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.