Acceleration of a Procedure to Generate Fractal Curves of a given Dimension through the Probabilistic Characterization of Execution Time

In a previous work [Ortega et al. 03], the authors have described the use of grammatical evolution to automatically generate L Systems (LS) representing fractal curves with a pre-determined fractal dimension. The experiments presented in this paper prove that the efficiency of this procedure is variable, with very different execution times for different executions with the same fractal dimension target. The paper shows that the probabilistic distribution of execution times belongs to a well-known family of random variables: heavytailed distributions. This analysis explains the erratic performance of the algorithm and suggests the use of a technique that corrects this variability and improves the efficiency about one order of magnitude. Acknowledgements: This paper has been sponsored by the Spanish Ministry of Science and Technology, project numbers TIC2001-0685-C02-01 and TIC 200201948. INTRODUCTION Our procedure to generate fractal curves with a required dimension consists of three parts: a) representation of fractals by means of L Systems (LS); b) computation of the fractal dimension from the grammar; c) application of a grammar-evolution based genetic algorithm to get a grammar representing a fractal with the required dimension. LS provide a powerful tool to represent fractals in the class of recursive transformations. The iterator may be represented by means of production rules, while the initiator corresponds to the axiom. The fractal curve is generated by the sequence of words derived from the axiom, by means of a representation scheme: vector graphics or turtle graphics. In a previous work [Alfonseca and Ortega 01] we have described an algorithm that estimates the fractal dimension of a non-trivial set of these fractals from their equivalent LS by means of symbolic manipulation, without the need of graphical procedures. In [Ortega et al. 03] we applied Grammar Evolution (GE) [O’Neill and Ryan 2001] to obtain the LS equivalent to a fractal with the required dimension. The proposed procedure has a clear biological inspiration acting on three different levels: a genotype (a vector of integers) an intermediate level, equivalent to proteins (LS) and a phenotype (the fractal curve). See Figure 1.