On Voronoi visibility maps of 1.5D terrains with multiple viewpoints

TL;DR

This paper introduces an efficient algorithm for computing Voronoi visibility maps on 1.5D terrains with multiple viewpoints, improving previous methods and extending to generalized cases with practical applications.

Contribution

The paper presents a faster algorithm for Voronoi visibility maps on 1.5D terrains, generalizes it to higher order maps and other distances, and establishes bounds relating visibility map complexities.

Findings

Algorithm runs in O(n+(m^2+k_c)log n) time, improving previous solutions.

Generalization to higher order Voronoi visibility maps and other distance metrics.

Application demonstrated in limited sight range problems.

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.