article
The complexity of many cells in arrangements of planes and related problems
published
yes
Herbert
Edelsbrunner
author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833
Leonidas
Guibas
author
Micha
Sharir
author
We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m 2/3 n logn +n 2); we can calculatem such cells specified by a point in each, in worst-case timeO(m 2/3 n log3 n+n 2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m 2/3 n logn+n 2), but this number is onlyO(m 3/5– n 4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m 3/4– n 3/4+3 +m) log2 n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m 3/4– n 3/4+3 +m] log2 n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m 2/3 n d/3 logn+n d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.
Springer1990
eng
Discrete & Computational Geometry
0179-5376
1432-044410.1007/BF02187785
51197 - 216
yes
H. Edelsbrunner, L. Guibas, M. Sharir, Discrete & Computational Geometry 5 (1990) 197–216.
Edelsbrunner, H., Guibas, L., & Sharir, M. (1990). The complexity of many cells in arrangements of planes and related problems. <i>Discrete & Computational Geometry</i>. Springer. <a href="https://doi.org/10.1007/BF02187785">https://doi.org/10.1007/BF02187785</a>
Edelsbrunner H, Guibas L, Sharir M. 1990. The complexity of many cells in arrangements of planes and related problems. Discrete & Computational Geometry. 5(1), 197–216.
H. Edelsbrunner, L. Guibas, and M. Sharir, “The complexity of many cells in arrangements of planes and related problems,” <i>Discrete & Computational Geometry</i>, vol. 5, no. 1. Springer, pp. 197–216, 1990.
Edelsbrunner, Herbert, et al. “The Complexity of Many Cells in Arrangements of Planes and Related Problems.” <i>Discrete & Computational Geometry</i>, vol. 5, no. 1, Springer, 1990, pp. 197–216, doi:<a href="https://doi.org/10.1007/BF02187785">10.1007/BF02187785</a>.
Edelsbrunner H, Guibas L, Sharir M. The complexity of many cells in arrangements of planes and related problems. <i>Discrete & Computational Geometry</i>. 1990;5(1):197-216. doi:<a href="https://doi.org/10.1007/BF02187785">10.1007/BF02187785</a>
Edelsbrunner, Herbert, Leonidas Guibas, and Micha Sharir. “The Complexity of Many Cells in Arrangements of Planes and Related Problems.” <i>Discrete & Computational Geometry</i>. Springer, 1990. <a href="https://doi.org/10.1007/BF02187785">https://doi.org/10.1007/BF02187785</a>.
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