Finding Pairwise Intersections of Rectangles in a Query Rectangle

We consider the following problem: Preprocess a set S of n axis-parallel boxes in R so that given a query of an axis-parallel box in R, the pairs of boxes of 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( √ n log n+ k log n) query time using O(n √ n log n) space. For a constant d > 2, we present a data structure supporting O(n1−δ logd−1 n + k polylog n) query time for any constant 1/d ≤ δ < 1. The size of the data structure is O(nδd−2δ+1 log n).