  Our main result is that the Steiner Point Removal (SPR) problem can always be
solved with polylogarithmic distortion, which answers in the affirmative a
question posed by Chan, Xia, Konjevod, and Richa (2006). Specifically, we prove
that for every edge-weighted graph $G = (V,E,w)$ and a subset of terminals $T
\subseteq V$, there is a graph $G'=(T,E',w')$ that is isomorphic to a minor of
$G$, such that for every two terminals $u,v\in T$, the shortest-path distances
between them in $G$ and in $G'$ satisfy $d_{G,w}(u,v) \le d_{G',w'}(u,v) \le
O(\log^5|T|) \cdot d_{G,w}(u,v)$. Our existence proof actually gives a
randomized polynomial-time algorithm. Our proof features a new variant of
metric decomposition. It is well-known that every $n$-point metric space
$(X,d)$ admits a $\beta$-separating decomposition for $\beta=O(\log n)$, which
roughly means for every desired diameter bound $\Delta>0$ there is a randomized
partitioning of $X$, which satisfies the following separation requirement: for
every $x,y \in X$, the probability they lie in different clusters of the
partition is at most $\beta\,d(x,y)/\Delta$. We introduce an additional
requirement, which is the following tail bound: for every shortest-path $P$ of
length $d(P) \leq \Delta/\beta$, the number of clusters of the partition that
meet the path $P$, denoted $Z_P$, satisfies $\Pr[Z_P > t] \le 2e^{-\Omega(t)}$
for all $t>0$.
