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Discrete & Computational Geometry

, Volume 29, Issue 3, pp 375–393 | Cite as

On Levels in Arrangements of Curves

Abstract. 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^ 1-1/(9· 2 s-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 7/9 log 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.

Keywords

Span Tree Polynomial Function Fundamental Problem Minimum Span Tree Planar Arrangement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 2003

Authors and Affiliations

  •  Chan
    • 1
  1. 1.Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.caCA

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