Year: 2020
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Nekrich, Yakov
Institution: Michigan Technological University
Language: English
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3116584312