Finding the maximum, merging and sorting in a parallel computation model

Abstract

A model for synchronized parallel computation is described in which all p processors have access to a common memory. this model is used to solve the problems of finding the maximum, merging, and sorting by p processors.

The main results are:

• 1. Finding the maximum of n elements (1<p≦n) within a depth of 0(n/p+log log p) (optimal for p≦n/log log n).

• 2. Merging two sorted lists of length m and n (m≦n) within a depth of 0(n/p+log n) for p≦n (optimal for p≦n/log n), 0(log m/log p/n) for p≧n (=0(k) if p=[m^1/kn], k>1.

• 3. Sorting n elements within a depth of 0(n/p log n + log n log p) for p≦n, (optimal for p≦n/log n). 0(log² n/log p/n + log n) for p ≧ n (=0(k log n) if p=[n^1+1/k], k>1).

The depth of 0(k log n) for p = [n^1+1/k] processors was also achieved by Hirschberg [Hi78] and Preparata [P78]. Our algorithm is substantially simpler.

All the elementary operations including allocation of processors to their jobs are taken into account in deriving the depth complexity and not only comparisons.