Ultra - Fast Expected TimeParallel

It has been shown previously that sorting n items into n locations with a polynomial number of processors requires (log n= log log n) time. We sidestep this lower bound with the idea of Padded Sorting, or sorting n items into n+o(n) locations. Since many problems do not rely on the exact rank of sorted items, a Padded Sort is often just as useful as an unpadded sort. Our algorithm for Padded Sort runs on the Tolerant CRCW PRAM and takes (log log n= log log log n) expected time using n log log log n= log log n processors, assuming the items are taken from a uniform distribution. Using similar techniques we solve some computational geometry problems, including Voronoi Diagram, with the same processor and time bounds, assuming points are taken from a uniform distribution in the unit square. Further, we present an Arbitrary CRCW PRAM algorithm to solve the Closest Pair problem in constant expected time with n processors regardless of the distribution of points. All of these algorithms achieve linear speedup in expected time over their optimal serial counterparts.