## 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 Δ∈C, the number of intersection points between the segments of A and those of B that lie in Δ. 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 et al. (2021) [3]. 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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Article number | 101945 |

Journal | Computational Geometry: Theory and Applications |

Volume | 109 |

DOIs | |

State | Published - Feb 2023 |

### Bibliographical note

Funding Information:Work by B.A. was partially supported by NSF grants CCF-15-40656 and CCF-20-08551 , and by grant 2014/170 from the U.S-Israel Binational Science Foundation . Work by M.d.B. was partially supported by the Dutch Research Council (NWO) through Gravitation Grant NETWORKS (project no. 024.002.003 ). Work by J.C. was partially supported by the F.R.S.-FNRS ( Fonds National de la Recherche Scientifique ) under CDR Grant J.0146.18 . Work by E.E. was partially supported by NSF CAREER under grant CCF:AF-1553354 and by grant 824/17 from the Israel Science Foundation . Work by J.I. was partially supported by Fonds de la Recherche Scientifique FNRS under grant no. MISU F 6001 1 . Work by M.S. was 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.

Publisher Copyright:

© 2022 The Author(s)

## Keywords

- 3SUM-hard problems
- Algebraic decision-tree model
- Order type
- Point location
- Polynomial partitions