Data Structures for Approximate Range Counting

We present new data structures for approximately counting the number of points in 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 $k^{1/c}$ in $O(\log \log U\cdot\log \log n)$ 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 $k^{\rho}$ in $O(\log \log U+ (1/\rho)\log\log n)$ time for any $\rho>0$. We can estimate the number of points in a three-dimensional orthogonal range up to an additive term $k^{\rho}$ in $O(\log \log U + (\log\log n)^3+ (3^v)\log\log n)$ time for $v=\log \frac{1}{\rho}/\log {3/2}+2$.