Almost-Optimal Sublinear Additive Spanners

Abstract: 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 $f: \mathbb{R}<em>+ \rightarrow \mathbb{R}</em>+$, if for every pair $s, t$ of vertices in $V$, $$\text{dist}<em>{H}(s, t)\le f(\text{dist}</em>{G}(s, t))$$. When $f(d) = d + o(d)$, $H$ is called a sublinear additive spanner; when $f(d) = d + o(n)$, $H$ is called an \emph{additive spanner}, and $f(d) - d$ is usually called the \emph{additive stretch} of $H$. As our primary result, we show that for any constant $\delta&gt;0$ and constant integer $k\geq 2$, every graph on $n$ vertices has a sublinear additive spanner with stretch function $f(d)=d+O(d<sup>{1-1/k})$ and $$O\big(n<sup>{1+\frac{1+\delta}{2<sup>{k+1}-1}}\big)$$ edges. When $k = 2$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d<sup>{1/2})$ and $\tilde{O}(n<sup>{1+3/17})$ edges; for any constant integer $k\geq 3$, this improves upon the previous spanner construction with stretch function $f(d) = d + O(d<sup>{1-1/k})$ and $$O\bigg(n^{{1+\frac{(3/4)^{k-2}}{7} - 2\cdot (3/4)<sup>{k-2}}}\bigg)$$ edges. Most importantly, the size of our spanners almost matches the lower bound of $\Omega\big(n<sup>{1+\frac{1}{2<sup>{k+1}-1}}\big)$, which holds for all compression schemes achieving the same stretch function. As our second result, we show a new construction of additive spanners with stretch $O(n<sup>{0.403})$ and $\tilde{O}(n)$ edges, which slightly improves upon the previous stretch bound of $O(n<sup>{3/7+\varepsilon})$ achieved by linear-size spanners. An additional advantage of our spanner is that it admits a subquadratic construction runtime of $\tilde{O}(m + n<sup>{13/7})$, while the previous construction requires all-pairs shortest paths computation which takes $O(\min{mn, n<sup>{2.373}})$ time.