Geodesic farthest-point Voronoi diagram in linear time

Abstract

Let $P$ be a simple polygon with $n$ vertices. For any two points in $P$, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in $P$. Given a set $S$ of $m$ sites being a subset of the vertices of $P$, we present a randomized algorithm to compute the geodesic farthest-point Voronoi diagram of $S$ in $P$ running in expected $O(n + m)$ time. That is, a partition of $P$ into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic distance. In particular, this algorithm can be extended to run in expected $O(n + m\log m)$ time when $S$ is an arbitrary set of $m$ sites contained in $P$, thereby solving the open problem posed by Mitchell in Chapter 27 of the Handbook of Computational Geometry.