Ultafast Expected Time Parallel Algorithms

It has been shown previously that sorting n items into n locations with a polynomial number of processors requires (logn= log logn) 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 logn) expected time using n log log log n= log logn 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 CRCWPRAM 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. Research done while at the University of Michigan and supported by an AT&T Fellowship. Supported by NSF/DARPA grant CCR-9004727.