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Discrete & Computational Geometry

, Volume 34, Issue 1, pp 11–24 | Cite as

On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences

  • Timothy M. Chan
Article

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(n2-1/2s) complexity. This answers one of the main open problems from the author’s previous paper [DCG 29, 375-393 (2003)], which provided a weaker upper bound for a restricted class of curves only (graphs of degree-s polynomials). 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(n3/2 bound for parabolas.

Keywords

Computational Mathematic Open Problem Fixed Number Short Proof Restricted Class 
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 2005

Authors and Affiliations

  1. 1.School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1Canada

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