Abstract
In this paper we study a cell of the subdivision induced by a union ofn half-lines (or rays) in the plane. We present two results. The first one is a novel proof of theO(n) bound on the number of edges of the boundary of such a cell, which is essentially of methodological interest. The second is an algorithm for constructing the boundary of any cell, which runs in optimal Θ(n logn) time. A by-product of our results are the notions of skeleton and of skeletal order, which may be of interest in their own right.
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P. Alevizos, J. D. Boissonnat, and M. Yvinec: An OptimalO(n logn) Algorithm for Contour Reconstruction from Rays,Proc. 3rd ACM Symposium on Computational Geometry, Waterloo, pp. 162–170, June 1987.
M. Atallah: Dynamic Computational Geometry,Proc. 24th IEEE Symposium on Foundations of Computer Science, pp. 92–99, Oct. 1983.
B. Chazelle and L. Guibas: Visibility and Intersection Problems in Plane Geometry,Proc. 1st ACM Symposium on Computational Geometry, Baltimore, pp. 135–147, June 1985.
B. Chazelle, L. Guibas, and D. T. Lee: The Power of Geometric Duality,BIT 25, 76–90 (1985).
H. Edelsbrunner, L. J. Guibas, and M. Sharir: The Complexity of Many Faces in Arrangements of Lines and Segments,Proc. 4th ACM Symposium on Computational Geometry, Urbana, pp. 44–56, June 1988.
H. Edelsbrunner, J. O'Rourke, and R. Seidel: Constructing Arrangements of Lines and Hyper-planes with Applications,SIAM J. Comput. 15, 341–363 (1986).
L. J. Guibas, M. Sharir, and S. Sifrony: On the General Motion Planning Problem with Two Degrees of Freedom,Proc. 4th ACM Symposium on Computational Geometry, Urbana, pp. 319–329, June 1988.
S. Hart and M. Sharir: Non-Linearity of Davenport-Schinzel Sequences and of Generalized Path Compression Schemes,Combinatorica 6 (2), 151–177 (1986).
R. Pollack, M. Sharir, and S. Sifrony: Separating Two Simple Polygons by a Sequence of Translations,Discrete Comput. Geom. 3, 123–136 (1988).
A. Wiernik and M. Sharir: Planar Realizations of Nonlinear Davenport-Schinzel Sequences by Segments,Discrete Comput. Geom. 3, 15–47 (1988).
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Communicated by Bernard Chazelle.
This work was partly supported by CEE ESPRIT Project P-940, by the Ecole Normale Supérieure, Paris, and by NSF Grant ECS-84-10902.
This work was done in part while this author was visiting the Ecole Normale Supérieure, Paris, France.
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Alevizos, P., Boissonnat, JD. & Preparata, F.P. An optimal algorithm for the boundary of a cell in a union of rays. Algorithmica 5, 573–590 (1990). https://doi.org/10.1007/BF01840405
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DOI: https://doi.org/10.1007/BF01840405