Visibility Queries in Simple Polygons

TL;DR

This paper introduces new data structures for visibility queries in simple polygons, balancing space and query time, and proposes a novel polygon decomposition method.

Contribution

It presents several improved data structures for visibility queries with different space-time trade-offs and introduces a new polygon decomposition technique.

Findings

Achieves $O( log n + k)$ query time with $O(n^{2+psilon})$ space.

Improves query time to $O( log n log log n + k)$ with $O(n^2)$ space.

Provides a new polygon decomposition method of independent interest.

Abstract

Given a simple polygon $P$ with $n$ vertices, we consider the problem of constructing a data structure for visibility queries: for any query point $q \in P$, compute the visibility polygon of $q$ in $P$. To obtain $O(\log n + k)$ query time, where $k$ is the size of the visibility polygon of $q$, the previous best result requires $O(n^3)$ space. In this paper, we propose a new data structure that uses $O(n^{2+\epsilon})$ space, for any $\epsilon > 0$, while achieving the same query time. If only $O(n^2)$ space is available, the best known result provides $O(\log^2 n + k)$ query time. We improve this to $O(\log n \log \log n + k)$ time. When restricted to $o(n^2)$ space, the only previously known approach, aside from the $O(n)$-time algorithm that computes the visibility polygon without preprocessing, is an $O(n)$-space data structure that supports $O(k \log n)$-time queries. We construct a data structure using $O(n \log n)$ space that answers visibility queries in $O(n^{1/2+\epsilon} + k)$ time. In addition, for the special case in which $q$ lies on the boundary of $P$, we build a data structure of $O(n \log n)$ space supporting $O(\log^2 n + k)$ query time; alternatively, we achieve $O(\log n + k)$ query time using $O(n^{1+\epsilon})$ space. To achieve our results, we propose a new method for decomposing simple polygons, which may be of independent interest.