A Theoretical and Empirical Analysis of -ary Landscapes for Genetic Algorithms a Theoretical and Empirical Analysis of -ary Landscapes for Genetic Algorithms

Genetic algorithms (GAs) are probabilistic search algorithms that are loosely based on biological evolution. Analyzing genetic algorithms has proven di cult, for a variety of reasons, but a landscape paradigm that rigorously models the search of GAs has become increasingly popular in their analysis. So far much of this analysis concerns binary representations, where each member of the population is a binary string. In this thesis we consider using -ary representations|using strings where each character can be one of characters|and conduct a theoretical and empirical study of the resulting -ary landscapes. In terms of theory, we discuss the types of landscape graphs produced by various -ary crossover and -ary mutation operators. We relate these landscapes to a common class of graphs, the -ary hypercubes. We then generalize a binary mutationcrossover isomorphism to higher alphabets and use this isomorphism to show that -ary crossover can simulate -ary mutation. Since crossover can simulate mutation, crossover must be at least as powerful as mutation, in the sense of computational power. Because this -ary mutation-crossover isomorphism is closely related to an -ary Gray code, extending our simulation to \hyper orders" is a generalization of iterating the landscape's representation (and thus the landscape) using this -ary Gray code. If we repeatedly apply this Gray code, the landscapes will eventually cycle. We prove a theorem on the maximum number of landscapes produced when this Gray code is iterated. We explore the long path problem for -ary mutation and -ary crossover. We construct exponentially long distance-preserving paths for -ary mutation, improving on a previous method. We also construct exponentially long distance-preserving paths for crossover between two complementary binary strings, and discuss the problem of creating long distance-preserving paths for populations greater than two. In Chapter 7, we create the schizophrenic function, a function designed to discriminate between mutation and crossover. This function has two optima classes, and is constructed in such a way that mutation should nd one optimum, crossover the other. Empirical tests show that the schizophrenic function is a useful but imperfect tool. More importantly, these tests o er insight into how crossover works. Finally, we show how to generate an -ary Gray code e ciently in sequence (constant amortized time per codeword). We generalize this Gray code to \multary" strings, and generalize our algorithm to generate this code in constant amortized time per codeword. Acknowledgements There are several people who made a direct contribution to this thesis, all deserving thanks. First, and foremost, is my supervisor, Joseph Culberson, for time spent deciphering and editing earlier versions of this thesis, and for helping to focus my research in the right direction and for giving me some of his research problems. I thank the members of my examining committee, Maziar Shirvani and Lorna Stewart, for time spent reading my thesis and their helpful suggestions. I'd also like to thank Jim Hoover for chairing my examining committee. Several chapters of this thesis were based on papers, either published or submitted, and the comments of the anonymous referees helped me to polish these portions of my thesis. I also thank Kevin Charter, Kurt Lichtner, and Basil Vandegriend for reading my thesis and giving helpful advice.