Thorup-Zwick Emulators are Universally Optimal Hopsets.

A (β,ϵ)-hopset is, informally, a weighted edge set that, when added to a graph, allows one to get from point a to point b using a path with at most β edges (“hops”) and length (1+ϵ)dist(a,b). In this paper we observe that Thorup and Zwick's sublinear additive emulators are also actually (O(k/ϵ)k,ϵ)-hopsets for every ϵ>0, and that with a small change to the Thorup–Zwick construction, the size of the hopset can be made O(n1+12k+1−1). As corollaries, we also shave “k” factors off the size of Thorup and Zwick's [20] sublinear additive emulators and the sparsest known (1+ϵ,O(k/ϵ)k−1)-spanners, due to Abboud, Bodwin, and Pettie [1].