## Orthogonal Range Searching on the RAM, Revisited (2011)

Citations: | 15 - 4 self |

### BibTeX

@MISC{Chan11orthogonalrange,

author = {Timothy M. Chan and Kasper Green Larsen and Mihai Pǎtrascu},

title = {Orthogonal Range Searching on the RAM, Revisited},

year = {2011}

}

### OpenURL

### 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: 1. 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 lg ε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. Our second data structure uses O(n) space and answers queries in O(lg ε 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. 2. We give a data structure for 3-d orthogonal range reporting with O(n lg 1+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 lg 3 n) space and O(lg lg n + k) query time, or with O(n lg 1+ε n) space and O(lg 2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.