Almost-Optimal Sublinear Additive Spanners

Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}+ \rightarrow \mathbb{R}+$, iff for every pair $s, t$ of vertices in $V$, $\textsf{dist}{H}(s, t)\le f(\textsf{dist}{G}(s, t))$. 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 $k\geq 2$, every graph on $n$ vertices has a sublinear additive spanner with stretch function $f(d)=d+O(d^{1-1/k})$ and $O\big(n^{{1+\frac{1+\delta}{2{k+1}-1}}\big)$} edges. When $k = 2$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1/2})$ and $\tilde{O}(n^{1+3/17})$ edges [Chechik, 2013]; for any constant integer $k\geq 3$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d^{1-1/k})$ and $O\bigg(n^{{1+\frac{(3/4){k-2}}{7} - 2\cdot (3/4)^{{k-2}}}\bigg)$} edges [Pettie, 2009]. Most importantly, the size of our spanners almost matches the lower bound of $\Omega\big(n^{{1+\frac{1}{2{k+1}-1}}\big)$} [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 $\tilde{O}(m + n^{13/7})$, 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.