## Abstract

Let P be a set of m points in R^{2}, let Σ be a set of n semi-algebraic sets of constant complexity in R^{2}, let (S, +) be a semigroup, and let w : P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P ∩ σ) for every σ ∈ Σ in overall expected 2s time O^{∗(m5s−4 n5 5 s s − − 6 4} + m^{2}/^{3}n^{2}/^{3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^{∗}(·) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an 2s−2 s on-line manner. The latter takes O^{∗}(m^{2}s^{−1} n^{2}s^{−1} + m + n) time. Let Φ: Σ × P → {0, 1} be the Boolean predicate (of constant complexity) such that Φ(σ, p) = 1 if p ∈ σ and 0 otherwise, and let Σ Φ P = {(σ, p) ∈ Σ × P | Φ(σ, p) = 1}. Our algorithm actually computes a partition B_{Φ} of Σ Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes 2s of the vertex sets of its bicliques) O^{∗(m5s−4 n5 5 s s − − 4 6} + m^{2}/^{3}n^{2}/^{3} + m + n). It is straightforward to compute w(P ∩ σ) for all σ ∈ Σ from B_{Φ}. Similarly, if η : Σ → S is a weight function on the regions of Σ, ^{P}_{σ}∈_{Σ:p∈σ} η(σ), for every point p ∈ P, can be computed from B_{Φ} in a straightforward manner.

Original language | English |
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Title of host publication | 40th International Symposium on Computational Geometry, SoCG 2024 |

Editors | Wolfgang Mulzer, Jeff M. Phillips |

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

ISBN (Electronic) | 9783959773164 |

DOIs | |

State | Published - Jun 2024 |

Event | 40th International Symposium on Computational Geometry, SoCG 2024 - Athens, Greece Duration: 11 Jun 2024 → 14 Jun 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 293 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 40th International Symposium on Computational Geometry, SoCG 2024 |
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Country/Territory | Greece |

City | Athens |

Period | 11/06/24 → 14/06/24 |

### Bibliographical note

Publisher Copyright:© Pankaj K. Agarwal, Esther Ezra, and Micha Sharir.

## Keywords

- Range-searching
- duality
- geometric cuttings
- pseudo-lines
- semi-algebraic sets