article

On counting point-hyperplane incidences

Authors:

Christian KnauerAuthors Info & Claims

Computational Geometry: Theory and Applications, Volume 25, Issue 1-2

Pages 13 - 20

https://doi.org/10.1016/S0925-7721(02)00127-X

Published: 01 May 2003 Publication History

Abstract

In this paper we discuss three closely related problems on the incidence structure between n points and m hyperplanes in d-dimensional space: the maximal number of incidences if there are no big bipartite subconfigurations, a compressed representation for the incidence structure, and a lower bound for any algorithm that determines the number of incidences (counting version of Hopcroft's problem). For this we give a construction of a special point-hyperplane configuration, giving a lower bound, which almost meets the best upper bound known thus far.

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Cited By

Sudakov BTomon I(2024)Evasive Sets, Covering by Subspaces, and Point-Hyperplane IncidencesDiscrete & Computational Geometry10.1007/s00454-023-00601-172:3(1333-1347)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s00454-023-00601-1
Balko MCibulka JValtr P(2019)Covering Lattice Points by Subspaces and Counting Point---Hyperplane IncidencesDiscrete & Computational Geometry10.1007/s00454-018-9970-761:2(325-354)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s00454-018-9970-7
Rudnev M(2018)On the Number of Incidences Between Points and Planes in Three DimensionsCombinatorica10.1007/s00493-016-3329-638:1(219-254)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1007/s00493-016-3329-6
Show More Cited By

Index Terms

On counting point-hyperplane incidences
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
      2. Enumeration
    2. Graph theory
      1. Graph algorithms
      2. Hypergraphs

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Information

Published In

cover image Computational Geometry: Theory and Applications

Computational Geometry: Theory and Applications Volume 25, Issue 1-2

Special issue: The European workshop on computational geometry -- CG01

May 2003

165 pages

ISSN:0925-7721

Issue’s Table of Contents

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 May 2003

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Cited By

Sudakov BTomon I(2024)Evasive Sets, Covering by Subspaces, and Point-Hyperplane IncidencesDiscrete & Computational Geometry10.1007/s00454-023-00601-172:3(1333-1347)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s00454-023-00601-1
Balko MCibulka JValtr P(2019)Covering Lattice Points by Subspaces and Counting Point---Hyperplane IncidencesDiscrete & Computational Geometry10.1007/s00454-018-9970-761:2(325-354)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s00454-018-9970-7
Rudnev M(2018)On the Number of Incidences Between Points and Planes in Three DimensionsCombinatorica10.1007/s00493-016-3329-638:1(219-254)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1007/s00493-016-3329-6
Sharir MSolomon NKlein P(2017)Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distancesProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039849(2456-2475)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039849
Sharir M(2017)Computational Geometry Column 65ACM SIGACT News10.1145/3106700.310671148:2(68-85)Online publication date: 12-Jun-2017
https://dl.acm.org/doi/10.1145/3106700.3106711
Solymosi JTÓth C(2008)On a question of bourgain about geometric incidencesCombinatorics, Probability and Computing10.1017/S096354830800912717:4(619-625)Online publication date: 1-Jul-2008
https://dl.acm.org/doi/10.1017/S0963548308009127
Dumitrescu ATÓth C(2008)On the number of tetrahedra with minimum, unit, and distinct volumes in three-spaceCombinatorics, Probability and Computing10.1017/S096354830700884X17:2(203-224)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1017/S096354830700884X
Dumitrescu ATóth CGabow H(2007)On the number of tetrahedra with minimum, unit, and distinct volumes in three-spaceProceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms10.5555/1283383.1283503(1114-1123)Online publication date: 7-Jan-2007
https://dl.acm.org/doi/10.5555/1283383.1283503
Laba ISolymosi J(2007)Incidence Theorems for PseudoflatsDiscrete & Computational Geometry10.1007/s00454-006-1279-237:2(163-174)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.1007/s00454-006-1279-2
Elekes GTóth CMitchell JRote G(2005)Incidences of not-too-degenerate hyperplanesProceedings of the twenty-first annual symposium on Computational geometry10.1145/1064092.1064098(16-21)Online publication date: 6-Jun-2005
https://dl.acm.org/doi/10.1145/1064092.1064098

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