Alden H. Wright

Efficient Linkage Discovery by Limited Probing

This paper addresses the problem of discovering the structure of a fitness function from binary strings to the reals under the assumption of bounded epistasis. Two loci (string positions) are epistatically linked if the effect of changing the allele (value) at one locus depends on the allele at the other locus. Similarly, a group of loci are epistatically linked if the effect of changing the allele at one locus depends on the alleles at all other loci of the group. Under the assumption that the size of such groups of loci are bounded, and assuming that the function is given only as a black box function, this paper presents and analyzes a randomized algorithm that finds the complete epistatic structure of the function in the form of the Walsh coefficients of the function.

Simple genetic algorithms with linear fitness

A general form of stochastic search is described (random heuristic search), and some of its general properties are proved. This provides a framework in which the simple genetic algorithm (SGA) is a special case. The framework is used to illuminate relationships between seemingly different probabilistic perspectives of SGA behavior. Next, the SGA is formalized as an instance of random heuristic search. The formalization then used to show expected population fitness is a Lyapunov function in the infinite population model when mutation is zero and fitness is linear. In particular, the infinite population algorithm must converge, and average population fitness increases from one generation to the next. The consequence for a finite population SGA is that the expected population fitness increases from one generation to the next. Moreover, the only stable fixed point of the expected next population operator corresponds to the population consisting entirely of the optimal string. This result is then extended by way of a perturbation argument to allow nonzero mutation.

The simple genetic algorithm and the walsh transform: Part i, theory

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(cl log2 3), where c is arity of the alphabet and l is the string length. This is in contrast to the time of O(c3l) for the algorithm as represented in the standard basis. There are two orthogonal decompositions of population space that are invariant under mixing. The sequel to this paper will apply the basic theoretical results obtained here to inverse problems and asymptotic behavior.

The simple genetic algorithm and the walsh transform: Part ii, the inverse

This paper continues the development, begun in Part I, of the relationship between the simple genetic algorithm and the Walsh transform. The mixing scheme (comprised of crossover and mutation) is essentially triangularized when expressed in terms of the Walsh basis. This leads to a formulation of the inverse of the expected next generation operator. The fixed points of the mixing scheme are also determined, and a formula is obtained giving the fixed point corresponding to any starting population. Geiringers theorem follows from these results in the special case corresponding to zero mutation.

The computational complexity of N-K fitness functions

N-K fitness landscapes have been used widely as examples and test functions in the field of evolutionary computation. We investigate the computational complexity of the problem of optimizing the N-K fitness functions and related fitness functions. We give an algorithm to optimize adjacent-model N-K fitness functions, which is polynomial in N. We show that the decision problem corresponding to optimizing random-model N-K fitness functions is NP-complete for K>1, and is polynomial for K=1. If the restriction that the ith component function depends on the ith bit is removed, then the problem is NP-complete, even for K=1. We also give a polynomial-time approximation algorithm for the arbitrary-model N-K optimization problem.

Group properties of crossover and mutation

It is supposed that the finite search space has certain symmetries that can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries, then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on to induce a group structure on itself.

Approximate string matching using within-word parallelism

Given a text string, a pattern string, and an integer k, the problem of approximate string matching with k differences is to find all substrings of the text string whose edit distance from the pattern string is less than k. The edit distance between two strings is defined as the minimum number of differences, where a difference can be a substitution, insertion, or deletion of a single character. An implementation of the dynamic programming algorithm for this problem is given that packs several characters and mod‐4 integers into a computer word. Thus, it is a parallelization of the conventional implementation that runs on ordinary processors. Since a small alphabet means that characters have short binary codes, the degree of parallelism is greatest for small alphabets and for processors with long words. For an alphabet of size 8 or smaller and a 64 bit processor, a 21‐fold parallelism over the conventional algorithm can be obtained. Empirical comparisons to the basic dynamic programming algorithm, to a version of Ukkonens algorithm, to the algorithm of Galil and Park, and to a limited implementation of the Wu‐Manber algorithm are given.

Allele Diffusion in Linear Genetic Programming and Variable-Length Genetic Algorithms with Subtree Crossover

In this paper we study, theoretically, the search biases produced by GP subtree crossover when applied to linear representations, such as those used in linear GP or in variable length GAs. The study naturally leads to generalisations of Geiringers theorem and of the notion of linkage equilibrium, which, until now, were applicable only to fixed-length representations. This indicates the presence of a diffusion process by which, even in the absence of selective pressure and mutation, the alleles in a particular individual tend not just to be swapped with those of other individuals in the population, but also to diffuse within the representation of each individual. More precisely, crossover attempts to push the population towards distributions of primitives where each primitive is equally likely to be found in any position in any individual.

Form Invariance and Implicit Parallelism

Hollands schema theorem (an inequality) may be viewed as an attempt to understand genetic search in terms of a coarse graining of the state space. Stephens and Waelbroeck developed that perspective, sharpening the schema theorem to an equality. Of particular interest is a form invariance of their equations; the form is unchanged by the degree of coarse graining. This paper establishes a similar form invariance for the more general model of Vose et al. and uses the attendant machinery as a springboard for an interpretation and discussion of implicit parallelism.

An Estimation of Distribution Algorithm Based on Maximum Entropy

Estimation of distribution algorithms (EDA) are similar to genetic algorithms except that they replace crossover and mutation with sampling from an estimated probability distribution. We develop a framework for estimation of distribution algorithms based on the principle of maximum entropy and the conservation of schema frequencies. An algorithm of this type gives better performance than a standard genetic algorithm (GA) on a number of standard test problems involving deception and epistasis (i.e. Trap and NK).