Data Structures for Approximate Range Counting

TL;DR

This paper introduces new data structures for approximate orthogonal range counting that achieve fast query and update times with controlled additive error, improving efficiency in multidimensional data analysis.

Contribution

The paper presents deterministic, space-efficient data structures for approximate range counting with improved update and query times across multiple dimensions.

Findings

Supports O(1) update time in 1D with additive error k^{1/c}

Provides 2D range count estimates with additive error k^{ ho} in polylogarithmic time

Extends to 3D range counting with similar approximation guarantees and efficient query times

Abstract

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 $v=\log \frac{1}{\rho}/\log {3/2}+2$.