An Improved Constant-Factor Approximation Algorithm for Planar Visibility Counting Problem

Given a set S of n disjoint line segments in R-2, the visibility counting problem (VCP) is to preprocess S such that the number of segments in S visible from any query point p can be computed quickly. This problem can trivially be solved in logarithmic query time using O(n(4)) preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a trade-off between the space and the query time. They answer any query in O epsilon(n (1-alpha)) with O epsilon (n(2+2 alpha)) of preprocessing time and space, where alpha is a constant 0 <= alpha <= 1, epsilon > 0 is another constant that can be made arbitrarily small, and O epsilon(f(n)) = O(f( n)n(epsilon)). In this paper, we propose a randomized approximation algorithm for VCP with a tradeoff between the space and the query time. We will show that for an arbitrary constants 0 <= beta <= 2/3 and 0 < delta < 1, the expected preprocessing time, the expected space, and the query time of our algorithm are O( n(4-3 beta) log n), O(n(4-3 beta)), and O(1/delta(3)n(beta) log n), respectively. The algorithm computes the number of visible segments from p, or mp, exactly if m(p) <= (1)/(3)(delta)n(beta) log n.Otherwise, it computes a ( 1+ delta)-approximation m'(p) with the probability of at least 1- 1/log n, where m(p) <= m'(p) <= ( 1 + delta) m(p).