# The time complexity of genetic algorithms and the theory of recombination operators

## Description

Genetic Algorithms (GAs) are a directed randomized parallel search method with optimal time in a generic search space. Because GAs are directed by domain information, the speed of solution is a function of the characteristics of the domain. This study finds the relationship between the domain characteristic termed the fitness ratio and the time complexity of GAs with proportional selection. The fitness ratio is a measure of the relative merit of one sampled set of solutions over the relative merit of the other sampled solutions. The relationship between fitness ratio and time complexity is proved first with induction and second with methods from finite difference equations for binary and nonbinary GAs. Each domain uses some procedure to evaluate merit of a candidate solution. Let $O$(e) denote the complexity of this procedure. The complexity of GAs is on the order of ($O$(e))(m ln m)/(ln (fitness ratio)), where m is the size of the array processed GAs use randomized operators called crossover operators to combine partial solutions. This study extends GA theory to accommodate the uniform crossover operator. It is shown that GAs using the uniform crossover operator process on the order of m$\sp3$ schemata, which validates that the uniform crossover operator preserves implicit parallelism. Further, the schema theorem is altered by modifying the disruption factor to accommodate uniform crossover. This suggests that research should focus on low order schemata, versus short defining length schemata. Two properties of crossover operators are identified and quantified. Allele positional bias is the relationship between the position within the string of a partial solution and its frequency of exchange with other candidate solutions. Allele neighborhood bias is the relationship between the likelihood of alleles remaining together after crossover and distance between alleles. The allele neighborhood bias of the simple crossover operator is shown to cause premature convergence. Extending GA theory to accommodate the uniform crossover operator is significant because this operator is more efficient in a number of domains