Parallel Algorithms for the Classes of ±2b DESCEND and ASCEND Computations on a SIMD Hypercube

Authors

David Nassimi, New Jersey Institute of Technology

Document Type

Article

Publication Date

1-1-1993

Abstract

There is a wide class of parallel algorithms which require communications in the form of a 2^b permutation and — 2^b permutation. (A 2^b permutation of X elements, where N = 2ⁿ, moves the element from position i to position i + 2^b mod N. The inverse is called a -2^b permutation.) We first examine the complexity of performing a 2^b permutation of X elements on an X -PE SIMD hypercube. (The hypercube model assumes that all data movements are restricted to a single dimension at a time.) We prove that, given any mapping of the elements to processors, log N - b steps are needed to perform a 2^bpermutation. This lower bound suggests that if a parallel algorithm requires f(N) communication steps of type ±2^bit may require Q(f(N) log N) steps on a SIMD hypercube. We identify an important class of parallel computations, called ±2^b descend, which includes Batcher’s odd-even merge and many other algorithms. A computation in this class performs log X iterations on an N -element input A[0: N - 1]. For b =log N - 1. • • •. 0. iteration b computes the new value of each A[i] as a function of A[i]. A[i + 2^b] and A[i — 2^b]. We obtain an efficient O(log N) algorithm for this class on a SIMD hypercube. We discuss several problems which are solved elegantly using this general algorithm. We also study a related class of parallel computations called ±2^b -ascend. A computation in this class has a structure similar to ±2^b-descend, except that the iterations run in the increasing order of b. We develop a simple O(log²N/log log N) algorithm for this class on a SIMD hypercube, assuming Ω(log log N) space per PE. It remains open whether this class can be implemented on a SIMD hypercube in O(log N) time. Index Terms— Cyclic shift, odd-even merge, parallel prefix, ±2^b -ascend, ±2^b -descend, 2^b permutation, PM2I interconnection, SIMD hypercube. © 1993 IEEE

Identifier

0027885676 (Scopus)

Publication Title

IEEE Transactions on Parallel and Distributed Systems

External Full Text Location

https://doi.org/10.1109/71.250118

ISSN

10459219

First Page

1372

Last Page

1381

Issue

12

Volume

4

Grant

SBR-421980

Recommended Citation

Nassimi, David, "Parallel Algorithms for the Classes of ±2b DESCEND and ASCEND Computations on a SIMD Hypercube" (1993). Faculty Publications. 17109.
https://digitalcommons.njit.edu/fac_pubs/17109