A General Reduction from Near-Additive Emulators to Near-Exact Hopsets

2026-07-08

AI Summary

This work resolves an open problem posed by Elkin and Neiman by presenting the first general black-box reduction from near-additive emulators to near-exact hopsets. Given any $(1+\varepsilon',\beta)$-emulator, the method constructs a $(1+\varepsilon, O(\beta^2/\varepsilon^2 \cdot \log(n/\varepsilon)))$-hopset for weighted undirected graphs with size $O((S_A(n + m\beta/\varepsilon^2, \varepsilon/294, \beta)/\varepsilon + n) \cdot \log(n/\varepsilon))$. The reduction leverages parameter scaling and graph transformation techniques to preserve comparable stretch, hopbound, and size, thereby strengthening the theoretical connection between these two classes of distance approximation structures—particularly in sparse graphs.

Abstract

Graph emulators and hopsets are two fundamental concepts for distance approximation. When the multiplicative stretch is $1+ε$ for arbitrarily small $ε>0$, these structures are known as near-additive emulators and near-exact hopsets, respectively. Prior work showed that there is a remarkable similarity between the constructions and guarantees of these two objects. In their survey on this topic, Elkin and Neiman [Bull. EATCS 130, 2020] explicitly asked whether one can obtain a general reduction between near-additive emulators and near-exact hopsets. Following that, Kogan and Parter [FOCS, 2022] provided a general reduction from hopsets to emulators and spanners. In this paper, we address the reverse direction and show that any construction for a near-additive emulator for undirected unweighted graphs can be leveraged as a black box to construct a hopset for an undirected weighted graph with comparable size, stretch, and a hopbound comparable to the emulator's additive stretch. Specifically, we show that any algorithm that constructs a $(1+ε',β)$-emulator, with $0 \le ε' \le 1$ and $β\ge 1$, of size $S_{\mathcal{A}}(n, ε',β)$, can be used to obtain a $(1+ε, O(\frac{β^2}{ε^2} \ln(\frac{n}ε)))$-hopset of size $O((S_{\mathcal{A}}(n+m\fracβ{ε^2}, \fracε{294},β) \frac{1}ε + n)\ln(\frac{n}ε))$, for any $0 < ε\le 1$. Therefore, our reduction answers the question of Elkin and Neiman [Bull. EATCS 130, 2020] for sparse graphs and further advances the understanding of the formal connection between these two structures. Designing a reduction resulting in a hopset size that does not depend on $m$ remains an intriguing open question.

Problem

• near-additive emulators
• near-exact hopsets
• graph distance approximation
• reduction
• sparse graphs

Innovation

• near-additive emulators
• near-exact hopsets
• graph sparsification
• black-box reduction
• distance approximation