Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back

We revisit the orthogonal range searching problem and the exact \(\ell _\infty \) nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of \(\log n\). Specifically, the preprocessing time and space are \(O(n^{1+\delta })\) for any constant \(\delta >0\), and the expected query time is \(n^{1-1/O(c\log c)}\) for \(d=c\log n\). The data structure is simple and is based on a new “augmented, randomized, lopsided” variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [(Impagliazzo et al. in arXiv:1401.5512, 2014) and (Chan in Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication. In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an \(n^{2-1/O(\log c)}\)-time non-combinatorial algorithm [Abboud et al. in Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015]. This reduction is also simple and is based on range trees. Finally, we use a similar approach to obtain a small improvement to Indyk’s data structure (J Comput Syst Sci 63(4):627–638, 2001) for approximate\(\ell _\infty \) nearest neighbor search when \(d=c\log n\).