Adaptive and approximate orthogonal range counting

Abstract

We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model.• It is well known that there are linear-space data structures for 2-D orthogonal range counting with worst-case optimal query time O(logwn). We give an O(n log log n)-space adaptive data structure that improves the query time to O(log log n + logwk), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan, Larsen, and Patrascu, SoCG 2011].• We give an O(n log log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in O(log log n) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem.• Lastly, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jorgensen and Larsen [SODA 2011] presented a linear-space adaptive data structure with query time O(log log n + logwk). We give a new linear-space structure that improves the query time to O(1 + logwk), exactly matching the lower bound proved by Jorgensen and Larsen.