Near-Optimal Spanners for General Graphs in (Nearly) Linear Time

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 Erd\H{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 $O(m\alpha(m,n) \log(1/\epsilon)/\epsilon) + SORT(m))$, 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 $O(m\alpha(m,n) \mathrm{poly}(1/\epsilon) + SORT(m))$ 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 $O(\min{m(n^{1+1/k}) + n\log n,k n^{2+1/k}})$, 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 $O(m(n^{1+1/k} + n\log n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.