On the average complexity of the $k$-level
$\newcommand{\LL}{\mathcal{L}}\newcommand{\SS}{\mathcal{S}}$Let \(\LL\) be an arrangement of \(n\) lines in the Euclidean plane. The \(k\)-level of \(\LL\) consists of all vertices \(v\) of the arrangement which have exactly \(k\) lines of \(\LL\) passing below \(v\). The complexity (the maximum siz...
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ftdatacite:10.20382/jocg.v11i1a19 2023-05-15T18:22:08+02:00 On the average complexity of the $k$-level Chiu, Man-Kwun Felsner, Stefan Scheucher, Manfred Schnider, Patrick Steiner, Raphael Valtr, Pavel 2020 https://dx.doi.org/10.20382/jocg.v11i1a19 https://jocg.org/index.php/jocg/article/view/3109 en eng Journal of Computational Geometry Text Article article-journal ScholarlyArticle 2020 ftdatacite https://doi.org/10.20382/jocg.v11i1a19 2021-11-05T12:55:41Z $\newcommand{\LL}{\mathcal{L}}\newcommand{\SS}{\mathcal{S}}$Let \(\LL\) be an arrangement of \(n\) lines in the Euclidean plane. The \(k\)-level of \(\LL\) consists of all vertices \(v\) of the arrangement which have exactly \(k\) lines of \(\LL\) passing below \(v\). The complexity (the maximum size) of the \(k\)-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of \(O(n\cdot (k+1)^{1/3})\). Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the \(k\)-level denotes the vertices at distance \(k\) to a marked cell, the south pole. We prove an upper bound of \(O((k+1)^2)\) on the expected complexity of the \((\le k)\)-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great \((d-1)\)-spheres on the \(d\)-sphere \(\SS^d\) which are orthogonal to a set of random points on \(\SS^d\). In this model, we prove that the expected complexity of the \(k\)-level is of order \(\Theta((k+1)^{d-1})\). In both scenarios, our bounds are independent of $n$, showing that the distribution of arrangements under our sampling methods differs significantly from other methods studied in the literature, where the bounds do depend on $n$. : Journal of Computational Geometry, Vol. 11 No. 1 (2020) Text South pole DataCite Metadata Store (German National Library of Science and Technology) South Pole |
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$\newcommand{\LL}{\mathcal{L}}\newcommand{\SS}{\mathcal{S}}$Let \(\LL\) be an arrangement of \(n\) lines in the Euclidean plane. The \(k\)-level of \(\LL\) consists of all vertices \(v\) of the arrangement which have exactly \(k\) lines of \(\LL\) passing below \(v\). The complexity (the maximum size) of the \(k\)-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of \(O(n\cdot (k+1)^{1/3})\). Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the \(k\)-level denotes the vertices at distance \(k\) to a marked cell, the south pole. We prove an upper bound of \(O((k+1)^2)\) on the expected complexity of the \((\le k)\)-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great \((d-1)\)-spheres on the \(d\)-sphere \(\SS^d\) which are orthogonal to a set of random points on \(\SS^d\). In this model, we prove that the expected complexity of the \(k\)-level is of order \(\Theta((k+1)^{d-1})\). In both scenarios, our bounds are independent of $n$, showing that the distribution of arrangements under our sampling methods differs significantly from other methods studied in the literature, where the bounds do depend on $n$. : Journal of Computational Geometry, Vol. 11 No. 1 (2020) |
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Text |
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Chiu, Man-Kwun Felsner, Stefan Scheucher, Manfred Schnider, Patrick Steiner, Raphael Valtr, Pavel |
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Chiu, Man-Kwun Felsner, Stefan Scheucher, Manfred Schnider, Patrick Steiner, Raphael Valtr, Pavel On the average complexity of the $k$-level |
author_facet |
Chiu, Man-Kwun Felsner, Stefan Scheucher, Manfred Schnider, Patrick Steiner, Raphael Valtr, Pavel |
author_sort |
Chiu, Man-Kwun |
title |
On the average complexity of the $k$-level |
title_short |
On the average complexity of the $k$-level |
title_full |
On the average complexity of the $k$-level |
title_fullStr |
On the average complexity of the $k$-level |
title_full_unstemmed |
On the average complexity of the $k$-level |
title_sort |
on the average complexity of the $k$-level |
publisher |
Journal of Computational Geometry |
publishDate |
2020 |
url |
https://dx.doi.org/10.20382/jocg.v11i1a19 https://jocg.org/index.php/jocg/article/view/3109 |
geographic |
South Pole |
geographic_facet |
South Pole |
genre |
South pole |
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South pole |
op_doi |
https://doi.org/10.20382/jocg.v11i1a19 |
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1766201494231580672 |