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 polylog n) and with query time O((k + 1) polylog n), 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 ℝ3, we present a data structure of size O(n √ n polylog n) and query time O((√ n + k) polylog n). When the objects and query are fat, we obtain O((k +1) polylog n) query time using O(n polylog n) storage.