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

TL;DR

This paper establishes a connection between hopsets and spanners, leading to new bounds and insights for graph distance approximation structures, advancing understanding of their relationships and limitations.

Contribution

It introduces a reduction from hopsets to spanners, emulator, and distance preserver schemes, yielding improved bounds and new lower bounds for hopsets.

Findings

New upper bounds for spanners, emulators, and distance preservers.

New lower bounds for hopsets.

A reduction scheme linking hopsets to other graph compression structures.

Abstract

Hopsets and spanners are fundamental graph structures, playing a key role in shortest path computation, distributed communication, and more. A (near-exact) hopset for a given graph $G$ is a (small) subset of weighted edges $H$ that when added to the graph $G$ reduces the number of hops (edges) of near-exact shortest paths. Spanners and distance preservers, on the other hand, ask for removing many edges from the graph while approximately preserving shortest path distances. We provide a general reduction scheme from graph hopsets to the known metric compression 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 significant progress on the tantalizing open problem concerning the formal connection between hopsets and spanners, e.g., as posed by Elkin and Neiman [Bull. EATCS 2020].