Ultrafast Expected Time Parallel Algorithms

Abstract

It has been shown previously that sorting n items into n locations with a polynomial number of processors requires Ω(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 Θ(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.

Approximate parallel prefix computation and its applications

“…Specifically, the approximate prefix sums produced are strictly non-decreasing in such a way that the difference between consecutive computed prefix sums is always not smaller than the difference between the exact prefix sums . We show the utility of this result by deriving a fast randomized parallel algorithm for a "relaxedln but still quite natural, version of the integer sorting problem, known as padded integer sorting [16]. In this version of integer sorting we allow for gaps in the ordered listing, so long as the total space needed for the array containing these elements is still linear.…”