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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© 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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