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Coding and Counting Arrangements of Pseudolines

  • Stefan Felsner
  • Pavel Valtr
Article

Abstract

Arrangements of lines and pseudolines are important and appealing objects for research in discrete and computational geometry. We show that there are at most \(2^{0.657\> n^{2}}\) simple arrangements of n pseudolines in the plane. This improves on previous work by Knuth who proved an upper bound of \(3^{\binom{n}{2}} \cong 2^{0.792\> n^{2}}\) in 1992 and the first author, who obtained \(2^{0.697\> n^{2}}\) in 1997. The argument uses surprisingly little geometry. The main ingredient is a lemma that was already central to the argument given by Knuth.

Keywords

Combinatorial geometry Enumeration Pseudoline Cutpath 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of Applied Mathematics and Institute for Theoretical Comp. Sci. (ITI)Charles UniversityPraha 1Czech Republic

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