In the range closest pair problem, we want to construct a data structure storing a set S of n points in the plane, such that for any axes-parallel query rectangle R, the closest pair in the set R∩S can be reported. The currently best result for this problem is by Xue et al. (SoCG 2018). Their data structure has size O(nlog 2 ⁡n) and query time O(log 2 ⁡n). We show that a data structure of size O(nlog⁡n) can be constructed in O(nlog⁡n) time, such that queries can be answered in O(log⁡n+flog⁡f) time, where f is the aspect ratio of R. Thus, for fat query rectangles, the query time is O(log⁡n). This result is obtained by reducing the range closest pair problem to standard range searching problems on the points of S.

Closest pairs, Data structures, Fat rectangles, Range searching, Yao graph
Computational Geometry
Computational Geometry Lab

Bae, S.W. (Sang Won), & Smid, M. (2019). Closest-pair queries in fat rectangles. Computational Geometry. doi:10.1016/j.comgeo.2019.05.003