On Voronoi visibility maps of 1.5D terrains with multiple viewpoints

Abstract: Given an $n$-vertex 1.5D terrain $\T$ and a set $\A$ of $m<n$ viewpoints, the Voronoi visibility map $\vorvis(\T,\A)$ is a partitioning of $\T$ into regions such that each region is assigned to the closest (in Euclidean distance) visible viewpoint. The colored visibility map $\colvis(\T,\A)$ is a partitioning of $\T$ into regions that have the same set of visible viewpoints. In this paper, we propose an algorithm to compute $\vorvis(\T,\A)$ that runs in $O(n+(m^2+k_c)\log n)$ time, where $k_c$ and $k_v$ denote the total complexity of $\colvis(\T,\A)$ and $\vorvis(\T,\A)$, respectively. This improves upon a previous algorithm for this problem. We also generalize our algorithm to higher order Voronoi visibility maps, and to Voronoi visibility maps with respect to other distances. Finally, we prove bounds relating $k_v$ to $k_c$, and we show an application of our algorithm to a problem on limited range of sight.