IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 9, NO. 5, OCTOBER 2005 489 Selection Intensity in Cellular Evolutionary Algorithms for Regular Lattices Mario Giacobini, Student Member, IEEE, Marco Tomassini, Andrea G. B. Tettamanzi, and Enrique Alba Abstract—In this paper, we present quantitative models for the selection pressure of cellular evolutionary algorithms on regular one- and two-dimensional (2-D) lattices. We derive models based on probabilistic difference equations for synchronous and several asynchronous cell update policies. The models are validated using two customary selection methods: binary tournament and linear ranking. Theoretical results are in agreement with experimental values, showing that the selection intensity can be controlled by using different update methods. It is also seen that the usual lo- gistic approximation breaks down for low-dimensional lattices and should be replaced by a polynomial approximation. The depen- dence of the models on the neighborhood radius is studied for both topologies. We also derive results for 2-D lattices with variable grid axes ratio. Index Terms—Asynchronous dynamics, cellular evolutionary al- gorithms (cEAs), regular lattices, selection intensity, synchronous dynamics. I. INTRODUCTION C ELLULAR evolutionary algorithms (cEAs) use popula- tions that are structured according to a lattice topology. The structure may also be an arbitrary graph, but more com- monly it is a one-dimensional (1-D) or two-dimensional (2-D) grid. This kind of spatially structured EA has been introduced in [1] and [2]. A distinctive feature of cEAs is slow diffusion of good individuals through the population and, thus, for a given selection method, they are more explorative than panmictic (i.e., standard mixing population) EAs. These aspects have been found useful for multimodal and other kinds of problems (see, for instance, [3]). Cellular evolutionary algorithms have become popular because they are easy to implement on par- allel hardware [4], [5]. However, it is clear that the search is performed by the model, not by its implementation. Thus, in this paper, we will focus on cEA models and on their properties without worrying about implementation issues. Several results have appeared on selection pressure and con- vergence speed in cEAs. Sarma and De Jong performed empir- Manuscript received October 8, 2004; revised January 17, 2005. The work of M. Giacobini and M. Tomassini was supported in part by the Fonds National Suisse pour la Recherche Scientifique under Contract 200021-103732/1. The work of E. Alba was supported in part by the Spanish Ministry of Education and European FEDER under Contract TIC2002-04498-C05-02 (the TRACER Project, http://tracer.lcc.uma.es). M. Giacobini and M. Tomassini are with the Information Systems Depart- ment, University of Lausanne, CP1 1015 Dorigny-Lausanne, Switzerland (e-mail: mario.giacobini@unil.ch; marco.tomassini@unil.ch). A. G. B. Tettamanzi is with the Information Technologies Department, University of Milano, IT-26013 Crema, Italy (e-mail: andrea.tettamanzi@ unimi.it). E. Alba is with the Department of Computer Science, University of Málaga, ES-29071 Málaga, Spain (e-mail: eat@lcc.uma.es). Digital Object Identifier 10.1109/TEVC.2005.850298 ical analyses of the dynamical behavior of cellular genetic al- gorithms (cGAs) [6], [7], focusing on the effect that the local selection method, and the neighborhood size and shape have on the global induced selection pressure. Rudolph and Sprave [8] have shown how cGAs can be modeled by a probabilistic au- tomata network and have provided proofs of complete conver- gence to a global optimum based on Markov chain analysis for a model including a fitness threshold. Recently, Giacobini et al. [9] have successfully modeled the selection pressure curves in cEAs on 1-D ring structures. Also, a preliminary study of 2-D, torus-shaped grids has appeared in [10] and [11]. Our purpose here is to investigate in detail, selection pressure in 1-D and 2-D population structures for two kinds of dynam- ical systems: synchronous and asynchronous. These two kinds of systems differ in the policies used to update the population at every step of the search. Our main contribution, thus, lies in providing mathematical models for the different update policies (and several selection strategies) that more accurately predict the experimentally observed takeover time curves with simple difference equations describing the propagation of the best in- dividual under probabilistic conditions. The paper proceeds as follows. In Section II, we summarize the main features relevant to our study. We discuss the takeover time concept in Section III, followed in Section IV by the in- troduction of the background ideas sustaining our mathemat- ical models. Section V contains a discussion on the limitations of directly applying the logistic model to cEAs. This justifies the necessity of new mathematical models that are developed in the next two sections, for a ring (Section VI) and for a torus (Section VII). In Section VIII, we validate our models exper- imentally for two standard selection mechanisms: binary tour- nament and linear ranking. Also, because of the high interest of rectangular toroidal structures for the population in cEAs, we devote Section IX to such shape to enlarge the spectrum of in- terest of this paper. We conclude our analysis by exploring the theoretical implications of changing the radius of the neighbor- hood in the considered models (Section X). Section XI offers our concluding remarks and future research lines. II. SYNCHRONOUS AND ASYNCHRONOUS CEAS Let us begin with a description of how a cEA works. A cEA maintains a population whose individuals are spatially distributed in cells. Each cell is occupied by one individual; therefore, the terms cell and individual may be used inter- changeably without possibility of confusion. A cEA starts with the cells in a random state and proceeds by successively updating them using evolutionary operators, until a termination condition is met. Updating a cell in a cEA means 1089-778X/$20.00 © 2005 IEEE