  In a directed graph $G$ with non-correlated edge lengths and costs, the
\emph{network design problem with bounded distances} asks for a cost-minimal
spanning subgraph subject to a length bound for all node pairs. We give a
bi-criteria $(2+\varepsilon,O(n^{0.5+\varepsilon}))$-approximation for this
problem. This improves on the currently best known linear approximation bound,
at the cost of violating the distance bound by a factor of at
most~$2+\varepsilon$.
  In the course of proving this result, the related problem of \emph{directed
shallow-light Steiner trees} arises as a subproblem. In the context of directed
graphs, approximations to this problem have been elusive. We present the first
non-trivial result by proposing a
$(1+\varepsilon,O(|R|^{\varepsilon}))$-ap\-proxi\-ma\-tion, where $R$ are the
terminals.
  Finally, we show how to apply our results to obtain an
$(\alpha+\varepsilon,O(n^{0.5+\varepsilon}))$-approximation for
\emph{light-weight directed $\alpha$-spanners}. For this, no non-trivial
approximation algorithm has been known before. All running times depends on $n$
and $\varepsilon$ and are polynomial in $n$ for any fixed $\varepsilon>0$.
