  In low distortion metric embeddings, the goal is to embed a host "hard"
metric space into a "simpler" target space while approximately preserving
pairwise distances. A highly desirable target space is that of a tree metric.
Unfortunately, such embedding will result in a huge distortion. A celebrated
bypass to this problem is stochastic embedding with logarithmic expected
distortion. Another bypass is Ramsey-type embedding, where the distortion
guarantee applies only to a subset of the points. However, both these solutions
fail to provide an embedding into a single tree with a worst-case distortion
guarantee on all pairs. In this paper, we propose a novel third bypass called
\emph{clan embedding}. Here each point $x$ is mapped to a subset of points
$f(x)$, called a \emph{clan}, with a special \emph{chief} point $\chi(x)\in
f(x)$. The clan embedding has multiplicative distortion $t$ if for every pair
$(x,y)$ some copy $y'\in f(y)$ in the clan of $y$ is close to the chief of $x$:
$\min_{y'\in f(y)}d(y',\chi(x))\le t\cdot d(x,y)$. Our first result is a clan
embedding into a tree with multiplicative distortion $O(\frac{\log
n}{\epsilon})$ such that each point has $1+\epsilon$ copies (in expectation).
In addition, we provide a "spanning" version of this theorem for graphs and use
it to devise the first compact routing scheme with constant size routing
tables.
  We then focus on minor-free graphs of diameter prameterized by $D$, which
were known to be stochastically embeddable into bounded treewidth graphs with
expected additive distortion $\epsilon D$. We devise Ramsey-type embedding and
clan embedding analogs of the stochastic embedding. We use these embeddings to
construct the first (bicriteria quasi-polynomial time) approximation scheme for
the metric $\rho$-dominating set and metric $\rho$-independent set problems in
minor-free graphs.
