Title: Light Spanners for High Dimensional Norms via Stochastic Decompositions.

Authors: Arnold Filtser and Ofer Neiman.

Abstract: 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.