  A set of one dimensional interfaces involving attachment and detachment of
$k$-particle neighbors is studied numerically using both large scale
simulations and finite size scaling analysis. A labeling algorithm introduced
by Barma and Dhar in related spin Hamiltonians enables to characterize the
asymptotic behavior of the interface width according to the initial state of
the substrate. For equal deposition--evaporation probability rates it is found
that in most cases the initial conditions induce regimes of saturated width. In
turn, scaling exponents obtained for initially flat interfaces indicate power
law growths which depend on $k$. In contrast, for unequal probability rates the
interface width exhibits a logarithmic growth for all $k > 1$ regardless of the
initial state of the substrate.
