Approximate parallel prefix computation and its applications

Abstract

In this paper we address two fundamental problems in parallel algorithm design-parallel prefix sums and integer sorting-and show that both of them can be approximately solved very quickly on a randomized CRCW PRAM. In the case of prefix sums the approximation is in terms of the accuracy of the sums and in the case of integer sorting it is in terms of allowing some gaps between consecutive elements in the ordered list. By introducing approximation in these ways we are able to solve these problems in o(lg lg n) time, and thus avoid the near-logarithmic lower bounds by Beame and H b t a d that hold for the exact versions of these problems. Nevertheless, we demonstrate that these approximations are strong enough to be used as subroutines in fast randomized algorithms for some well-known problems in parallel computational geometry. Perhaps the most succinct way to describe the power of the new tools which are presented is by observing that prior to this work it was known how to solve the interval allocation problem fast. In the present work we show how to solve the ordered version of the problem.

Improved Parallel Integer Sorting without Concurrent Writing

“…While re cent research has concentrated on algorithms for the CRCW PRAM (Rajasekaran and Reif, 1989;Rajasekaran and Sen, 1992;Bhatt et al, 1991;Vishkin, 1991a, 1991b;Raman, , 1991aRaman, , 1991bHagerup, 1991;Gi! et al, 1991;Hagerup andRaman, 1992, 1993;Bast and Hagerup, 1993;Goodrich et al, 1993Goodrich et al, , 1994, in this paper we are interested in getting the most out of the weaker EREW and CREW PRAM models. Consider the problem of sorting n integers in the range 1 .. m. In view of sequential radix sorting, which works in linear time if m = nO(l), a parallel algorithm for this most interesting range of m is optimal only if its timeprocessor product is O(n).…”

Practical trade‐offs for the prefix‐sum problem

“…Third, a different line of research studied the problem in the parallel‐computing setting where the solution is computed using p > 1 processors 19,20 or using specialized/dedicated hardware 21,22 . As already mentioned in Section 1, we entirely focus on single‐core solutions that should run on commodity hardware.…”