On modeling genetic algorithms for functions of unitation

Abstract

We discuss a novel model for analyzing the working of Genetic Algorithms (GAs), when the objective function is a function of unitation. The model is exact (not approximate), and is valid for infinite populations. Functions of unitation depend only on the number of 1's in any string. Hence, we only need to model the variations in the distribution of strings with respect to the number of 1's in the strings. We introduce the notion of a Binomial Distributed Population (BDP) as the building block of our model, and we show that the effect of uniform crossover on BDPs is to generate two other BDPs. We demonstrate that a population with any general distribution may be decomposed into several BDPs. We also show that a general multipoint crossover may be considered as a composition of several uniform crossovers. Based on these results, the effects of mutation and crossover on the distribution of strings have been characterized, and the model has been defined. GASIM-a Genetic Algorithm Simulator for functions of unitation-has been implemented based on the model, and the exactness of the results obtained from GASIM has been verified using actual Genetic Algorithm runs. The time complexity of the GA simulator derived from the model is O(l³) (where l is the string length), a significant improvement over previous models with exponential time complexities.