  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 ($\mu$+$\lambda$) 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 $\mu$ and $\lambda$. 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 $(1+\sqrt{5})/2 \cdot 1/n \approx 1.618/n$. 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.
