A Unified Framework for Hopsets and Spanners

TL;DR

This paper presents a unified algorithmic framework that constructs optimal or near-optimal spanners and hopsets for graphs, improving previous results and resolving an open problem on the trade-offs between stretch and hopbound.

Contribution

The authors develop a single versatile algorithm that attains all known state-of-the-art spanners and hopsets, and establish a lower bound on the product of stretch and hopbound.

Findings

The unified algorithm matches or improves previous bounds for various regimes.

A lower bound shows the product of stretch and hopbound must be at least proportional to k.

The work resolves an open question about the optimality of hopset size and parameters.

Abstract

Given an undirected graph $G=(V,E)$, an {\em $(\alpha,\beta)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,v\in V$, $d_H(u,v)\le \alpha\cdot d_G(u,v)+\beta$. An $(\alpha,\beta)$-hopset is a graph $H=(V,E")$, so that adding its edges to $G$ guarantees every pair has an $\alpha$-approximate shortest path that has at most $\beta$ edges (hops), that is, $d_G(u,v)\le d_{G\cup H}^{(\beta)}(u,v)\le \alpha\cdot d_G(u,v)$. Given the usefulness of spanners and hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter $\alpha$. In this work we develop a single algorithm that can attain all state-of-the-art spanners and hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In \cite{BP20}, given a parameter $k$, a $(O(k^{\epsilon}),O(k^{1-\epsilon}))$-hopset of size $\tilde{O}(n^{1+1/k})$ was shown for any $n$-vertex graph and parameter $0<\epsilon<1$, and they asked whether this result is best possible. We resolve this open problem, showing that any $(\alpha,\beta)$-hopset of size $O(n^{1+1/k})$ must have $\alpha\cdot \beta\ge\Omega(k)$.