  We study the non--equilibrium motion of an elastic string in a two
dimensional pinning landscape using Langevin dynamics simulations. The
relaxation of a line, initially flat, is characterized by a growing length,
$L(t)$, separating the equilibrated short length scales from the flat long
distance geometry that keep memory of the initial condition. We show that, in
the long time limit, $L(t)$ has a non--algebraic growth with a universal
distribution function. The distribution function of waiting times is also
calculated, and related to the previous distribution. The barrier distribution
is narrow enough to justify arguments based on scaling of the typical barrier.
