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Tight Bounds on the Number of Closest Pairs in Vertical Slabs

par Biniaz, Ahmad;Bose, Prosenjit ;Chung, Chaeyoon;De Carufel, Jean Lou;Iacono, John ;Maheshwari, Anil;Odak, Saeed;Smid, Michiel M.;Tóth, Csaba Dégi
Référence arXiv.org, arXiv:2502.17600
Publication Publié, 2025-03-30

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Résumé :

Let be a set of points in , where is a constant, and let be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly points of are between any two successive hyperplanes. Let be the number of different closest pairs in the vertical slabs that are bounded by and , over all . We prove tight bounds for the largest possible value of , over all point sets of size , and for all values of . As a result of these bounds, we obtain, for any constant , a data structure of size , such that for any vertical query slab , the closest pair in the set can be reported in time. Prior to this work, no linear space data structure with sublinear query time was known.