Finding Pairwise Intersections of Rectangles in a Query Rectangle

TL;DR

This paper develops efficient data structures for finding pairs of intersecting axis-parallel boxes within a query box in multi-dimensional space, improving query times for 2D and extending solutions to higher dimensions.

Contribution

It introduces a new data structure for 2D with faster query time and generalizes the approach to higher dimensions with sublinear query time complexities.

Findings

2D query time improved to O(log n + k)

Higher dimensions achieve sublinear query times

Data structure size is polynomial in n for fixed dimensions

Abstract

We consider the following problem: Preprocess a set $\mathcal{S}$ of $n$ axis-parallel boxes in $\mathbb{R}^d$ so that given a query of an axis-parallel box in $\mathbb{R}^d$, the pairs of boxes of $\mathcal{S}$ whose intersection intersects the query box can be reported efficiently. For the case that $d=2$, we present a data structure of size $O(n\log n)$ supporting $O(\log n+k)$ query time, where $k$ is the size of the output. This improves the previously best known result by de Berg et al. which requires $O(\log n+ k\log n)$ query time using $O(n\log n)$ space. There has been no result known for this problem for higher dimensions, except that for $d=3$, the best known data structure supports $O(\sqrt{n}\log^2n+k\log^2n)$ query time using $O(n\sqrt {n}\log n)$ space. For a constant $d>2$, we present a data structure supporting $O(n^{1-\delta}\log^{d-1} n + k \text{ polylog } n)$ query time for any constant $1/d\leq\delta<1$. The size of the data structure is $O(n^{\delta d - 2\delta + 1}\log n)$.