Planar Visibility: Testing and Counting

In this paper we consider query versions of visibility testing and visibility counting. Let $S$ be a set of $n$ disjoint line segments in $\R^2$ and let $s$ be an element of $S$. Visibility testing is to preprocess $S$ so that we can quickly determine if $s$ is visible from a query point $q$. Visibility counting involves preprocessing $S$ so that one can quickly estimate the number of segments in $S$ visible from a query point $q$. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, $V_S(s)$, of $\R^2$ that is weakly visible from a segment $s$ can be represented as the union of a set, $C_S(s)$, of $O(n^2)$ triangles, even though the complexity of $V_S(s)$ can be $\Omega(n^4)$. We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of $S$ visible from a point $p$ to the number of triangles in $\bigcup_{s\in S} C_S(s)$ that contain $p$.Comment: 22 page

Efficient computation of query point visibility in polygons with holes

In this paper, we consider the problem of computing the visibility polygon of a query point inside polygons with holes. The goal is to perform this computation efficiently per query with more cost in the preprocessing phase. Our algorithm is based on solutions in [12] and [13] proposed for simple polygons. In our solution, the preprocessing is done in time O(n 3 log(n)) to construct a data structure of size O(n 3). It is then possible to report the visibility polygon of any query point q in time O((1 + h ′)log n + |V (q)|), in which n and h are the number of the vertices and holes of the polygon respectively, |V (q) | is the size of the visibility polygon of q, and h ′ is an output and preprocessing sensitive parameter of at most min(h, |V (q)|). This is claimed to be the best query-time result on this problem so far