Near-Additive Spanners and Near-Exact Hopsets, A Unified View

Abstract

Given an {\em unweighted} undirected graph $G = (V,E)$, and a pair of parameters $\epsilon > 0$, $\beta = 1,2,\ldots$, a subgraph $G' =(V,H)$, $H \subseteq E$, of $G$ is a {\em $(1+\epsilon,\beta)$-spanner} (aka, a {\em near-additive spanner}) of $G$ if for every $u,v \in V$, $$d_{G'}(u,v) \le (1+\epsilon)d_G(u,v) + \beta~.$$ It was shown in \cite{EP01} that for any $n$-vertex $G$ as above, and any $\epsilon > 0$ and $\kappa = 1,2,\ldots$, there exists a $(1+\epsilon,\beta)$-spanner $G'$ with $O_{\epsilon,\kappa}(n^{1+1/\kappa})$ edges, with $$\beta = \beta_{EP} = \left({{\log \kappa} \over \epsilon}\right)^{\log \kappa - 2}~.$$ This bound remains state-of-the-art, and its dependence on $\epsilon$ (for the case of small $\kappa$) was shown to be tight in \cite{ABP18}. Given a {\em weighted} undirected graph $G = (V,E,\omega)$, and a pair of parameters $\epsilon > 0$, $\beta = 1,2,\ldots$, a graph $G'= (V,H,\omega')$ is a {\em $(1+\epsilon,\beta)$-hopset} (aka, a {\em near-exact hopset}) of $G$ if for every $u,v \in V$, $$d_G(u,v) \le d_{G\cup G'}^{(\beta)}(u,v) \le (1+\epsilon)d_G(u,v)~,$$ where $ d_{G\cup G'}^{(\beta)}(u,v)$ stands for a $\beta$-(hop)-bounded distance between $u$ and $v$ in the union graph $G \cup G'$. It was shown in \cite{EN16} that for any $n$-vertex $G$ and $\epsilon$ and $\kappa$ as above, there exists a $(1+\epsilon,\beta)$-hopset with $\tilde{O}(n^{1+1/\kappa})$ edges, with $\beta = \beta_{EP}$. Not only the two results of \cite{EP01} and \cite{EN16} are strikingly similar, but so are also their proof techniques. Moreover, Thorup-Zwick's later construction of near-additive spanners \cite{TZ06} was also shown in \cite{EN19,HP17} to provide hopsets with analogous (to that of \cite{TZ06}) properties. In this survey we explore this intriguing phenomenon, sketch the basic proof techniques used for these results, and highlight open questions.