Orthogonal range searching in linear and almost-linear space

Authors:
Yakov Nekrich
Affiliations:
Department of Computer Science, University of Bonn, Germany
Venue:
Computational Geometry: Theory and Applications
Year:
2009

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Abstract

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(logn+klog^@en) time, where k is the size of the answer. Our data structure also supports emptiness and one-reporting queries in O(logn) 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((logn/loglogn)^2) time and a dynamic O(nlogn/loglogn) space data structure for semigroup range queries with O((logn/loglogn)^2) query time.