Abstract
We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ε>0, anO(n>1+ε) algorithm for computing the diameter of a point set in 3-space, anO(8/5+ε) algorithm for computing the width of such a set, and onO(n>8/5+ε) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic.
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Work by Bernard Chazelle was supported by NSF Grant CCR-90-02352. Work by Herbert Edelsbrunner was supported by NSF Grant CCR-89-21421. Work by Leonidas Guibas and Micha Sharir was supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Micha Sharir was also supported by ONR Grant N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.
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Chazelle, B., Edelsbrunner, H., Guibas, L. et al. Diameter, width, closest line pair, and parametric searching. Discrete Comput Geom 10, 183–196 (1993). https://doi.org/10.1007/BF02573973
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DOI: https://doi.org/10.1007/BF02573973