research-article

A fast algorithm for well-spaced points and approximate delaunay graphs

Authors:

Gary L. Miller,

Donald R. Sheehy,

Ameya VelingkerAuthors Info & Claims

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

Pages 289 - 298

https://doi.org/10.1145/2462356.2462404

Published: 17 June 2013 Publication History

Abstract

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2^O(d)(n log n + m)), where n is the input size, m is the output point set size, and d is the ambient dimension. The constants only depend on the desired element quality bounds.

To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on d is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size 2^O(d)m graph in 2^O(d)(n log n + m) expected time. If m is superlinear in n, then we can produce a hierarchically well-spaced superset of size 2^O(d)n in 2^O(d)n log n expected time.

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Cited By

Boissonnat JDevillers ODutta KGlisse M(2020)Randomized Incremental Construction of Delaunay Triangulations of Nice Point SetsDiscrete & Computational Geometry10.1007/s00454-020-00235-7Online publication date: 8-Sep-2020
https://doi.org/10.1007/s00454-020-00235-7
Mitchell SEbeida MAwad MPark CPatney ARushdi ASwiler LManocha DWei L(2018)Spoke-Darts for High-Dimensional Blue-Noise SamplingACM Transactions on Graphics10.1145/319465737:2(1-20)Online publication date: 12-May-2018
https://dl.acm.org/doi/10.1145/3194657
Cohen MFasy BMiller GNayyeri ASheehy DVelingker A(2015)Approximating Nearest Neighbor DistancesAlgorithms and Data Structures10.1007/978-3-319-21840-3_17(200-211)Online publication date: 28-Jul-2015
https://doi.org/10.1007/978-3-319-21840-3_17
Show More Cited By

Index Terms

A fast algorithm for well-spaced points and approximate delaunay graphs
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

June 2013

472 pages

ISBN:9781450320313

DOI:10.1145/2462356

General Chairs:
Guilherme D. da Fonseca
UNIRIO
,
Thomas Lewiner
PUC-Rio
,
Luis Peñaranda
IMPA
,
Program Chairs:
Timothy Chan
University of Waterloo
,
Rolf Klein
University of Bonn

Copyright © 2013 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 17 June 2013

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SoCG '13

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SoCG '13: Symposium on Computational Geometry 2013

June 17 - 20, 2013

Rio de Janeiro, Brazil

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SoCG '13 Paper Acceptance Rate 48 of 137 submissions, 35%;

Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Boissonnat JDevillers ODutta KGlisse M(2020)Randomized Incremental Construction of Delaunay Triangulations of Nice Point SetsDiscrete & Computational Geometry10.1007/s00454-020-00235-7Online publication date: 8-Sep-2020
https://doi.org/10.1007/s00454-020-00235-7
Mitchell SEbeida MAwad MPark CPatney ARushdi ASwiler LManocha DWei L(2018)Spoke-Darts for High-Dimensional Blue-Noise SamplingACM Transactions on Graphics10.1145/319465737:2(1-20)Online publication date: 12-May-2018
https://dl.acm.org/doi/10.1145/3194657
Cohen MFasy BMiller GNayyeri ASheehy DVelingker A(2015)Approximating Nearest Neighbor DistancesAlgorithms and Data Structures10.1007/978-3-319-21840-3_17(200-211)Online publication date: 28-Jul-2015
https://doi.org/10.1007/978-3-319-21840-3_17
Ebeida MAwad MGe XMahmoud AMitchell SKnupp PWei L(2014)Improving spatial coverage while preserving the blue noise of point setsComputer-Aided Design10.1016/j.cad.2013.08.01546(25-36)Online publication date: 1-Jan-2014
https://dl.acm.org/doi/10.1016/j.cad.2013.08.015
Miller GSheehy D(2014)A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay TriangulationsDiscrete & Computational Geometry10.1007/s00454-014-9629-y52:3(476-491)Online publication date: 1-Oct-2014
https://dl.acm.org/doi/10.1007/s00454-014-9629-y
Miller GSheehy Dda Fonseca GLewiner TPeñaranda LChan TKlein R(2013)A new approach to output-sensitive voronoi diagrams and delaunay triangulationsProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462372(281-288)Online publication date: 17-Jun-2013
https://dl.acm.org/doi/10.1145/2462356.2462372

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