On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Discrete and Computational Geometry Year : 2012

On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions

Marc Glisse

Abstract

We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among $n$ disjoint \emph{unit} balls has complexity $\Omega(n^4)$, which matches the trivial $O(n^4)$ upper bound. This improves the trivial $\Omega(n^2)$ bound and also the $\Omega(n^3)$ lower bound for the restricted setting of arbitrary-size balls [Devillers and Ramos, 2001]. This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an $\Omega(n^3)$ lower bound on the complexity of the set of non-occluded lines among $n$ balls of arbitrary radii, improving on the trivial $\Omega(n^2)$ bound. This new bound almost matches the recent $O(n^{3+\epsilon})$ upper bound [Rubin, 2010].
Fichier principal
Vignette du fichier
dcg_revised.pdf (244.56 Ko) Télécharger le fichier
Vignette du fichier
2012 Free Lines Among Balls.png (55.88 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Format Figure, Image
Loading...

Dates and versions

hal-00643880 , version 1 (23-11-2011)

Identifiers

Cite

Marc Glisse, Sylvain Lazard. On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions. Discrete and Computational Geometry, 2012, 47 (4), pp.756-772. ⟨10.1007/s00454-012-9414-8⟩. ⟨hal-00643880⟩
250 View
166 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More