Finding Pairwise Intersections of Rectangles in a Query Rectangle

Abstract

We consider the following problem: Preprocess a set S of n axis-parallel boxes in \mathbb{R}^d so that given a query of an axis-parallel box Q in \mathbb{R}^d, 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\log^* n+ k\log n) query time using O(n\log n) space.There has been no known result for this problem for higher dimensions, except that for d=3, the best known data structure supports
O(\sqrt{n}+k\log^2\log^* n) 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 \polylog n) query time for any constant 1/d\leq\delta<1.The size of the data structure is O(n^{\delta d}\log n) if 1/d\leq\delta<1/2, or O(n^{\delta d-2\delta+1}) if 1/2\leq \delta<1.