New Data Structures for Orthogonal Range Reporting and Range Minima Queries

TL;DR

This paper introduces new space-efficient data structures for orthogonal range minima and reporting queries, achieving improved query times and extending to higher dimensions, advancing the efficiency of multidimensional range searching.

Contribution

It presents novel data structures that improve query times and space complexity for orthogonal range minima and reporting problems, including the first optimal-time structure for 4D range reporting.

Findings

Range minima queries in 2D with O(n) space and O(log^ε n) time

Modified data structure with O(n log log n) space and O(log log n) time

Optimal O(log n / log log n + k) query time for 4D range reporting

Abstract

In this paper we present new data structures for two extensively studied variants of the orthogonal range searching problem. First, we describe a data structure that supports two-dimensional orthogonal range minima queries in $O(n)$ space and $O(\log^{\varepsilon} n)$ time, where $n$ is the number of points in the data structure and $\varepsilon$ is an arbitrarily small positive constant. Previously known linear-space solutions for this problem require $O(\log^{1+\varepsilon} n)$ (Chazelle, 1988) or $O(\log n\log \log n)$ time (Farzan et al., 2012). A modification of our data structure uses space $O(n\log \log n)$ and supports range minima queries in time $O(\log \log n)$. Both results can be extended to support three-dimensional five-sided reporting queries. Next, we turn to the four-dimensional orthogonal range reporting problem and present a data structure that answers queries in optimal $O(\log n/\log \log n + k)$ time, where $k$ is the number of points in the answer. This is the first data structure that achieves the optimal query time for this problem. Our results are obtained by exploiting the properties of three-dimensional shallow cuttings.