Orthogonal range searching on the RAM, revisited

Abstract

We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: We present two data structures for 2-d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves the previous results by Alstrup, Brodal, and Rauhe (FOCS'00), with O(n lgε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg2lg n) query time. Our second data structure uses O(n) space and answers queries in O(lgε n) time. The best previous O(n)-space data structure, due to Nekrich (WADS'07), answers queries in O(lg n/lg lg n) time. We give a data structure for 3-d orthogonal range reporting with O(n lg1+ε n) space and O(lg lg n + k) query time for points in rank space, for any constant ε>0. This improves the previous results by Afshani (ESA'08), Karpinski and Nekrich (COCOON'09), and Chan (SODA'11), with O(n lg3 n) space and O(lg lg n + k) query time, or with O(n lg1+εn) space and O(lg2lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.