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

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 $\varOmega (n+m\log m)$, 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 $O(n+m\log m\log ^2n)$ time, which are optimal for $m=\varOmega (n)$ and $m=O(\frac{n}{\log ^3n})$, 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.