  Spanners for low dimensional spaces (e.g. Euclidean space of constant
dimension, or doubling metrics) are well understood. This lies in contrast to
the situation in high dimensional spaces, where except for the work of
Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any $n$-point
Euclidean metric has an $O(t)$-spanner with $\tilde{O}(n^{1+1/t^2})$ edges,
little is known.
  In this paper we study several aspects of spanners in high dimensional normed
spaces. First, we build spanners for finite subsets of $\ell_p$ with $1<p\le
2$. Second, our construction yields a spanner which is both sparse and also
{\em light}, i.e., its total weight is not much larger than that of the minimum
spanning tree. In particular, we show that any $n$-point subset of $\ell_p$ for
$1<p\le 2$ has an $O(t)$-spanner with $n^{1+\tilde{O}(1/t^p)}$ edges and
lightness $n^{\tilde{O}(1/t^p)}$.
  In fact, our results are more general, and they apply to any metric space
admitting a certain low diameter stochastic decomposition. It is known that
arbitrary metric spaces have an $O(t)$-spanner with lightness $O(n^{1/t})$. We
exhibit the following tradeoff: metrics with decomposability parameter
$\nu=\nu(t)$ admit an $O(t)$-spanner with lightness $\tilde{O}(\nu^{1/t})$. For
example, $n$-point Euclidean metrics have $\nu\le n^{1/t}$, metrics with
doubling constant $\lambda$ have $\nu\le\lambda$, and graphs of genus $g$ have
$\nu\le g$. While these families do admit a ($1+\epsilon$)-spanner, its
lightness depend exponentially on the dimension (resp. $\log g$). Our
construction alleviates this exponential dependency, at the cost of incurring
larger stretch.
