The Mixing Rate of Different Crossover Operators

Abstract

This chapter illustrates the mixing effect of crossover with the help of a simple shuffling problem. The time taken for the strings in a population to become mixed is calculated for different crossover procedures. An order parameter was defined to measure the degree of mixing within the population. As the strings become mixed the order parameter decays to the value of a totally mixed population. The mixing rate is the asymptotic rate at which the order parameter decays. The mixing rates for single-point, two-point and uniform crossover are calculated analytically. The effect of single-point crossover is calculated with a new approach. The strings were considered to be made up from regions or blocks of genes where the regions are bounded by the points where crossover has occurred. Analytic expressions for the mixing rate for uniform, single-point and two-point crossover were plotted on a graph for a population size and string length of 100. The curve obtained for uniform crossover is exact and those for single-point and two-point crossover are approximate. Results suggest that uniform crossover mix the population fastest, while single-point crossover causes very slow mixing. Two-point crossover extrapolates between uniform crossover and single-point crossover.