A distance between populations for one-point crossover in genetic algorithms

Genetic algorithms use transformation operators on the genotypic structures of the individuals to carry out a search. These operators define a neighborhood. To analyze various dynamics of the search process, it is often useful to define a distance in this space. In fact, using an operator-based distance can make the analysis more accurate and reliable than using distances which have no relationship with the genetic operators. In this paper we define a distance which is based on the standard one-point crossover. Given that the population strongly affects the neighborhood induced by the crossover, we first define a crossover-based distance between populations. Successively, we show that it is naturally possible to derive from this function a family of distances between individuals. Finally, we also introduce an algorithm to compute this distance efficiently.