A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
Journal
Algorithmica
Date Issued
2019
Author(s)
Abstract
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+ mlog 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+ mlog mlog 2n) time, which are optimal for m= Ω(n) and m=O(nlog3n), respectively. In this paper, we give a construction algorithm with O(n+ m(log m+ log 2n)) 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+ mlog 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. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Subjects
Computational geometry; 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
journal article