  The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a
natural generalization of the well-studied minimum-weight spanning tree (MST)
problem, and it has received considerable attention in the area of network
design. The latter problem asks for a minimum-weight subgraph with an edge
connectivity of $1$ between each pair of vertices while the former strengthens
this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS
problem is considerably more complex than MST. While MST admits a linear-time
centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized
approximation algorithm for it (that runs in polynomial time) gives a
$2$-approximation.
  In this paper, we give a deterministic distributed algorithm with round
complexity of $\widetilde{O}(D+\sqrt{n})$ that computes a
$(5+\epsilon)$-approximation of 2-ECSS, for any constant $\epsilon>0$. Up to
logarithmic factors, this complexity matches the
$\widetilde{\Omega}(D+\sqrt{n})$ lower bound that can be derived from Das Sarma
et al. [STOC'11], as shown by Censor-Hillel and Dory [OPODIS'17]. Our result is
the first distributed constant approximation for 2-ECSS in the nearly optimal
time and it improves on a recent randomized algorithm of Dory [PODC'18], which
achieved an $O(\log n)$-approximation in $\widetilde{O}(D+\sqrt{n})$ rounds.
  We also present an alternative algorithm for $O(\log n)$-approximation, whose
round complexity is linear in the low-congestion shortcut parameter of the
network, following a framework introduced by Ghaffari and Haeupler [SODA'16].
This algorithm has round complexity $\widetilde{O}(D+\sqrt{n})$ in worst-case
networks but it provably runs much faster in many well-behaved graph families
of interest. For instance, it runs in $\widetilde{O}(D)$ time in planar
networks and those with bounded genus, bounded path-width or bounded
tree-width.
