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 \(m_p\le m'_p\le (1+\delta )m_p\).
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Alipour, S., Ghodsi, M., Jafari, A. (2016). An Improved Constant-Factor Approximation Algorithm for Planar Visibility Counting Problem. In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_17
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DOI: https://doi.org/10.1007/978-3-319-42634-1_17
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