Volume
99
Pages
581-5814
Date Issued
2018
Author(s)
Abstract
The geodesic Voronoi diagram ofm 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 + mlogm), 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 + mlogmlog2 n) time, which are optimal for m = Ω(n) and m = O(n/log3 n), respectively. In this paper, we give a construction algorithm with O(n + m(logm + log2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(logn) 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(logn) time. © Chih-Hung Liu; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).
Subjects
Geodesic distance; Simple polygons; Voronoi diagrams
Other Subjects
Geodesy; Graphic methods; Construction algorithms; Construction time; Geodesic distances; Geodesic voronoi diagram; Optimal algorithm; Simple polygon; State of the art; Voronoi diagrams; Computational geometry
Type
conference paper