New Bounds on the Size of Optimal Meshes

The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents...

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Bibliographic Details
Published in:Computer graphics forum Vol. 31; no. 5; pp. 1627 - 1635
Main Author: Sheehy, Donald R.
Format: Journal Article
Language:English
Published: Oxford, UK Blackwell Publishing Ltd 01.08.2012
Subjects:
Computer graphics
F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-Geometrical problems and computations
Initiatives
Integrals
Lower bounds
Matching
Mathematical analysis
Mathematical functions
Mathematical models
Optimization
Sizing
Studies
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary:The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕdn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(n+nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).
Bibliography:ark:/67375/WNG-DFD2RKFS-V
ArticleID:CGF3168
istex:E4E9CE791060A97C2B4FD83570F7BAD16D886055
This work was partially supported by the National Science Foundation under grant number CCF‐1065106, by GIGA grant ANR‐09‐BLAN‐0331‐01, and by the European project CG‐Learning No. 255827.
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ISSN:0167-7055
1467-8659
DOI:10.1111/j.1467-8659.2012.03168.x