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 $O_ε(n^{1-α})$ with $O_ε(n^{2+2α})$ of preprocessing time and space, where $α$ is a constant $0\leq α\leq 1$, $ε> 0$ is another constant that can be made arbitrarily small, and $O_ε(f(n))=O(f(n)n^ε)$. 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 $0\leq β\leq \frac{2}{3}$ 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 $O(\frac{1}{δ^3}n^β\log n)$, respectively. The algorithm computes the number of visible segments from $p$, or $m_p$, exactly if $m_p\leq \frac{1}{δ^3}n^β\log n$. Otherwise, it computes a $(1+δ)$-approximation $m'_p$ with the probability of at least $1-\frac{1}{\log n}$, where $m_p\leq m'_p\leq (1+δ)m_p$.