Visibility testing and counting

For a set of n disjoint line segments S in R2, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from a query point p can be computed quickly. For this configuration, the visibility testing problem (VTP) is to test whether p sees a fixed segment s. These problems can be solved in logarithmic query time by using O(n4) preprocessing time and space. In this paper, we approximately solve this problem using quadratic preprocessing time and space. Our methods are superior to current approximation algorithms in terms of both approximation factor and preprocessing cost. In this paper, we propose a 2-approximation algorithm for the VCP using at most quadratic preprocessing time and space. The query time of this method is Oe(n2/√k) where k is the preprocessing time and Oe(f(n)) = O(f(n)ne). We also solve the VTP in expected logarithmic query time using quadratic time and space.