The rectangle enclosure and point-dominance problems revisited

We consider the problem of reporting the pairwise enclosures in a set of n axes-parallel rectangles in IR², which is equivalent to reporting dominance pairs in a set of n points in IR⁴. Over a decade ago, Lee and Preparata [LP82] gave an O(n log² n + k)-time and O(n)-space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithm which runs in O (n log n log log n + k log log n) time and uses O(n) space. Thus, although our result is not a strict improvement over the Lee-Preparata algorithm for the full range of k, it is, nevertheless, the first result since [LP82] to make any progress on this long-standing open problem. Our algorithm is based on the divide-and-conquer paradigm. The heart of the algorithm is the solution to a red-blue dominance reporting problem (the "merge" step). We give a novel solution for this problem which is based on the iterative application of a sequence of non-trivial sweep routines. This solution technique should be of independent interest. We also present another algorithm whose bounds match the bounds given in [LP82], but which is simpler. Finally, we consider the special case where the rectangles have at most a constant number, a, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(αnlog n + k) time and uses O(n) space.