Computing the Visibility Polygon of an Island in a Polygonal Domain

Abstract

Given a set $${\mathcal {P}}$$P of h pairwise-disjoint polygonal obstacles with a total of n vertices in the plane, we study the problem of computing the (weak) visibility polygon from a polygonal obstacle $$P^*$$Pź (an island) in $${\mathcal {P}}$$P. The problem was previously solved in $$O(n^4)$$O(n4) time, which has been proved worst-case optimal. However, since h may be much smaller than n, it is desirable to have an algorithm whose running time is also a function of h. In this paper, we present such an algorithm of $$O(n^2h^2)$$O(n2h2) time, and our algorithm improves the previous result when $$h=o(n)$$h=o(n). In addition, when all obstacles in $${\mathcal {P}}$$P (including $$P^*$$Pź) are convex, our algorithm runs in $$O(n+h^4)$$O(n+h4) time.