Finding Pairwise Intersections Inside a Query Range

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 $O(n(\mathrm{polylog}n))$ and with query time $O((k+1)(\mathrm{polylog}n))$ 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 $O(n\sqrt{n}(\mathrm{polylog}n))$ and query time $O((\sqrt{n}+k)(\mathrm{polylog}n))$. When the objects and query are fat, we obtain $O((k+1)(\mathrm{polylog}n))$ query time using $O(n(\mathrm{polylog}n))$ storage.