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41st Annual Symposium on Foundations of Computer Science
On levels in arrangements of curves
Redondo Beach, California
November 12-November 14
ISBN: 0-7695-0850-2
T.M. Chan, Dept. of Comput. Sci., Waterloo Univ., Ont., Canada
Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk/sup 1-2/3/*)) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s=1 and s=2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk/sup 7/9/log/sup 2/3/ k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.
Index Terms:
computational complexity; computational geometry; polynomials; worst-case complexity; planar arrangement; combinatorial geometry; subquadratic upper bound; polynomial functions; pseudo-parabolas; pseudo-segments; kinetic minimum spanning trees
T.M. Chan, "On levels in arrangements of curves," focs, pp.219, 41st Annual Symposium on Foundations of Computer Science, 2000
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