Article

Slimming down by adding; selecting heavily covered points

Authors:

Bernard Chazelle,

Herbert Edelsbrunner,

Leonidas J. Guibas,

John E. Hershberger,

Raimund Seidel, and

Micha SharirAuthors Info & Claims

SCG '90: Proceedings of the sixth annual symposium on Computational geometry

May 1990

Pages 116 - 127

https://doi.org/10.1145/98524.98551

Published: 01 May 1990 Publication History

Abstract

We show that for any set Π of n points in three-dimensional space there is a set Q of 𝒪(n^1/2 log³ n) points so that the Delaunay triangulation of Π ∪ Q has at most 𝒪(n^3/2 log³ n) edges — even though the Delaunay triangulation of Π may have Ω(n²) edges. The main tool of our construction is the following geometric covering result: For any set Π of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in Π, there exists a point x, not necessarily in Π, that is enclosed by Ω(m²/n² log³ n²/m) of the spheres in S.

Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.

Cited By

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Khoury IToussaint GCiampi AAntoniano I(2009)Proximity-Graph-Based Tools for DNA ClusteringEncyclopedia of Data Warehousing and Mining, Second Edition10.4018/978-1-60566-010-3.ch248(1623-1631)Online publication date: 2009
https://doi.org/10.4018/978-1-60566-010-3.ch248
Erickson J(2005)Dense Point Sets Have Sparse Delaunay Triangulations or "... But Not Too Nasty"Discrete & Computational Geometry10.5555/3115459.311569133:1(83-115)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.5555/3115459.3115691
Battista GLenhart WLiotta G(2005)Proximity drawability: A survey extended abstractGraph Drawing10.1007/3-540-58950-3_388(328-339)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-58950-3_388
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Index Terms

Slimming down by adding; selecting heavily covered points
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

SCG '90: Proceedings of the sixth annual symposium on Computational geometry

May 1990

371 pages

ISBN:0897913620

DOI:10.1145/98524

Chairman:
Raimund Seidel

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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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 May 1990

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SOCG

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SOCG: 6th Annual Conference on Computational Geometry

June 7 - 9, 1990

California, Berkley, USA

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

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Cited By

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Khoury IToussaint GCiampi AAntoniano I(2009)Proximity-Graph-Based Tools for DNA ClusteringEncyclopedia of Data Warehousing and Mining, Second Edition10.4018/978-1-60566-010-3.ch248(1623-1631)Online publication date: 2009
https://doi.org/10.4018/978-1-60566-010-3.ch248
Erickson J(2005)Dense Point Sets Have Sparse Delaunay Triangulations or "... But Not Too Nasty"Discrete & Computational Geometry10.5555/3115459.311569133:1(83-115)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.5555/3115459.3115691
Battista GLenhart WLiotta G(2005)Proximity drawability: A survey extended abstractGraph Drawing10.1007/3-540-58950-3_388(328-339)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-58950-3_388
Chazelle BEdelsbrunner HGrigni MGuibas LSharir MWelzl E(1995)Improved bounds on weak ź-nets for convex setsDiscrete & Computational Geometry10.1007/BF0257402513:1(1-15)Online publication date: 1-Dec-1995
https://dl.acm.org/doi/10.1007/BF02574025
Chazelle BEdelsbrunner HGrigni MGuibas LSharir MWelzl EKosaraju RJohnson DAggarwal A(1993)Improved bounds on weak ε-nets for convex setsProceedings of the twenty-fifth annual ACM symposium on Theory of Computing10.1145/167088.167222(495-504)Online publication date: 1-Jun-1993
https://dl.acm.org/doi/10.1145/167088.167222
Melissaratos ESouvaine DAvis D(1992)Coping with inconsistenciesProceedings of the eighth annual symposium on Computational geometry10.1145/142675.142719(202-211)Online publication date: 1-Jul-1992
https://dl.acm.org/doi/10.1145/142675.142719
Jaromczyk JToussaint G(1992)Relative neighborhood graphs and their relativesProceedings of the IEEE10.1109/5.16341480:9(1502-1517)Online publication date: Jan-1992
https://doi.org/10.1109/5.163414
Aronov BChazelle BEdelsbrunner HGuibas LSharir MWenger R(1991)Points and triangles in the plane and halving planes in spaceDiscrete & Computational Geometry10.5555/2805868.28336766:1(435-442)Online publication date: 1-Dec-1991
https://dl.acm.org/doi/10.5555/2805868.2833676
Aronov BChazelle BEdelsbrunner HGuibas LSharir MWenger R(1991)Points and triangles in the plane and halving planes in spaceDiscrete & Computational Geometry10.1007/BF025747006:3(435-442)Online publication date: 1-Dec-1991
https://dl.acm.org/doi/10.1007/BF02574700

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Selecting Heavily Covered Points

Koenderink Corner Points

Direct reconstruction of a displaced subdivision surface from unorganized points

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Selecting Heavily Covered Points

Koenderink Corner Points

Direct reconstruction of a displaced subdivision surface from unorganized points

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