O(log log n)-time integer geometry on the CRCW PRAM

We study problems in computational geometry on PRAMs under the assumption that input objects are specified by points with O(log n)-bit coordinates, or, equivalently, with polynomially bounded integer coordinates. We show that in this setting many geometric problems can be solved in time O(log log n). The following five specific problems are investigated:closest pair of points, intersection of convex polygons, intersection of manhattan line segments, dominating set, and largest empty square. Algorithms solving them are developed which operate in time O(log log n) on the arbitrary CRCW PRAM. The number of processors used is either O(n) or O(n log n).