From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices

Alexander Pilz, Emo Welzl, Manuel Wettstein

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

Abstract

A set P = H ⊃ {w} of n + 1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2n/2. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(nd-1) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(nd log n). Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n = d + k points in general position in Rd in time O(nmax{ω,k-2) where ω ≈ 2.373, even though the asymptotic number of facets may be as large as nk.

Original languageEnglish
Title of host publication33rd International Symposium on Computational Geometry, SoCG 2017
PublisherSchloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH
Pages541-5416
Number of pages4876
Volume77
ISBN (Electronic)9783959770385
DOIs
Publication statusPublished - 1 Jun 2017
Externally publishedYes
Event33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia
Duration: 4 Jul 20177 Jul 2017

Conference

Conference33rd International Symposium on Computational Geometry, SoCG 2017
CountryAustralia
CityBrisbane
Period4/07/177/07/17

Keywords

  • Gale transform
  • Geometric graph
  • Polytope
  • Simplicial depth
  • Wheel set

ASJC Scopus subject areas

  • Software

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