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. [STOC11], as shown by Censor-Hillel and Dory [OPODIS17]. 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 [PODC18], 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 [SODA16]. 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.
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