The Simple Genetic Algorithm and the Walsh Transform: part I, Theory
Michael D. Vose
Computer Science Dept.
The University of Tennessee
Knoxville, TN 37996-1301
vose@cs.utk.edu
Alden H. Wright
Computer Science Dept.
The University of Montana
Missoula, MT 59812-1008
wright@cs.umt.edu
ABSTRACT
This paper is the first part of a two part series. It proves a
number of direct relationships between the Fourier transform and the
simple genetic algorithm. (For a binary representation, the Walsh
transform is the Fourier transform.) The results are of a theoretical
nature and are based on the analysis of mutation and crossover.
The Fourier transform of the mixing matrix is shown to be sparse.
An explicit formula is given for the spectrum of the differential of the
mixing transformation. By using the Fourier representation and the
fast Fourier transform, one generation of the infinite population
simple genetic algorithm can be computed in time $O(c^{\ell\log_2 3})$,
where $c$ is arity of the alphabet and $\ell$ is the string length.
This is in contrast to the time of $O(c^{3\ell})$ for the algorithm
as represented in the standard basis. There are two orthogonal
decompositions of population space which are invariant under mixing.
The sequel to this paper will apply the basic theoretical results
obtained here to inverse problems and asymptotic behavior.