A Unified Framework for Hopsets and Spanners

Given an undirected graph $G=(V,E)$, an {\em $(\alpha,\beta)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,v\in V$, $d_H(u,v)\le \alpha\cdot d_G(u,v)+\beta$. An {\em $(\alpha,\beta)$-hopset} is a graph $H=(V,E")$, so that adding its edges to $G$ guarantees every pair has an $\alpha$-approximate shortest path that has at most $\beta$ edges (hops), that is, $d_G(u,v)\le d_{G\cup H}^{(\beta)}(u,v)\le \alpha\cdot d_G(u,v)$. Given the usefulness of spanners and hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter $\alpha$. In this work we develop a single algorithm that can attain all state-of-the-art spanners and hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In \cite{BP20}, given a parameter $k$, a $(O(k^{\epsilon}),O(k^{1-\epsilon}))$-hopset of size $\tilde{O}(n^{1+1/k})$ was shown for any $n$-vertex graph and parameter $0<\epsilon<1$, and they asked whether this result is best possible. We resolve this open problem, showing that any $(\alpha,\beta)$-hopset of size $O(n^{1+1/k})$ must have $\alpha\cdot \beta\ge\Omega(k)$.