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

Abstract

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 tradeoff between the space and the query time. They answer any query in O ǫ (n 1−α ) with O ǫ (n 2+2α ) of preprocessing time and space, where α is a constant 0 ≤ α ≤ 1, ǫ > 0 is another constant that can be made arbitrarily small, and O ǫ (f (n)) = O(f (n)n ǫ ).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 ≤ β ≤ 2 3 and 0 < δ < 1, the expected preprocessing time, the expected space, and the query time of our algorithm are O(n 4−3β log n), O(n 4−3β ), and O( 1 δ 3 n β log n), respectively. The algorithm computes the number of visible segments from p, or m p , exactly if m p ≤ 1 δ 3 n β log n. Otherwise, it computes a (1 + δ)-approximation m ′ p with the probability of at least 1 − 1 log n , where m p ≤ m ′ p ≤ (1 + δ)m p .