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Fitness Probability Distribution of Bit-Flip Mutation.

Francisco Chicano1, Andrew M Sutton2, L Darrell Whitley3

  • 1Departamento de Lenguajes y Ciencias de la Computación, Universidad de Málaga, Spain chicano@lcc.uma.es.

Evolutionary Computation
|June 3, 2014
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Summary

We precisely calculate the fitness distribution for binary strings under bit-flip mutation using landscape theory and Krawtchouk polynomials. This provides exact formulas for problems like Onemax and MAX-SAT, aiding evolutionary algorithm analysis.

Keywords:
Bit-flip mutationcombinatorial optimizationevolutionary algorithmslandscape theoryrandomized algorithms

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Area of Science:

  • Computer Science
  • Artificial Intelligence
  • Optimization

Background:

  • Bit-flip mutation is a standard operator in evolutionary algorithms for binary string optimization.
  • Understanding fitness value distributions is crucial for analyzing algorithm performance.

Purpose of the Study:

  • To exactly compute the probability distribution of fitness values for binary strings under uniform bit-flip mutation.
  • To analyze this distribution using Krawtchouk polynomials and landscape theory.

Main Methods:

  • Leveraging the theory of evolutionary computation landscapes.
  • Applying Krawtchouk polynomials to derive exact probability distributions.
  • Analyzing derived polynomials for specific optimization problems.

Main Results:

  • The probability distribution of fitness values is proven to be a polynomial in the bit-flip probability (p).
  • Closed-form expressions for the distribution are derived for the Onemax and MAX-SAT problems.
  • Connections between these results and the runtime analysis of evolutionary algorithms are discussed.

Conclusions:

  • The developed framework provides exact analytical tools for understanding bit-flip mutation.
  • The findings offer insights into the behavior of evolutionary algorithms on different problem classes.
  • This work contributes to the theoretical foundation of evolutionary computation.