## Abstract

The collinearity testing problem is a basic problem in computational geometry, in which, given three sets A, B, C in the plane, of n points each, the task is to detect a collinear triple of points in A×B×C or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the collinearity indexing problem, in which we are given two sets A and B, each of n points in the plane, and our goal is to preprocess A and B into a data structure, so that, for any query point q∈R^{2}, we can determine whether q is collinear with a pair of points (a,b)∈A×B. We provide a solution to the problem for the case where the points of A, B lie on an integer grid, and the query points lie on a vertical line, with a data structure of subquadratic storage and sublinear query time. We then extend our result to the case where the query points lie on the graph of a polynomial of constant degree. Our solution is based on the function-inversion technique of Fiat and Naor [11].

Original language | English |
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Article number | 101963 |

Journal | Computational Geometry: Theory and Applications |

Volume | 110 |

DOIs | |

State | Published - Mar 2023 |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier B.V.

### Funding

Work by Boris Aronov 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 Esther Ezra was partially supported by NSF CAREER under grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation . Work by Micha Sharir was partially supported by Grant 260/18 from the Israel Science Foundation .

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

United States-Israel Binational Science Foundation | 824/17, AF-1553354 |

Israel Science Foundation | 260/18 |

## Keywords

- 3SUM-indexing
- Collinearity testing
- Computational geometry
- Function inversion