### Abstract

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in IR^{d}, is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time Ω(n log m + n^{2/3} m^{2/3} + m log n) in two dimensions, or Ω(n log m + n^{5/6} m^{1/2} + n^{1/2}m^{5/6} + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2^{O}(log^{∗(n+m))} of the best known upper bound, due to Matoušek. Previously, the best known lower bound, in any dimension, was Ω(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive lower bounds on the complexity of this representation. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover.

Original language | English (US) |
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Title of host publication | Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 |

Publisher | Association for Computing Machinery |

Pages | 127-137 |

Number of pages | 11 |

ISBN (Electronic) | 0897917243 |

DOIs | |

State | Published - Sep 1 1995 |

Externally published | Yes |

Event | 11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada Duration: Jun 5 1995 → Jun 7 1995 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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Volume | Part F129372 |

### Other

Other | 11th Annual Symposium on Computational Geometry, SCG 1995 |
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Country | Canada |

City | Vancouver |

Period | 6/5/95 → 6/7/95 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

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## Cite this

*Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995*(pp. 127-137). (Proceedings of the Annual Symposium on Computational Geometry; Vol. Part F129372). Association for Computing Machinery. https://doi.org/10.1145/220279.220293