Grammatical Evolution with Restarts for Fast Fractal Generation

TL;DR

This paper analyzes the heavy-tailed distribution of a grammatical evolution algorithm for fractal generation and introduces restart strategies that significantly improve its efficiency and stability.

Contribution

It provides statistical evidence of heavy tail behavior in the algorithm's execution time and proposes restart methods to mitigate this issue, reducing expected time and variance.

Findings

Heavy tail distribution causes erratic performance.

Restart strategies reduce expected execution time.

Variance of execution time becomes finite.

Abstract

In a previous work, the authors proposed a Grammatical Evolution algorithm to automatically generate Lindenmayer Systems which represent 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 performance of different executions of the algorithm. Three different restart strategies have been incorporated in the algorithm to mitigate the problems associated to heavy tail distributions: the first assumes full knowledge of the execution time probability distribution, the second and third assume no knowledge. These strategies exploit the fact that the probability 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.