  The metric sketching problem is defined as follows. Given a metric on $n$
points, and $\epsilon>0$, we wish to produce a small size data structure
(sketch) that, given any pair of point indices, recovers the distance between
the points up to a $1+\epsilon$ distortion. In this paper we consider metrics
induced by $\ell_2$ and $\ell_1$ norms whose spread (the ratio of the diameter
to the closest pair distance) is bounded by $\Phi>0$. A well-known
dimensionality reduction theorem due to Johnson and Lindenstrauss yields a
sketch of size $O(\epsilon^{-2} \log (\Phi n) n\log n)$, i.e., $O(\epsilon^{-2}
\log (\Phi n) \log n)$ bits per point. We show that this bound is not optimal,
and can be substantially improved to $O(\epsilon^{-2}\log(1/\epsilon) \cdot
\log n + \log\log \Phi)$ bits per point. Furthermore, we show that our bound is
tight up to a factor of $\log(1/\epsilon)$.
  We also consider sketching of general metrics and provide a sketch of size
$O(n\log(1/\epsilon)+ \log\log \Phi)$ bits per point, which we show is optimal.
