  Given an undirected weighted graph $G(V,E)$, a constrained sketch over a
terminal set $T\subset V$ is a subgraph $G'$ that connects the terminal
vertices while satisfying a given set of constraints. Examples include Steiner
trees (preserving connectivity among $T$) and subsetwise spanners (preserving
shortest path distances up to a stretch factor). Multi-level constrained
terminal sketches are generalizations in which terminal vertices require
different levels or grades of service and each terminal pair is connected with
edges of at least the minimum required level of the two terminals.
  This paper gives a flexible procedure for computing a broad class of
multi-level graph sketches, which encompasses multi-level graph spanners,
Steiner trees, and $k$--connected subgraphs as a few special cases. The
proposed procedure is modular, i.e., it relies on availability of a
single-level solver module (be it an oracle or approximation algorithm). One
key result is that an $\ell$--level constrained terminal sketch can be computed
with $\log\ell$ queries of the solver module while producing feasible solutions
with approximation guarantees independent of $\ell$.
  Additionally, a new polynomial time algorithm for computing a subsetwise
spanner is proposed. We show that for $k\in\N$, $\eps>0$, and $T\subset V$,
there is a subsetwise $(2k-1)(1+\eps)$--spanner with total weight
$O(|T|^\frac1kW(\ST(G,T)))$, where $W(\ST(G,T))$ is the weight of the Steiner
tree of $G$ over the subset $T$. This is the first algorithm and weight
guarantee for a multiplicative subsetwise spanner for nonplanar graphs.
Numerical experiments are also done to illustrate the performance of the
proposed algorithms.
