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On levels in arrangements of curves

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1 Author(s)
Chan, T.M. ; 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(nk1-2/3*)) 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(nk7/9log2/3 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

Published in:

Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on

Date of Conference:

2000