# Thorup-Zwick Emulators are Universally Optimal Hopsets

## TL;DR

This paper shows that Thorup-Zwick emulators are also universal hopsets with near-optimal size, improving bounds on hopset construction and related graph sparsification structures.

## Contribution

It demonstrates that Thorup-Zwick's sublinear additive emulators serve as universal hopsets and improves their size bounds with minor modifications.

## Key findings

- Thorup-Zwick emulators are also $(O(k/eta)^k,eta)$-hopsets.
- Modified construction reduces hopset size to $O(n^{1+1/(2^{k+1}-1)})$.
- Results improve bounds on sublinear additive emulators and graph spanners.

## Abstract

A $(\beta,\epsilon)$-$\textit{hopset}$ 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 $\beta$ edges ("hops") and length $(1+\epsilon)\mathrm{dist}(a,b)$. In this paper we observe that Thorup and Zwick's $\textit{sublinear additive}$ emulators are also actually $(O(k/\epsilon)^k,\epsilon)$-hopsets for every $\epsilon>0$, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made $O(n^{1+\frac{1}{2^{k+1}-1}})$. As corollaries, we also shave "$k$" factors off the size of Thorup and Zwick's sublinear additive emulators and the sparsest known $(1+\epsilon,O(k/\epsilon)^{k-1})$-spanners, due to Abboud, Bodwin, and Pettie.