Parent selection methods are widely used in evolutionary computation to accelerate the optimization process, yet their theoretical benefits are still poorly understood. In this paper, we address this gap by proposing a parent selection strategy for the $(μ+1)$ genetic algorithm (GA) that prioritizes the selection of maximally distant parents for crossover. We show that, with an appropriately chosen population size, the resulting algorithm solves the Jump$_k$ problem in $O(k4^kn\log(n))$ expected time. This bound is significantly smaller than the best known bound of $O(nμ\log(μ)+n\log(n)+n^{k-1})$ for any $(μ+1)$~GA using no explicit diversity-preserving mechanism and a constant crossover probability. To establish this result, we introduce a novel diversity metric that captures both the maximum distance between pairs of individuals in the population and the number of pairs achieving this distance. The main novelty of our analysis is that it relies on crossover as a mechanism for creating and maintaining diversity throughout the run, rather than using crossover only in the final step to combine already diversified individuals. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.

翻译：父代选择方法广泛应用于进化计算中以加速优化进程，但其理论优势仍未被充分理解。本文针对这一空白，提出了一种面向$(μ+1)$遗传算法(GA)的父代选择策略，该策略优先选择最大距离的父代进行交叉。研究表明，在适当选择种群规模的情况下，由此产生的算法能够在$O(k4^kn\log(n))$期望时间内求解Jump$_k$问题。这一上界显著优于已知的采用恒定交叉概率且无显式多样性保持机制的$(μ+1)$~GA的最佳上界$O(nμ\log(μ)+n\log(n)+n^{k-1})$。为证明该结论，我们引入了一种新型多样性度量指标，该指标既能刻画种群中个体对间的最大距离，又能统计达到该距离的个体对数量。本分析的主要创新在于：将交叉视为在整个运行过程中创造并维持多样性的机制，而非仅在最终阶段用于组合已分化的个体。本文分析所得洞见有助于从理论层面更深入理解遗传算法种群动力学中交叉的作用机制。