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.