4D Range Reporting in the Pointer Machine Model in Almost-Optimal Time

TL;DR

This paper introduces a nearly space-efficient data structure for four-dimensional orthogonal range reporting that achieves almost-optimal query times, with generalization to higher dimensions in the pointer machine model.

Contribution

It presents the first data structure with nearly-linear space that supports almost-optimal 4D range reporting queries, extending to higher dimensions.

Findings

Queries in 4D are answered in $O( ext{log} n ext{log} ext{log} n + k)$ time.

Space complexity is $O(n ext{log}^4 n)$, nearly linear in the number of points.

The approach generalizes to $d ext{≥} 4$ dimensions with similar query time bounds.

Abstract

In the orthogonal range reporting problem we must pre-process a set $P$ of multi-dimensional points, so that for any axis-parallel query rectangle $q$ all points from $q\cap P$ can be reported efficiently. In this paper we study the query complexity of multi-dimensional orthogonal range reporting in the pointer machine model. We present a data structure that answers four-dimensional orthogonal range reporting queries in almost-optimal time $O(\log n\log\log n + k)$ and uses $O(n\log^4 n)$ space, where $n$ is the number of points in $P$ and $k$ is the number of points in $q\cap P$ . This is the first data structure with nearly-linear space usage that achieves almost-optimal query time in 4d. This result can be immediately generalized to $d\ge 4$ dimensions: we show that there is a data structure supporting $d$-dimensional range reporting queries in time $O(\log^{d-3} n\log\log n+k)$ for any constant $d\ge 4$.