Space Efficient Multi-Dimensional Range Reporting

TL;DR

This paper introduces a space-efficient data structure for 3D range reporting that improves space usage over previous methods with a minimal increase in query time, enabling more scalable multi-dimensional data management.

Contribution

The authors present a novel data structure that reduces space complexity for 3D range reporting while maintaining near-optimal query times, improving upon prior work.

Findings

Supports 3D range queries in $O( ext{log log } U + ( ext{log log } n)^3 + k)$ time.

Uses $O(n ext{log}^{1+ ext{eps}} n)$ space, improving space efficiency.

Achieves faster static and incremental data structures for $d extgreater= 3$.

Abstract

We present a data structure that supports three-dimensional range reporting queries in $O(\log \log U + (\log \log n)^3+k)$ time and uses $O(n\log^{1+\eps} n)$ space, where $U$ is the size of the universe, $k$ is the number of points in the answer,and $\eps$ is an arbitrary constant. This result improves over the data structure of Alstrup, Brodal, and Rauhe (FOCS 2000) that uses $O(n\log^{1+\eps} n)$ space and supports queries in $O(\log n+k)$ time,the data structure of Nekrich (SoCG'07) that uses $O(n\log^{3} n)$ space and supports queries in $O(\log \log U + (\log \log n)^2 + k)$ time, and the data structure of Afshani (ESA'08) that uses $O(n\log^{3} n)$ space and also supports queries in $O(\log \log U + (\log \log n)^2 + k)$ time but relies on randomization during the preprocessing stage. Our result allows us to significantly reduce the space usage of the fastest previously known static and incremental $d$-dimensional data structures, $d\geq 3$, at a cost of increasing the query time by a negligible $O(\log \log n)$ factor.