Efficient computation of query point visibility in polygons with holes

Abstract

In this paper, we consider the problem of computing the visibility 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 [13] and [2] 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.

Space–Query-Time Tradeoff for Computing the Visibility Polygon

“…2). It is easy to arrange a scene such that the number of combinatorially different visibility polygons is Θ(n 4 ) (e.g., see [17]). Therefore it seems likely that the above result is the best we can reach for optimal query time.…”

“…Pocchiola and Vegter [13] then showed that if the boundary polygon of the scene and its holes are convex polygons, using visibility complex, V P (q) can be reported in time O(|V P (q)| log n) after preprocessing the scene in time O(n log n + E) time and O(E) space, where E is the number of edges in the visibility graph of the scene. Very recently Zarei and Ghodsi [17] proposed an algorithm that reports V P (q) in O((1 + min(h, |V P (q)|)) log n + |V P (q)|) query time. The preprocessing takes O(n 3 log n) time and O(n 3 ) space.…”

“…For example when h = Θ(n) and |V P (q)| = Θ(n) the algorithms of Pocchiola and Vegter [13] and Zarei and Ghodsi [17] degrade to the algorithms without any preprocessing time. For Vegter's algorithm, we can also generate instances of the problem that use more time than the required time to report V P (q).…”

Generalized Watchman Route Problem With Discrete View Cost

“…This implies that the properties of the cells defined in [21] are preserved here. Similar terminologies (not by exactly the same names) and results can also be found in [2,11].…”

“…Our decomposition is a "finer" version than that given in [21], i.e., each cell defined here is completely contained in a single cell defined in [21]. This implies that the properties of the cells defined in [21] are preserved here.…”