O(log logn)-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 withO(logn)-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 logn). The following five specific problems are investigated:closest pair of points, intersection of convex polygons, intersection of manhattan line segments, dominating set, andlargest empty square. Algorithms solving them are developed which operate in time O(log logn) on the arbitrary CRCW PRAM. The number of processors used is eitherO(n) orO(n logn).This research was supported in part by Grants KBN 2-2044-92-03, KBN 2-2043-92-03, and KBN 2-1190-91-01.