The rectangle enclosure and point-dominance problems revisited

We consider the problem of reporting the pairwise enclosures among a set of n axes parallel rectangles in IR which is equivalent to reporting dominance pairs in a set of n points in IR For more than ten years it has been an open problem whether these problems can be solved faster than in O n log n k time where k denotes the number of reported pairs First we give a divide and conquer algorithm that matches the O n space and O n log n k time bounds of the algorithm of Lee and Preparata LP but is simpler Then we give another algorithm that uses O n space and runs in O n logn log logn k log logn time For the special case where the rectangles have at most di erent aspect ratios we give an algorithm that runs in O n logn k time and uses O n space