## Abstract

We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ^{3} in nearly-quadratic time. This solves a problem posed by Pellegrini. Using a variant of the technique, one can represent the set of all κ triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. Our approach also applies to any collection of convex planar objects of constant description complexity in ℝ^{3}, with the same performance bounds. We also prove that this counting problem belongs to the 3SUM-hard family, and thus our algorithm is likely to be nearly optimal (since it is believed that 3SUM-hard problems cannot be solved in subquadratic time).

Original language | English |
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Pages | 210-219 |

Number of pages | 10 |

DOIs | |

State | Published - 2004 |

Externally published | Yes |

Event | Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States Duration: 9 Jun 2004 → 11 Jun 2004 |

### Conference

Conference | Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) |
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Country/Territory | United States |

City | Brooklyn, NY |

Period | 9/06/04 → 11/06/04 |

### Bibliographical note

Funding Information:Keywords: Triangles in three dimensions; Curve-sensitive cuttings; Counting intersections; Arrangements; 3SUM-hard problems ✩ Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israel Science Fund, Israeli Academy of Sciences, for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper has appeared in Proc. 20th Annu. ACM Sympos. Comput. Geom., 2004, pp. 210– 219. * Corresponding author. E-mail addresses: estere@post.tau.ac.il (E. Ezra), michas@post.tau.ac.il (M. Sharir).

## Keywords

- 3SUM-hard problems
- Arrangements
- Counting intersections
- Curve-sensitive cuttings
- Triangles in three dimensions