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

  • Sharareh AlipourEmail author
  • Mohammad Ghodsi
  • Amir Jafari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

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 \(m_p\le m'_p\le (1+\delta )m_p\).

Keywords

Computational geometry Visibility Randomized algorithm Approximation algorithm Graph theory 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Sharareh Alipour
    • 1
    Email author
  • Mohammad Ghodsi
    • 1
    • 2
  • Amir Jafari
    • 1
  1. 1.Sharif University of TechnologyTehranIran
  2. 2.Institute for Research in Fundamental Sciences (IPM)TehranIran

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