Acceleration of a procedure to generate fractal curves of a given dimension through the probabilistic analysis of execution time Manuel Cebri´ an, Manuel Alfonseca and Alfonso Ortega * Abstract In a previous work, the authors proposed a Grammatical Evolution algorithm to automatically generate Lindenmayer Systems which repre- sent fractal curves with a pre-determined fractal dimension. This paper gives strong statistical evidence that the probability distributions of the execution time of that algorithm exhibits a heavy tail with an hyperbolic probability decay for long executions, which explains the erratic perfor- mance of different executions of the algorithm. Three different restart strategies have been incorporated in the algorithm to mitigate the prob- lems associated to heavy tail distributions: the first assumes full knowl- edge of the execution time probability distribution, the second and third assume no knowledge. These strategies exploit the fact that the proba- bility of finding a solution in short executions is non-negligible and yield a severe reduction, both in the expected execution time (up to one order of magnitude) and in its variance, which is reduced from an infinite to a finite value. Keywords: Fractal Generation, Grammatical Evolution, Randomized Algo- rithm, Heavy Tail Distribution, Restart Strategy. 1 Introduction In the last decades, genetic algorithms, which emulate biological evolution in computer software, have been applied to ever wider fields of research and de- velopment and have given rise to a few astounding successes, together with a certain mount of disappointment, frequently related to the apparently inherent slowness of the procedure. This is not a surprise, as biological evolution, which serves as the source for most of the ideas used by the research in genetic algo- rithms, makes a extremely slow and difficult to experiment field, where actual processes require millions of years in many cases. This slowness is in part a consequence of the fact that randomness is a basic underlying of the search performed by genetic algorithms. For this reason, the discovery and proposal of procedures to accelerate their execution time is one of the most interesting open questions in this field. * The authors are with the Departamento de Ingenier´ ıa Inform´atica, Escuela Polit´ ecnica Superior, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain, fax number: (+34) 914 972 235, e-mail: {manuel.cebrian, manuel.alfonseca, alfonso.ortega}@uam.es. 1 arXiv:0805.1696v2 [cs.NE] 15 Oct 2010