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Visibility of disjoint polygons.(English)Zbl 0611.68062

Consider a collection of disjoint polygons in the plane containing a total of n edges. We show how to build, in \(O(n^ 2)\) time and space, a data structure from which in O(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed in \(O(n^ 2)\) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed in \(O(n^ 2)\) time, improving earlier \(O(n^ 2 \log n)\) results.

MSC:

68U99 Computing methodologies and applications
68Q25 Analysis of algorithms and problem complexity
51M20 Polyhedra and polytopes; regular figures, division of spaces

Cite

References:

[1]Asano, T., An efficient algorithms for finding the visibility polygons for a polygonal region with holes, Transaction of IECE of Japan, E-68, 557-559 (1985)
[2]K. Q. Brown, Geometric transforms for fast geometric algorithms. Ph.D. thesis, Department of Computer Science, Carnegie-Mellon University, 1980.
[3]B. Chazelle, Filtering search: a new approach to query-answering.Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, Tucson, 1983, pp. 122-132.
[4]B. Chazelle, L. J. Guibas and D. T. Lee, The power of geometric duality.Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, Tucson, 1983, pp. 217-225; also,BIT, Vol. 25 (1985), pp. 76-90. ·Zbl 0603.68072
[5]H. Edelsbrunner, J. O’Rourke and R. Seidel, Constructing arrangements of lines and hyperplanes with applications.Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, Tucson, 1983, pp. 83-91. ·Zbl 0603.68104
[6]H. Edelsbrunner, M. H. Overmars and D. Wood, Graphics in flatland: a case study. InAdvances in Computing Research (F. P. Preparata, ed.), Vol. 1, JAI Press Inc., 1983, pp. 35-59.
[7]El Gindy, H.; Avis, D., A linear algorithm for computing the visibility polygon from a point, Journal of Algorithms, 2, 186-197 (1981) ·Zbl 0459.68057 ·doi:10.1016/0196-6774(81)90019-5
[8]H. A. El Gindy and G. T. Toussaint, Efficient algorithms for inserting and deleting edges from triangulations. Manuscript, School of Computer Science, McGill University, 1984.
[9]H. N. Gabow and R. E. Tarjan, A linear-time algorithm for a special case of disjoint set union.Proceedings of the 15th Annual ACM Symposium on Theory of Computing, Boston, 1983, pp. 246-251; also,Journal of Computer and System Sciences, Vol. 30 (1985), pp. 209-221. ·Zbl 0572.68058
[10]Garey, M. R.; Johnson, D. S.; Preparata, F. P.; Tarjan, R. E., Triangulating a simple polygon, Information Processing Letters, 7, No. 4, 175-179 (1978) ·Zbl 0384.68040 ·doi:10.1016/0020-0190(78)90062-5
[11]D. Harel, A linear time algorithm for the lowest common ancestors problem.Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science, Syracuse, N.Y., 1980, pp. 308-319.
[12]D. T. Lee, Proximity and reachability in the plane. Ph.D. dissertation, University of Illinois at Urbana-Champaign, 1978.
[13]Lee, D. T., Visibility of a simple polygon, Computer Vision, Graphics, and Image Processing, 22, 207-221 (1983) ·Zbl 0532.68071 ·doi:10.1016/0734-189X(83)90065-8
[14]Lee, D. T.; Preparata, F. P., Euclidean shortest paths in the presence of rectilinear barriers, Networks, 14, 393-410 (1984) ·Zbl 0545.90098 ·doi:10.1002/net.3230140304
[15]Lozano-Perez, T.; Wesley, M. A., An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM, 22, 560-570 (1979) ·doi:10.1145/359156.359164
[16]M. Sharir and A. Schoorr, On shortest paths in polyhedral spaces.Proceedings of the 16th Annual ACM Symposium on Theory of Computing, Washington, D.C., 1984, pp. 144-153.
[17]Welzl, E., Constructing the visibility graph forn line segments inO(n^2) time, Information Processing Letters, 20, 167-171 (1985) ·Zbl 0573.68036 ·doi:10.1016/0020-0190(85)90044-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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