# Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

## TL;DR

This paper presents efficient data structures for orthogonal range reporting and rectangle stabbing problems specifically optimized for fat rectangles, achieving improved query times and space complexity compared to general cases.

## Contribution

The paper introduces novel data structures for orthogonal range reporting and rectangle stabbing tailored for fat rectangles, with significantly better performance than existing solutions.

## Key findings

- 2D range reporting queries answered in O(log log U + k) time with O(n) space.
- 3D range reporting queries answered in O(log log U + k) time with O(n log^ε U) space.
- Rectangle stabbing queries in 3D answered in O(log U log log U + k) time with O(n) space.

## Abstract

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.