How Crossover Speeds Up Building-Block Assembly in Genetic Algorithms
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We re-investigate a fundamental question: how effective is crossover in Genetic Algorithms in combining building blocks of good solutions? Although this has been discussed controversially for decades, we are still lacking a rigorous and intuitive answer. We provide such answers for royal road functions and OneMax, where every bit is a building block. For the latter we show that using crossover makes every (μ+λ) Genetic Algorithm at least twice as fast as the fastest evolutionary algorithm using only standard bit mutation, up to small-order terms and for moderate μ and λ. Crossover is beneficial because it effectively turns fitness-neutral mutations into improvements by combining the right building blocks at a later stage. Compared to mutation-based evolutionary algorithms, this makes multi-bit mutations more useful. Introducing crossover changes the optimal mutation rate on OneMax from 1/n to (1+√(5))/2 · 1/n ≈ 1.618/n. This holds both for uniform crossover and k-point crossover. Experiments and statistical tests confirm that our findings apply to a broad class of building-block functions.
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