Algorithms explained by symmetries

Abstract

Many of the linear transforms that are used in digital signalprocessing and other areas have a lot of symmetry properties. This makes the development of fast algorithms for them possible. The usual quadratic cost of multiplying with the matrix is reduced, in some cases to an almost linear complexity. Salient examples are the Fourier transform, the Cosine transform, linear maps with Toepliz matrices or convolutions. The article gives an exact definition of the notion of symmetry that leads to fast algorithms and presents a method to construct those algorithms automatically in the case of an existing symmetry with a soluble group. The results may serve to speed up the multiplication with a transform matrix and also to solve a linear system of equations with symmetry. Even though the construction is done at the level of abstract algebra, the derived algorithms for many linear transforms compare well with the best found in the literature [CoTu65, ElRa82, Ra68, RaYi90]. In most cases, where the new method was applicable, even the manually optimized algorithms [Nu81] were not better, while nothing more than the transform matrix and its symmetry were provided here to optain the results.

Research sponsored by DFG through the Graduiertenkolleg “Beherrschbarkeit komplexer Systeme”

The work presented here has become part of the IDEAS system that was written at the IAKS in Karlsruhe. That system is a tool to derive realizations for mathematically well structured problems on many implementation technologies such as C-programs for general purpose processors or VLSI layouts. I wish to thank my advisor T. Beth and my colleague A. Nückel for their support and comments.