Finding Pairwise Intersections Inside a Query Range

TL;DR

This paper introduces efficient data structures for quickly reporting intersecting object pairs within a query range across 2D and 3D spaces, optimizing for various object types and complexities.

Contribution

It presents novel data structures with near-linear size and polylogarithmic query time for detecting intersecting pairs inside a range in 2D and 3D.

Findings

Achieves $O(n({ m polylog} n))$ storage with $O((k+1)({ m polylog} n))$ query time for rectangles and small union complexity objects.

Provides $O(nrac{1}{2}({ m polylog} n))$ storage and $O((rac{1}{2} +k)({ m polylog} n))$ query time for 3D axis-aligned boxes.

Optimizes query time to $O((k+1)({ m polylog} n))$ for fat objects with linear storage.

Abstract

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$ time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog} n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm polylog} n))$ storage.