  A new \emph{spanner} construction algorithm is presented, working under the
\emph{LOCAL} model with unique edge IDs. Given an $n$-node communication graph,
a spanner with a constant stretch and $O (n^{1 + \varepsilon})$ edges (for an
arbitrarily small constant $\varepsilon > 0$) is constructed in a constant
number of rounds sending $O (n^{1 + \varepsilon})$ messages whp. Consequently,
we conclude that every $t$-round LOCAL algorithm can be transformed into an $O
(t)$-round LOCAL algorithm that sends $O (t \cdot n^{1 + \varepsilon})$
messages whp. This improves upon all previous message-reduction schemes for
LOCAL algorithms that incur a $\log^{\Omega (1)} n$ blow-up of the round
complexity.
