  With elementary means, we prove a stronger run time guarantee for the
univariate marginal distribution algorithm (UMDA) optimizing the LeadingOnes
benchmark function in the desirable regime with low genetic drift. If the
population size is at least quasilinear, then, with high probability, the UMDA
samples the optimum within a number of iterations that is linear in the problem
size divided by the logarithm of the UMDA's selection rate. This improves over
the previous guarantee, obtained by Dang and Lehre (2015) via the deep
level-based population method, both in terms of the run time and by
demonstrating further run time gains from small selection rates. With similar
arguments as in our upper-bound analysis, we also obtain the first lower bound
for this problem. Under similar assumptions, we prove that a bound that matches
our upper bound up to constant factors holds with high probability.
