On Voronoi visibility maps of 1.5D terrains with multiple viewpoints

Abstract

Given an n-vertex 1.5D terrain T and a set P of m<n viewpoints, the Voronoi visibility map VorVis(T,P) 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,P) 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,P) that runs in O(n+(m2+kc)log⁡n) time, where kc and kv denote the total complexity of ColVis(T,P) and VorVis(T,P), 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 kv to kc, and we show an application of our algorithm to a problem on limited range of sight.