  A partition $\mathcal{P}$ of a weighted graph $G$ is
$(\sigma,\tau,\Delta)$-sparse if every cluster has diameter at most $\Delta$,
and every ball of radius $\Delta/\sigma$ intersects at most $\tau$ clusters.
Similarly, $\mathcal{P}$ is $(\sigma,\tau,\Delta)$-scattering if instead for
balls we require that every shortest path of length at most $\Delta/\sigma$
intersects at most $\tau$ clusters. Given a graph $G$ that admits a
$(\sigma,\tau,\Delta)$-sparse partition for all $\Delta>0$, Jia et al. [STOC05]
constructed a solution for the Universal Steiner Tree problem (and also
Universal TSP) with stretch $O(\tau\sigma^2\log_\tau n)$. Given a graph $G$
that admits a $(\sigma,\tau,\Delta)$-scattering partition for all $\Delta>0$,
we construct a solution for the Steiner Point Removal problem with stretch
$O(\tau^3\sigma^3)$. We then construct sparse and scattering partitions for
various different graph families, receiving many new results for the Universal
Steiner Tree and Steiner Point Removal problems.
