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.

翻译：给定一个无向无权图 $G = (V, E)$，包含 $n$ 个顶点和 $m$ 条边，子图 $H\subseteq G$ 是 $G$ 的一个具有拉伸函数 $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ 的稀疏子图，当且仅当对于 $V$ 中任意一对顶点 $s, t$，有 $\textsf{dist}_{H}(s, t)\le f(\textsf{dist}_{G}(s, t))$。当 $f(d) = d + o(d)$ 时，$H$ 称为次线性加法稀疏子图；当 $f(d) = d + o(n)$ 时，$H$ 称为加法稀疏子图，并且 $f(d) - d$ 通常称为 $H$ 的加法拉伸。作为我们的主要结果，我们证明对于任意常数 $\delta>0$ 和任意常数整数 $k\geq 2$，每个具有 $n$ 个顶点的图都存在一个拉伸函数为 $f(d)=d+O(d^{1-1/k})$ 且边数为 $O\big(n^{1+\frac{1+\delta}{2^{k+1}-1}}\big)$ 的次线性加法稀疏子图。当 $k = 2$ 时，这改进了之前拉伸函数为 $f(d) = d + O(d^{1/2})$ 且边数为 $\tilde{O}(n^{1+3/17})$ 的稀疏子图构造 [Chechik, 2013]；对于任意常数整数 $k\geq 3$，这改进了之前拉伸函数为 $f(d) = d + O(d^{1-1/k})$ 且边数为 $O\bigg(n^{1+\frac{(3/4)^{k-2}}{7 - 2\cdot (3/4)^{k-2}}}\bigg)$ 的稀疏子图构造 [Pettie, 2009]。最重要的是，我们的稀疏子图的大小几乎匹配下界 $\Omega\big(n^{1+\frac{1}{2^{k+1}-1}}\big)$ [Abboud, Bodwin, Pettie, 2017]。作为我们的第二个结果，我们展示了一种新的加法稀疏子图构造，其拉伸为 $O(n^{0.403})$ 且边数为 $O(n)$，这略微改进了之前由线性大小稀疏子图实现的拉伸界 $O(n^{3/7+\epsilon})$ [Bodwin and Vassilevska Williams, 2016]。我们的稀疏子图还有一个额外优势，即它支持次二次构造运行时间 $\tilde{O}(m + n^{13/7})$，而之前 [Bodwin and Vassilevska Williams, 2016] 中的构造需要计算全源最短路径，耗时 $O(\min\{mn, n^{2.373}\})$。