  Near-additive (aka $(1+\epsilon,\beta)$-) emulators and spanners are a
fundamental graph-algorithmic construct, with numerous applications for
computing approximate shortest paths and related problems in distributed,
streaming and dynamic settings.
  Known constructions of near-additive emulators enable one to trade between
their sparsity (i.e., number of edges) and the additive stretch $\beta$.
Specifically, for any pair of parameters $\epsilon >0$, $ \kappa=1,2,\dots$,
one can have a $(1+\epsilon,\beta)$-emulator with $O(n^{1+1/\kappa})$ edges,
with $\beta = \left(\frac{\log \kappa}{\epsilon}\right)^{\log \kappa}$. At
their sparsest, these emulators employ $c\cdot n$ edges, for some constant
$c\geq 2$.
  We tighten this bound, and show that in fact precisely $n^{1+1/\kappa}$ edges
suffice.
  In particular, our emulators can be \emph{ultra-sparse}, i.e., we can have an
emulator with $n+o(n)$ edges and $\beta = \left(\frac{\log {\log n}}{\epsilon
}\right)^{{\log {\log n}}(1+o(1))}$.
  We also devise a distributed deterministic algorithm in the CONGEST model
that builds these emulators in low polynomial time (i.e., in $O(n^\rho)$ time,
for an arbitrarily small constant parameter $\rho >0$).
  Finally, we also improve the state-of-the-art distributed deterministic
\congest-model construction of
  $(1+\epsilon,\beta)$-spanners devised in the PODC'19 paper
  [ElkinM19]. Specifically, the spanners of [ElkinM19] have $O(\beta\cdot
n^{1+1/\kappa})$ edges, i.e., at their sparsest they employ
  $ O\left(\frac{\log {\log n}}{\epsilon }\right)^{{\log {\log n}}}\cdot n$
edges. In this paper, we devise an efficient distributed deterministic
CONGEST-model algorithm that builds such spanners with $O(n^{1+1/\kappa})$
edges for $\kappa = O\left(\frac{\log n}{\log ^{(3)}n}\right)$. At their
sparsest, these spanners employ only $O(n\cdot {\log {\log n}})$ edges.
