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

Let G = (V, E, w) be a weighted undirected graph on vertical bar V vertical bar = n vertices and vertical bar E vertical bar = m edges, let k >= 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,(1) which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdos' girth conjecture and disregarding the.-dependencies.) There are (deterministic) algorithms for constructing (2k - 1)(1+epsilon)-spanners for G with a near-optimal sparsity of O(n(1/k) center dot log(1/epsilon)/epsilon)). The first algorithm can be implemented in the pointer-machine model within time O(m alpha(m, n) center dot log(1/epsilon)/epsilon)+ SORT(m)), where alpha(center dot, center dot) 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(mlog(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) center dot 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) center dot poly(1/epsilon)+ SORT(m)) and in the Word RAM model within time O(m alpha(m, n) center dot poly(1/epsilon)). The previous fastest constructions of (2k - 1)(1 + epsilon)-spanners with near-optimal sparsity incur a run-time of is O(min{m(n(1+1/k)) + n log n, k center dot 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 (including the greedy spanner) is poor, even regardless of the sparsity and runtime.