The study of butterflies

Abstract

Butterfly networks arise in many signal processing circuits and in parallel algorithms for many sorts of message-passing computers. This paper attempts to explain why this should be, and what butterfly networks are, using a new and elegant formulation based on a language of relations.

Most of the material covered by this paper has appeared in a less tractable form in earlier papers [6, 7]. The novelty here is in the simplicity and elegance of the presentation, which derives from an appropriate choice of high-level structures. These structures are represented by functions which are used to compose circuits from components, and are chosen to have simple mathematical properties.

This presentation makes it easier to explain how the design comes about, showing that butterflies are natural implementations of divide-and-conquer algorithms. We are then able to go on to explain many of the properties of butterfly networks, and of their implementations.