Almost-Optimal Sublinear Additive Spanners

Given an undirected unweighted graph G = (V, E) on n vertices and m edges, a subgraph H subset of G is a spanner of G with stretch function f: R+ -> R+, iff for every pair s, t of vertices in V, dist(H)(s, t)<= f(dist(G)(s, t)). When f(d) = d + o(d), H is called a sublinear additive spanner; when f(d) = d + o(n), H is called an additive spanner, and f(d) - d is usually called the additive stretch of H. As our primary result, we show that for any constant delta>0 and constant integer k >= 2, every graph on n vertices has a sublinear additive spanner with stretch function f(d)=d+O(d(1-1)/k) and O(n(1)+1+delta/2(k+1)-1) edges. When k = 2, this improves upon the previous spanner construction with stretch function f(d) = d + O(d(1/2)) and (O) over tilde (n(1+3/17)) edges [Chechik, 2013]; for any constant integer k= 3, this improves upon the previous spanner construction with stretch function f(d) = d + O(d(1-1/k)) and O(n(1)+(3/4)(k-2)/7-2 center dot(3/4)(k-2)) edges [Pettie, 2009]. Most importantly, the size of our spanners almost matches the lower bound of Omega(n(1)+1/2(k+1)-1) [Abboud, Bodwin, Pettie, 2017]. As our second result, we show a new construction of additive spanners with stretch O(n(0.403)) and O(n) edges, which slightly improves upon the previous stretch bound of O(n(3/7+epsilon)) achieved by linear-size spanners [Bodwin and Vassilevska Williams, 2016]. An additional advantage of our spanner is that it admits a subquadratic construction runtime of (O) over tilde (m + n(13/7)), while the previous construction in [Bodwin and Vassilevska Williams, 2016] requires all-pairs shortest paths computation which takes O(min{mn, n(2.373)}) time.