Abstract
In this paper, we establish two combinatorial bounds related to the separation problem for sets of n pairwise disjoint translates of convex objects: 1) there exists a line which separates one translate from at least n — co√n translates, for some constant c that depends on the “shape” of the translates and 2) there is a function f such that there exists a line with orientation Θ or f(Θ) which separates one translate from at least ⌈3n⌉/4-4 translates, for any orientation Θ (f is defined only by the “shape” of the translate). We also present an O(n log (n+k)+k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst sets of n pairwise disjoint translates of convex k-gons.
This research was supported by FCAR, FODAR and NSERC.
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References
N. Alon, M. Katchalski, and W. Pulleyblank. Cutting disjoint disks by straight lines. Disc. Comp. Geom., 4:239–243, 1989.
M. Atallah. Some dynamic computational geometry problems. Comp. Math. Appl., 11:1171–1181, 1985.
B. Chazelle, H. Edelsbrunner, M. Gringi, L. J. Guibas, M. Sharir, and J. Snoeyink. Ray shooting in polygons using geodesic triangulations. In Proc. of the 18th Int. Coll. on Automata, Languages and Programming, pages 661–673, 1991.
J. Czyzowicz, E. Rivera-Campo, and J. Urrutia. A note on separation of convex sets. To appear in Disc. Math.
J. Czyzowicz, E. Rivera-Campo, J. Urrutia, and J. Zaks. Separating convex sets in the plane. In Proc. of the Sec. Can. Conf. on Comp. Geom., pages 50–54, 1990.
J. Czyzowicz, E. Rivera-Campo, J. Urrutia, and J. Zaks. Separating convex sets in the plane. Disc. Comp. Geom., 7:189–195, 1992.
H. Edelsbrunner, H.A. Maurer, F.P. Preparata, A.L. Rosenberg, E. Welzl, and D. Wood. Stabbing line segments. BIT, 22:274–281, 1982.
H. Everett, J.-M. Robert, and M. van Kreveld. An optimal algorithm for the (≤ k)-levels, with applications to separation and transversal. In Proc. of the 9th Annual ACM Symp. on Comp. Geom., pages 38–46, 1993.
R. Hope and M. Katchalski. Separating plane convex sets. Math. Scand., 66:44–46, 1990.
M. Katchalski, T. Lewis, and A. Liu. Geometric permutations of disjoint translates of convex sets. Disc. Math., 65:249–259, 1987.
T. Nishizeki and N. Chiba. Planar Graphs: Theory and Algorithms. North-Holland, 1988.
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© 1994 Springer-Verlag Berlin Heidelberg
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Czyzowicz, J., Everett, H., Robert, JM. (1994). Separating translates in the plane: Combinatorial bounds and an algorithm. In: Schmidt, E.M., Skyum, S. (eds) Algorithm Theory — SWAT '94. SWAT 1994. Lecture Notes in Computer Science, vol 824. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58218-5_10
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DOI: https://doi.org/10.1007/3-540-58218-5_10
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