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Hopsets and spanners are fundamental graph structures, playing a key role inshortest path computation, distributed communication, and more. A (near-exact)hopset for a given graph $G$ is a (small) subset of weighted edges $H$ thatwhen added to the graph $G$ reduces the number of hops (edges) of near-exactshortest paths. Spanners and distance preservers, on the other hand, ask forremoving many edges from the graph while approximately preserving shortest pathdistances.  We provide a general reduction scheme from graph hopsets to the known metriccompression schemes of spanners, emulators and distance preservers.Consequently, we get new and improved upper bound constructions for the latter,as well as, new lower bound results for hopsets. Our work makes a significantprogress on the tantalizing open problem concerning the formal connectionbetween hopsets and spanners, e.g., as posed by Elkin and Neiman [Bull. EATCS2020].

Having Hope in Hops: New Spanners, Preservers and Lower Bounds for Hopsets