A Unified Framework for Hopsets and Spanners

Given an undirected graph G = (V, E), an (α, β)-spanner H = (V, E' ) is a subgraph that approximately preserves distances; for every u, v ∈ V , d_H(u, v) ≤ α·d_G(u, v)+β. An (α, β)-hopset is a graph H = (V, E00), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, dG(u, v) ≤ d ^(β) _G∪H(u, v) ≤ α · 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 α. 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 [BLP20], given a parameter k, a (O(k ^∈, O(k^1−∈))-hopset of size O˜(n^1+1/k) was shown for any n-vertex graph and parameter 0 < ∈ < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α, β)-hopset of size O(n^1+1/k) must have α · β ≥ Ω(k).