The Rectangle Enclosure and Point-Dominance Problems Revisited

Abstract

We consider the problem of reporting the pairwise enclosures in a set of n axes-parallel rectangles in ℝ 2, which is equivalent to reporting dominance pairs in a set of n points in ℝ 4. Over a decade ago, Lee and Preparata7 gave an O(n log 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 uses O(n + k) space and runs in O(n log n log log n + k log log n) time. 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 Ref. (6) 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 Ref. (6), but which is simpler. Finally, we consider the special case where the rectangles have a small number, α, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(α n log n + k) time and uses O(n) space.