New $(\alpha,\beta)$ Spanners and Hopsets

An $f(d)$-spanner of an unweighted $n$-vertex graph $G=(V,E)$ is a subgraph $H$ satisfying that $dist_H(u, v)$ is at most $f(dist_G(u, v))$ for every $u,v \in V$. We present new spanner constructions that achieve a nearly optimal stretch of $O(\lceil k /d \rceil)$ for any distance value $d \in [1,k^{1-o(1)}]$, and $d \geq k^{1+o(1)}$. We show the following: 1. There exists an $f(d)$-spanner $H \subseteq G$ with $f(d)\leq 7k$ for any $d \in [1,\sqrt{k}/2]$ with expected size $O_{k}(n^{1+1/k})$. This in particular gives $(\alpha,\beta)$ spanners with $\alpha=O(\sqrt{k})$ and $\beta=O(k)$. 2. For any $\epsilon \in (0,1/2]$, there exists an $(\alpha,\beta)$-spanner with $\alpha=O(k^{\epsilon})$, $\beta=O_{\epsilon}(k)$ and of expected size $O_{k}(n^{1+1/k})$. This implies a stretch of $O(\lceil k/d \rceil)$ for any $d \in [\sqrt{k}/2, k^{1-\epsilon}]$, and for every $d\geq k^{1+\epsilon}$. In particular, it provides a constant stretch already for vertex pairs at distance $k^{1+o(1)}$ (improving upon $d=(\log k)^{\log k}$ that was known before). Up to the $o(1)$ factor in the exponent, and the constant factor in the stretch, this is the best possible by the girth argument. 3. For any $\epsilon \in (0,1)$ and integer $k\geq 1$, there is a $(3+\epsilon, \beta)$-spanner with $\beta=O_{\epsilon}(k^{\log(3+8/\epsilon)})$ and $O_{k,\epsilon}(n^{1+1/k})$ edges. We also consider the related graph concept of hopsets introduced by [Cohen, J. ACM '00]. We present a new family of $(\alpha,\beta)$ hopsets with $\widetilde{O}(k \cdot n^{1+1/k})$ edges and $\alpha \cdot \beta=O(k)$. Most notably, we show a construction of $(3+\epsilon,\beta)$ hopset with $\widetilde{O}_{k,\epsilon}(n^{1+1/k})$ edges and hop-bound of $\beta=O_{\epsilon}(k^{\log(3+9/\epsilon)})$, improving upon the state-of-the-art hop-bound of $\beta=O(\log k /\epsilon)^{\log k}$ by [Elkin-Neiman, '17] and [Huang-Pettie, '17].