  Starting from a unit mass on a vertex of a graph, we investigate the minimum
number of "\emph{controlled diffusion}" steps needed to transport a constant
mass $p$ outside of the ball of radius $n$. In a step of a controlled diffusion
process we may select any vertex with positive mass and topple its mass equally
to its neighbors. Our initial motivation comes from the maximum overhang
question in one dimension, but the more general case arises from optimal mass
transport problems.
  On $\mathbb{Z}^{d}$ we show that $\Theta( n^{d+2} )$ steps are necessary and
sufficient to transport the mass. We also give sharp bounds on the comb graph
and $d$-ary trees. Furthermore, we consider graphs where simple random walk has
positive speed and entropy and which satisfy Shannon's theorem, and show that
the minimum number of controlled diffusion steps is $\exp{( n \cdot h / \ell (
1 + o(1) ))}$, where $h$ is the Avez asymptotic entropy and $\ell$ is the speed
of random walk. As examples, we give precise results on Galton-Watson trees and
the product of trees $\mathbb{T}_d \times \mathbb{T}_k$.
