Abstract
Computing the visibility polygon, VP, of a point in a polygonal scene, is a classical problem that has been studied extensively. In this paper, we consider the problem of computing VP for any query point efficiently, with some additional preprocessing phase. The scene consists of a set of obstacles, of total complexity O(n). We show for a query point q, VP(q) can be computed in logarithmic time using O(n 4) space and O(n 4 logn) preprocessing time. Furthermore to decrease space usage and preprocessing time, we make a tradeoff between space usage and query time; so by spending O(m) space, we can achieve \(O(n^2 \log (\sqrt{m}/n) / \sqrt{m})\) query time, where n 2 ≤ m ≤ n 4. These results are also applied to angular sorting of a set of points around a query point.
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References
Agarwal, P.K.: Partitioning arrangements of lines, part I: An efficient deterministic algorithm. Discrete Comput. Geom. 5(5), 449–483 (1990)
Aronov, B., Guibas, L.J., Teichmann, M., Zhang, L.: Visibility queries and maintenance in simple polygons. Discrete Comput. Geom. 27(4), 461–483 (2002)
Asano, T.: Efficient algorithms for finding the visibility polygon for a polygonal region with holes. Manuscript, University of California at Berkeley
Asano, T., Asano, T., Guibas, L.J., Hershberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1(1), 49–63 (1986)
Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9, 145–158 (1993)
Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25(1), 76–90 (1985)
Clarkson, K.L.: New applications of random sampling in computational geometry. Discrete Comput. Geom. 2(2), 195–222 (1987)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational geometry: algorithms and applications. Springer, New York (1997)
Edelsbrunner, H., Seidel, R., Sharir, M.: On the zone theorem for hyperplane arrangements. SIAM J. Comput. 22(2), 418–429 (1993)
Heffernan, P.J., Mitchell, J.S.B.: An optimal algorithm for computing visibility in the plane. SIAM J. Comput. 24(1), 184–201 (1995)
Matoušek, J.: Construction of ε-nets. Disc. Comput. Geom. 5, 427–448 (1990)
Nouri, M., Zarei, A., Ghodsi, M.: Weak visibility of two objects in planar polygonal scenes. In: Gervasi, O., Gavrilova, M.L. (eds.) ICCSA 2007, Part I. LNCS, vol. 4705, pp. 68–81. Springer, Heidelberg (2007)
Pocchiola, M., Vegter, G.: The visibility complex. Int. J. Comput. Geometry Appl. 6(3), 279–308 (1996)
Sarnak, N., Tarjan, R.E.: Planar point location using persistent search trees. Commun. ACM 29(7), 669–679 (1986)
Suri, S., O’Rourke, J.: Worst-case optimal algorithms for constructing visibility polygons with holes. In: Symp. on Computational Geometry, pp. 14–23 (1986)
Vegter, G.: The visibility diagram: A data structure for visibility problems and motion planning. In: Gilbert, J.R., Karlsson, R. (eds.) SWAT 1990. LNCS, vol. 447, pp. 97–110. Springer, Heidelberg (1990)
Zarei, A., Ghodsi, M.: Efficient computation of query point visibility in polygons with holes. In: Proc. of the 21st symp. on Comp. geom, pp. 314–320 (2005)
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Nouri, M., Ghodsi, M. (2009). Space–Query-Time Tradeoff for Computing the Visibility Polygon. In: Deng, X., Hopcroft, J.E., Xue, J. (eds) Frontiers in Algorithmics. FAW 2009. Lecture Notes in Computer Science, vol 5598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02270-8_14
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DOI: https://doi.org/10.1007/978-3-642-02270-8_14
Publisher Name: Springer, Berlin, Heidelberg
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