## On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences (2003)

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Venue: | In Proc. 44th IEEE Sympos. Found. Comput. Sci |

Citations: | 9 - 2 self |

### BibTeX

@INPROCEEDINGS{Chan03onlevels,

author = {Timothy M. Chan},

title = {On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences},

booktitle = {In Proc. 44th IEEE Sympos. Found. Comput. Sci},

year = {2003},

pages = {544--550}

}

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### Abstract

We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degree-s polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n ) bound for parabolas.