Orthogonal range searching in linear and almost-linear space

Abstract

In this paper we describe space-efficient data structures for two-dimensional range searching problem. We present a dynamic linear space data structure that supports orthogonal range reporting queries in O(log n + k logε 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 linear space data structure that supports range reporting queries in O(log n/ log log n + k logε n) time and emptiness and one-reporting queries in O(log n/log log n) time. This is the first linear space data structure for these problems that achieves 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 semi-group range sum queries with query time O((log n/ log logn)2). © Springer-Verlag Berlin Heidelberg 2007.