Weak Visibility Queries of Line Segments in Simple Polygons and Polygonal Domains

Abstract

In this paper we consider the problem of computing the weak visibility polygon of any query line segment $pq$ (or $WVP(pq)$) inside a given polygon $P$. Our first non-trivial algorithm runs in simple polygons and needs $O(n^3 \log n)$ time and $O(n^3)$ space in the preprocessing phase to report $WVP(pq)$ of any query line segment $pq$ in time $O(\log n + |WVP(pq)|)$. We also give an algorithm to compute the weak visibility polygon of a query line segment in a non-simple polygon with $h$ pairwise-disjoint polygonal obstacles with a total of $n$ vertices. Our algorithm needs $O(n^2 \log n)$ time and $O(n^2)$ space in the preprocessing phase and computes $WVP(pq)$ in query time of $O(n\hbar \log n + k)$, in which $\hbar$ is an output sensitive parameter of at most $\min(h,k)$, and $k = O(n^2h^2)$ is the output size. This is the best query-time result on this problem so far.