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Tight Bounds on the Number of Closest Pairs in Vertical Slabs

Authors Ahmad Biniaz, Prosenjit Bose , Chaeyoon Chung, Jean-Lou De Carufel , John Iacono , Anil Maheshwari , Saeed Odak, Michiel Smid, Csaba D. Tóth

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  • Theory of computation → Communication complexity
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • closest pair
  • vertical slab
  • data structure

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Abstract

Let S be a set of n points in ℝ^d, where d ≥ 2 is a constant, and let H₁,H₂,…,H_{m+1} be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly n/m points of S are between any two successive hyperplanes. Let |A(S,m)| be the number of different closest pairs in the {(m+1) choose 2} vertical slabs that are bounded by H_i and H_j, over all 1 ≤ i < j ≤ m+1. We prove tight bounds for the largest possible value of |A(S,m)|, over all point sets of size n, and for all values of 1 ≤ m ≤ n. 
As a result of these bounds, we obtain, for any constant ε > 0, a data structure of size O(n), such that for any vertical query slab Q, the closest pair in the set Q ∩ S can be reported in O(n^{1/2+ε}) time. Prior to this work, no linear space data structure with sublinear query time was known.

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Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung, Jean-Lou De Carufel, John Iacono, Anil Maheshwari, Saeed Odak, Michiel Smid, and Csaba D. Tóth. Tight Bounds on the Number of Closest Pairs in Vertical Slabs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.8

Author Details

Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Chaeyoon Chung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, Republic of Korea
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
John Iacono
  • Department of Computer Science, Université libre de Bruxelles, Brussels, Belgium
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Odak
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

  • Biniaz, Ahmad: Research supported by NSERC.
  • Bose, Prosenjit: Research supported by NSERC.
  • Chung, Chaeyoon: Research supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT).
  • De Carufel, Jean-Lou: Research supported by NSERC.
  • Iacono, John: This work was supported by the Fonds de la Recherche Scientifique-FNRS.
  • Maheshwari, Anil: Research supported by NSERC.
  • Odak, Saeed: Research supported by NSERC.
  • Smid, Michiel: Research supported by NSERC.
  • Tóth, Csaba D.: Research supported in part by the NSF award DMS-2154347.

Acknowledgements

Research on this paper was conducted, in part, at the Eleventh Annual Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Barbados, March 8-15, 2024. The authors are grateful to the organizers and to the participants of this workshop.

References

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