Abstract
We consider the following problem. Given a polygon P, possibly with holes, and having n vertices, compute a pair of equal radius disks that do not intersect each other, are contained in P, and whose radius is maximized. Our main result is a simple randomized algorithm whose expected running time, on worst case input, is O(n log n). This is optimal in the algebraic decision tree model of computation.
This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Hong Kong Research Grant Council CERG grant HKUST6137/98E.
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References
P.K. Agarwal, J. MatouÃsek, and S. Suri Farthest neighbors, maximum spanning trees, and related problems in higher dimensions. Comput. Geom.: Theory & Appl., 4:189–201, 1992.
Helmut Alt and Otfried Schwarzkopf The Voronoi diagram of curved objects. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 89–97, 1995.
Boaz Ben-Moshe, Matthew J. Katz, and Michael Segal Obnoxious facility location: Complete service with minimal harm. In Proceedings of the 11th Canadian Conference on Computational Geometry (CCCG’99), pages 76–79, 1999.
S. Bespamyatnikh Draft: Efficient algorithm for finding two largest empty circles. In Proceedings of the 15th European Workshop on Computational Geometry (EuroCG’99), pages 37–38, 1999.
T.C. Biedl, E.D. Demaine, M.L. Demaine, A. Lubiw, and G.T. Toussaint Hiding disks in folded polygons. In Proceedings of the 10th Canadian Conference on Computational Geometry (CCCG’98), 1998.
P. Bose, J. Czyzowicz, E. Kranakis, and A. Maheshwari Algorithms for packing two circles in a convex polygon. In Proceedings of Japan Conference on Discrete and Computational Geometry (JCDCG’ 98), pages 93–103, 1998.
F. Chin, J. Snoeyink, and C.A. Wang Finding the medial axis of a simple polygon in linear time. Discrete and Computational Geometry, 21, 1999.
K.L. Clarkson and P.W. Shor Algorithms for diametral pairs and convex hulls that are optimal, randomized, and incremental. In Proceedings of the Fourth Annual Symposium on Computational Geometry (SoCG’88), pages 12–17, 1988.
Matthew J. Katz, Klara Kedem, and Michael Segal Improved algorithms for placing undesirable facilities. In Proceedings of the 11th Canadian Conference on Computational Geometry (CCCG’99), pages 65–67, 1999.
S.K. Kim and C.-S. Shin Placing two disks in a convex polygon. Information Processing Letters, 73, 2000.
N. Megiddo Applying parallel computation algorithms to the design of serial algorithms. Journal of the ACM, 30:852–865, 1983.
F.P. Preparata and M.I. Shamos Computational Geometry. Springer-Verlag, New York, 1985.
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Bose, P., Morin, P., Vigneron, A. (2001). Packing Two Disks into a Polygonal Environment. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_16
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DOI: https://doi.org/10.1007/3-540-44679-6_16
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