## Abstract

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle ∆ ∈ C, the number of intersection points between the segments of A and those of B that lie in ∆. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n^{2}/log^{2} n) log^{O(1)} log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n^{60}/31+^{ε}), for any ε > 0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.

Original language | English |
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Title of host publication | 32nd International Symposium on Algorithms and Computation, ISAAC 2021 |

Editors | Hee-Kap Ahn, Kunihiko Sadakane |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772143 |

DOIs | |

State | Published - 1 Dec 2021 |

Event | 32nd International Symposium on Algorithms and Computation, ISAAC 2021 - Fukuoka, Japan Duration: 6 Dec 2021 → 8 Dec 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 212 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd International Symposium on Algorithms and Computation, ISAAC 2021 |
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Country/Territory | Japan |

City | Fukuoka |

Period | 6/12/21 → 8/12/21 |

### Bibliographical note

Publisher Copyright:© Boris Aronov, Mark de Berg, Jean Cardinal, Esther Ezra, John Iacono, and Micha Sharir.

### Funding

Micha Sharir: Partially supported by ISF Grant 260/18, by Grant 1367/2016 from the German-Israeli Science Foundation (GIF), and by Blavatnik Research Fund in Computer Science at Tel Aviv University. Funding Boris Aronov: Partially supported by NSF Grants CCF-15-40656 and CCF-20-08551, and by Grant 2014/170 from the US-Israel Binational Science Foundation. Mark de Berg: Partially supported by the Dutch Research Council (NWO) through Gravitation Grant NETWORKS (project no. 024.002.003). Jean Cardinal: Partially supported by the F.R.S.-FNRS (Fonds National de la Recherche Scientifique) under CDR Grant J.0146.18. Esther Ezra: Partially supported by NSF CAREER under Grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation. John Iacono: Partially supported by Fonds National de la Recherche Scientifique (FNRS) under Grant no. MISU F.6001.1.

Funders | Funder number |
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Blavatnik Research Fund in Computer Science | |

German–Israeli Science Foundation | |

National Science Foundation | CCF-20-08551, 2014/170, CCF-15-40656 |

United States-Israel Binational Science Foundation | |

Fonds De La Recherche Scientifique - FNRS | J.0146.18, 824/17, AF-1553354 |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek | 024.002.003 |

Israel Science Foundation | 1367/2016, 260/18 |

Tel Aviv University |

## Keywords

- Algebraic decision-tree model
- Computational geometry
- Hierarchical partitions
- Order types
- Point location
- Polynomial partitioning
- Primal-dual range searching