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

Abstract

Given a set S of n disjoint line segments in $\mathbb {R}^{2}$, 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 $O_{\epsilon }(n^{1-\alpha })$ with $O_{\epsilon }(n^{2+2\alpha })$ of preprocessing time and space, where $\alpha$ is a constant $0\le \alpha \le 1, \epsilon > 0$ 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 $0\le \beta \le \frac{2}{3}$ 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 $O(\frac{1}{\delta ^3}n^{\beta }\log n)$, respectively. The algorithm computes the number of visible segments from p, or $m_p$, exactly if $m_p\le \frac{1}{\delta ^3}n^{\beta }\log n$. Otherwise, it computes a $(1+\delta )$-approximation $m'_p$ with the probability of at least $1-\frac{1}{\log n}$, where $m_p\le m'_p\le (1+\delta )m_p$.