Fast Digit‐Index Permutations

Abstract

We introduce a tensor sum which is useful for the design and analysis of digit-index permutations (DIPs) algorithms. Using this operation we obtain a new high-performance algorithm for the family of DIPs. We discuss an implementation in the applicative language Sisal and show how different choices of parameters yield different DIPs. The efficiency of the special case of digit reversal is illustrated with performance results on a Cray C-90.

Parallel fast Fourier transform in SPMD style of Cilk

“…Data reordering in FFT program using bit-reverse of array index has been widely studied (Bollman et al, 1996;Karp, 1996;Lokhmotov and Mycroft, 2007;Rodriguez, 1988;Rubio et al, 2002;Seguel et al, 2000;Zhang and Zhang, 2000). Most of algorithms proposed were designed to be executed on single processor platforms (Karp, 1996;Lokhmotov and Mycroft, 2007;Rodriguez, 1988;Rubio et al, 2002).…”

A framework for the design and implementation of FFT permutation algorithms

“…Then, the tensor sum [2] of and is defined to be The Hadamard product is associative and commutative. Kronecker products and tensor sums are both associative, but not commutative.…”

“…In fact, specializing rev ns and rev ps to this case gives the algorithms of Elster [5] and the ªtree digitreversalº (tdr) algorithm of the authors [2], respectively. Method 4 is also applicable in this case and gives the ªrecursive divide and conquerº (rd) digit-reversal of the authors [11].…”

“…The first parallel bit-reversal algorithm was developed by Elster [5], whose algorithm requires ylog n parallel steps and yn processors. The authors [11], [2], [12] have developed bit-reversal algorithms that require only ylog log n parallel steps and yn processors and have shown [2] that the vector representation of any index-digit permutation can be computed using no more than ylog log n parallel steps and yn processors.…”

“…The mathematical semantics of functional languages greatly simplify the programming of mathematical formulations. Moreover, Sisal's high-performance, optimizing compiler eliminates the overhead of functional semantics and automatically parallelizes and vectorizes source code [2], [3], [6]. Using Sisal permits us to evaluate the algorithms' true parallel and vector performance with minimal programming effort.…”