Data Structures for Approximate Range Counting

We present new data structures for approximately counting the number of points in an orthogonal range. There is a deterministic linear space data structure that supports updates in O(1) time and approximates the number of elements in a 1-D range up to an additive term k in O(log logU · log log n) time, where k is the number of elements in the answer, U is the size of the universe and c is an arbitrary fixed constant. We can estimate the number of points in a two-dimensional orthogonal range up to an additive term k in O(log logU+(1/ρ) log log n) time for any ρ > 0. We can estimate the number of points in a threedimensional orthogonal range up to an additive term k in O(log logU + (log log n) + (3) log log n) time for v = log 1 ρ / log 3 2 + 2.