The Rectangle Enclosure and Point-Dominance Problems Revisited

We consider the problem of reporting the pairwise enclosures in a set of n axesparallel rectangles in IR2, which is equivalent to reporting dominance pairs in a set of n points in IR4. Over a decade ago, Lee and Preparata6 gave an O(nlog2 n +k){time and O(n){space algorithmfor these problems, where k is the numberof reportedpairs. Since that time, the question of whether there is a faster algorithmhas remained an intriguing open problem. Inthispaper,we giveanalgorithmwhichuses O(n)spaceandrunsin O(nlognloglogn+ k loglogn) 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. (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 a...

• Type: Conference
• Year: 1995
• Where: COMPGEOM
• Authors: Prosenjit Gupta, Ravi Janardan, Michiel H. M. Smid, Bhaskar DasGupta
• Keywords: Algorithm, COMPGEOM 1995, Discrete Geometry, Intriguing Open Problem, Open Problem