Query point visibility computation in polygons with holes

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 considering the cost of the preprocessing phase. Our algorithm is based on solutions in [A.L.P. Bose, J.I. Munro, Efficient visibility queries in simple polygons, Computational Geometry: Theory and Applications 23 (3) (2002) 313–335] and [M.T.B. Aronov, L. Guibas, L. Zhang, Visibility queries and maintenance in simple polygons, Discrete and Computational Geometry 27 (4) (2002) 461–483] proposed for simple polygons. In our solution, the preprocessing is done in time O(n3 log n) to construct a data structure of size O(n3). 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, andh ′ 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