## Abstract

We consider a range-search variant of the closest-pair problem. Let Γ be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Γ, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Γ is a polygon (possibly with holes) and when Γ is a general convex body whose boundary is smooth. When Γ is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Γ is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log^{2} n) query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].

Original language | English (US) |
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Pages (from-to) | 26-61 |

Number of pages | 36 |

Journal | Journal of Computational Geometry |

Volume | 11 |

Issue number | 2 |

State | Published - 2020 |

### Bibliographical note

Funding Information:A preliminary version of the paper appeared in the Proceedings of the 35th International Symposium on Computational Geometry (SoCG 2019). The research of Jie Xue was supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota.

Funding Information:

∗A preliminary version of the paper appeared in the Proceedings of the 35th International Symposium on Computational Geometry (SoCG 2019). The research of Jie Xue was supported, in part, by a Doctoral Dissertation Fellowship from the Graduate School of the University of Minnesota. †University of Minnesota - Twin Cities, MN, USA, xuexx193@umn.edu ‡Facebook Inc., Seattle, WA, USA, lydxlx@fb.com §University of Illinois at Urbana-Champaign, IL, USA, saladi.rahul@gmail.com ¶University of Minnesota - Twin Cities, MN, USA, janardan@umn.edu

Publisher Copyright:

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