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)


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


Computational geometry Visibility Randomized algorithm Approximation algorithm Graph theory 


  1. 1.
    Alipour, S., Zarei, A.: Visibility testing and counting. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 343–351. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Aronov, B., Guibas, L.J., Teichmann, M., Zhang, L.: Visibility queries and maintenance in simple polygons. Discret. Comput. Geom. 27, 461–483 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asano, T.: An efficient algorithm for finding the visibility polygon for a polygonal region with holes. IEICE Trans. 68(9), 557–589 (1985)Google Scholar
  4. 4.
    Bose, P., Lubiw, A., Munro, J.I.: Efficient visibility queries in simple polygons. Comput. Geom. Theory Appl. 23(7), 313–335 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fischer, M., Hilbig, M., Jahn, C., Meyer auf der Heide F., Ziegler M.: Planar visibility counting. CoRR, abs/0810.0052 (2008)Google Scholar
  6. 6.
    Fischer, M., Hilbig, M., Jahn, C., Meyer auf der Heide F., Ziegler M.: Planar visibility counting. In: Proceedings of the 25th European Workshop on Computational Geometry (EuroCG 2009), pp. 203–206 (2009)Google Scholar
  7. 7.
    Ghosh, S.K., Mount, D.: An output sensitive algorithm for computing visibility graphs. SIAM J. Comput. 20, 888–910 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gudmundsson, J., Morin, P.: Planar visibility: testing and counting. In: Annual Symposium on Computational Geometry, pp. 77–86 (2010)Google Scholar
  10. 10.
    Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nouri, M., Ghodsi, M.: Space/query-time tradeoff for computing the visibility polygon. Comput. Geom. 46(3), 371–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pocchiola, M., Vegter, G.: The visibility complex. Int. J. Comput. Geom. Appl. 6(3), 279–308 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Suri, S., O’Rourke, J.: Worst-case optimal algorithms for constructing visibility polygons with holes. In: Proceedings of the Second Annual Symposium on Computational Geometry (SCG 86), pp. 14–23 (1986)Google Scholar
  14. 14.
    Vegter, G.: The visibility diagram: a data structure for visibility problems and motion planning. In: Gilbert, J.R., Karlsson, R. (eds.) SWAT 90. LNCS, vol. 447, pp. 97–110. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  15. 15.
    Zarei, A., Ghodsi, M.: Efficient computation of query point visibility in polygons with holes. In: Proceedings of the 21st Annual ACM Symposium on Computational Geometry (SCG 2005) (2005)Google Scholar

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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