Counting crossing-free structures

Victor Alvarez, Karl Bringmann, Radu Curticapean, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30n) and at least Ω(2.43n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n O(k), where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414n). Given that there are several well-studied configurations of points with at least Ω(3.464n) triangulations, and some even with Ω(8n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W [2]-hard in the parameter k. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

Original languageEnglish (US)
Title of host publicationProceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012
Pages61-68
Number of pages8
DOIs
StatePublished - 2012
Event28th Annual Symposuim on Computational Geometry, SCG 2012 - Chapel Hill, NC, United States
Duration: Jun 17 2012Jun 20 2012

Publication series

NameProceedings of the Annual Symposium on Computational Geometry

Other

Other28th Annual Symposuim on Computational Geometry, SCG 2012
CountryUnited States
CityChapel Hill, NC
Period6/17/126/20/12

Keywords

  • Counting
  • Onion layers
  • Parameterized complexity
  • Triangulations

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics

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