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.


Span Tree Polynomial Function Fundamental Problem Minimum Span Tree Planar Arrangement 
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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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