  Our theoretical understanding of crossover is limited by our ability to
analyze how population diversity evolves. In this study, we provide one of the
first rigorous analyses of population diversity and optimization time in a
setting where large diversity and large population sizes are required to speed
up progress. We give a formal and general criterion which amount of diversity
is necessary and sufficient to speed up the $(\mu+1)$ Genetic Algorithm on
LeadingOnes. We show that the naturally evolving diversity falls short of
giving a substantial speed-up for any $\mu=O(\sqrt{n}/\log^2 n)$. On the other
hand, we show that even for $\mu=2$, if we simply break ties in favor of
diversity then this increases diversity so much that optimization is
accelerated by a constant factor.
