THE RECTANGLE ENCLOSURE AND POINT DOMINANCE PROBLEMS REVISITED PROSENJIT GUPTA and RAVI JANARDAN

We consider the problem of reporting the pairwise enclosures in a set of n axes parallel rectangles in IR which is equivalent to reporting dominance pairs in a set of n points in IR Over a decade ago Lee and Preparata 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 algorithmwhich usesO n space and runs inO n logn log logn k log logn 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 rst result since Ref 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 but which is simpler Finally we consider the special case where the rectangles have a small number of di erent aspect ratios which is often the case in practice For this problem we give an algorithm which runs in O n logn k time and uses O n space