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

TL;DR

This paper presents a nearly optimal algorithm for constructing the geodesic Voronoi diagram in a simple polygon, reducing the problem to efficiently computing a single Voronoi vertex, thus approaching theoretical lower bounds.

Contribution

The authors develop an algorithm with near-optimal construction time, linking the overall complexity to the efficient computation of a single Voronoi vertex.

Findings

Achieves O( n+m( log m+ log^2 n) ) construction time.

Reduces the problem to computing a single Voronoi vertex in O( log n ) time.

Approaches the theoretical lower bound for the problem.

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 Omega( 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=Omega(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.