Visibility Testing and Counting

For a set of n disjoint line segments S in R, the visibility testing problem (VTP) is to test whether the query point p sees a query segment s ∈ S. For this configuration, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from any query point p can be computed quickly. In this paper, we solve VTP in expected logarithmic query time using quadratic preprocessing time and space. Moreover, we propose a (1+δ)-approximation algorithm for VCP using at most quadratic preprocessing time and space. The query time of this method is O ( 1 δ2 √ n) where O (f(n)) = O(f(n)n ) and > 0 is an arbitrary constant number.