Abstract
For a set of n disjoint line segments S in R 2, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from a query point p can be computed quickly. For this configuration, the visibility testing problem (VTP) is to test whether p sees a fixed segment s. These problems can be solved in logarithmic query time by using O(n 4) preprocessing time and space. In this paper, we approximately solve this problem using quadratic preprocessing time and space. Our methods are superior to current approximation algorithms in terms of both approximation factor and preprocessing cost. In this paper, we propose a 2-approximation algorithm for the VCP using at most quadratic preprocessing time and space. The query time of this method is \(O_{\varepsilon}(n^{2}/\sqrt{k})\) where k is the preprocessing time and O ε (f(n) ) = O(f(n)n ε). We also solve the VTP in expected logarithmic query time using quadratic time and space.
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References
Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Compu- tational Geometry, Contemporary Mathematics, vol. 223, pp. 1–56. American Mathematical Society Press, Providence (1999)
Aronov, B., Guibas, L.J., Teichmann, M., Zhang, L.: Visibility queries and maintenance in simple polygons. Discrete and Computational Geometry 27, 461–483 (2002)
Asano, T.: An efficient algorithm for finding the visibility polygon for a polygonal region with holes. IEICE Transactions, 557–589 (1985)
Bose, P., Lubiw, A., Munro, J.I.: Eficient visibility queries in simple polygons. Computational Geometry Theory and Applications 23(7), 313–335 (2002)
Fischer, M., Hilbig, M., Jahn, C., auf der Heide, F.M., Ziegler, M.: Planar visibility counting. CoRR, abs/0810.0052 (2008)
Fischer, M., Hilbig, M., Jahn, C., auf der Heide, F.M, and Ziegler, M.: Planar visibility counting. In: Proceedings of the 25th European Workshop on Computational Geometry(EuroCG 2009), pp. 203–206 (2009)
Ghosh, S.K., Mount, D.: An output sensitive algorithm for computing visibility graphs. SIAM Journal on Computing 20, 888–910 (1991)
Ghosh, S.K.: Visibility algorithms in the plane. Cambridge University Press, Cambridge (2007)
Gudmundsson, J., Morin, P.: Planar visibility: testing and counting. In: Annual Symposium on Computational Geometry, pp. 77–86 (2010)
Matoutsek, J.: Effcient partition trees. Discrete and Computational Geometry 8, 315–334 (1992)
Pocchiola, M., Vegter, G.: The visibility complex. International Journal of Computational Geometry and Applications 6(3), 279–308 (1996)
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 1984), pp. 14–23 (1984)
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)
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Alipour, S., Zarei, A. (2011). Visibility Testing and Counting. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_36
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DOI: https://doi.org/10.1007/978-3-642-21204-8_36
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