  Miller et al. \cite{MPVX15} devised a distributed\footnote{They actually
showed a PRAM algorithm. The distributed algorithm with these properties is
implicit in \cite{MPVX15}.} algorithm in the CONGEST model, that given a
parameter $k = 1,2,\ldots$, constructs an $O(k)$-spanner of an input unweighted
$n$-vertex graph with $O(n^{1+1/k})$ expected edges in $O(k)$ rounds of
communication. In this paper we improve the result of \cite{MPVX15}, by showing
a $k$-round distributed algorithm in the same model, that constructs a
$(2k-1)$-spanner with $O(n^{1+1/k}/\epsilon)$ edges, with probability $1-
\epsilon$, for any $\epsilon>0$. Moreover, when $k = \omega(\log n)$, our
algorithm produces (still in $k$ rounds) {\em ultra-sparse} spanners, i.e.,
spanners of size $n(1+ o(1))$, with probability $1- o(1)$. To our knowledge,
this is the first distributed algorithm in the CONGEST or in the PRAM models
that constructs spanners or skeletons (i.e., connected spanning subgraphs) that
sparse. Our algorithm can also be implemented in linear time in the standard
centralized model, and for large $k$, it provides spanners that are sparser
than any other spanner given by a known (near-)linear time algorithm.
  We also devise improved bounds (and algorithms realizing these bounds) for
$(1+\epsilon,\beta)$-spanners and emulators. In particular, we show that for
any unweighted $n$-vertex graph and any $\epsilon > 0$, there exists a $(1+
\epsilon, ({{\log\log n} \over \epsilon})^{\log\log n})$-emulator with $O(n)$
edges. All previous constructions of $(1+\epsilon,\beta)$-spanners and
emulators employ a superlinear number of edges, for all choices of parameters.
  Finally, we provide some applications of our results to approximate shortest
paths' computation in unweighted graphs.
