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.
Originalsprache | englisch |
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Titel | 33rd International Symposium on Computational Geometry, SoCG 2017 |
Herausgeber (Verlag) | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Seiten | 541-5416 |
Seitenumfang | 4876 |
Band | 77 |
ISBN (elektronisch) | 9783959770385 |
DOIs | |
Publikationsstatus | Veröffentlicht - 1 Juni 2017 |
Extern publiziert | Ja |
Veranstaltung | 33rd International Symposium on Computational Geometry: SoCG 2017 - The University of Queensland, St Lucia, Brisbane, Australien Dauer: 4 Juli 2017 → 7 Juli 2017 http://socg2017.smp.uq.edu.au/ |
Konferenz
Konferenz | 33rd International Symposium on Computational Geometry |
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Kurztitel | SoCG |
Land/Gebiet | Australien |
Ort | Brisbane |
Zeitraum | 4/07/17 → 7/07/17 |
Internetadresse |
ASJC Scopus subject areas
- Software