Binary Representations of Integers and the Performance of Selectorecombinative Genetic Algorithms

When using representations for genetic algorithms (GAs) every optimization problem can be separated into a genotype-phenotype and a phenotype-fitness mapping. The genotype-phenotype mapping is the used representation and the phenotype-fitness mapping is the problem that should be solved. This paper investigates how the use of different binary representations of integers influences the performance of selectorecombinative GAs using only crossover and no mutation. It is illustrated that the used representation strongly influences the performance of GAs. The binary and Gray encoding are two examples for assigning bitstring genotypes to integer phenotypes. Focusing the investigation on these two encodings reveals that for the easy integer one-max problem selectorecombinative GAs perform better using binary encoding than using Gray encoding. This is surprising as binary encoding is affected with problems due to the Hamming cliff and because there are proofs that show the superiority of Gray encoding. However, the performance of selectorecombinative GAs using binary representations of integers is determined by the resulting building blocks and not by the structure of the search space resulting from the Hamming distances between the individuals. Therefore, the performance difference between the encodings can be explained by analyzing the fitness of the resulting schemata.