## On Levels in Arrangements of Curves (2002)

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Venue: | Proc. 41st IEEE |

Citations: | 23 - 3 self |

### BibTeX

@INPROCEEDINGS{Chan02onlevels,

author = {Timothy M. Chan},

title = {On Levels in Arrangements of Curves},

booktitle = {Proc. 41st IEEE},

year = {2002},

pages = {219--227}

}

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### 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 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 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.