Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k ge 1$ be any integer, and let $epsilon < 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under ErdH{o}s girth conjecture and disregarding the $epsilon$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+epsilon)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} log(1/epsilon)/epsilon))$. The first algorithm can be implemented in the pointer-machine model within time $O(malpha(m,n) log(1/epsilon)/epsilon) + SORT(m))$, where $alpha( , )$ is the two-parameter inverse-Ackermann function and $SORT(m)$ is the time needed to sort $m$ integers. The second algorithm can be implemented in the WORD RAM model within time $O(m log(1/epsilon)/epsilon))$. - There is a (deterministic) algorithm for constructing a $(2k-1)(1+epsilon)$-spanner for $G$ that achieves a near-optimal bound of $O(n^{1/k}mathrm{poly}(1/epsilon))$ on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time $O(malpha(m,n) mathrm{poly}(1/epsilon) + SORT(m))$ and in the WORD RAM model within time $O(m alpha(m,n) mathrm{poly}(1/epsilon))$. The previous fastest constructions of $(2k-1)(1+epsilon)$-spanners with near-optimal sparsity incur a runtime of is $O(min{m(n^{1+1/k}) + nlog n,k n^{2+1/k}})$, even regardless of the lightness. Importantly, the greedy spanner for stretch $2k-1$ has sparsity $O(n^{1/k})$ -- with no $epsilon$-dependence whatsoever, but its runtime is $O(m(n^{1+1/k} + nlog n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.
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