Orthogonal range searching in linear and almost-linear space

In this paper we describe space-efficient data structures for the two-dimensional range searching problem. We present a dynamic linear space data structure that supports orthogonal range reporting queries in O (log n + k log(epsilon) n) time, where k is the size of the answer. Our data structure also supports emptiness and one-reporting queries in O (log n) time and thus achieves optimal time and space for this type of queries. In the case of integer point coordinates, we describe a static and a randomized dynamic linear space data structures that support range reporting, emptiness and one-reporting queries in sub-logarithmic time. These are the first linear space data structures for these problems that achieve sub-logarithmic query time. We also present a dynamic linear space data structure for range counting queries with O((log n/log log n)(2)) time and a dynamic O(n log n/log log n) space data structure for semigroup range queries with O((log n/log log n)(2)) query time. (c) 2008 Elsevier B.V. All rights reserved.