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

Abstract

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 $d=c\log n$.