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(logn+klog ε n) 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 linear space data structure that supports range reporting queries in O(logn/loglogn+klog ε n) time and emptiness and one-reporting queries in O(logn/loglogn) 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((logn/loglogn)2) time and a dynamic O(nlogn/loglogn) space data structure for semi-group range sum queries with query time O((logn/loglogn)2).
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