A Unified Framework for Hopsets and Spanners

2021-08-22

arXiv.org

AI Summary

This paper addresses the unified optimization problem of efficiently constructing (α,β)-spanners and hopsets for undirected graphs, proposing the first tunable, general-purpose framework. Methodologically, it designs a greedy algorithm based on hierarchical clustering and probabilistic sampling, integrating distance labeling and bounded-hop shortest-path analysis. Theoretical contributions include: (1) the first unified construction for both spanners and hopsets; (2) a complete resolution of the open problem posed by Bernstein and Patt-Shamir (BP20), proving that any hopset of size O(n^{1+1/k}) must satisfy α·β = Ω(k), thereby establishing a tight lower bound; and (3) reproducibility and improvement upon all major prior results—achieving superior trade-offs among size, stretch, and hop-bound across diverse parameter regimes, and, for the first time, attaining a matching lower bound on the product of hopset size and (stretch × hop).

Abstract

Given an undirected graph $G=(V,E)$, an {em $(alpha,eta)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,vin V$, $d_H(u,v)le alphacdot d_G(u,v)+eta$. An $(alpha,eta)$-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 $eta$ edges (hops), that is, $d_G(u,v)le d_{Gcup H}^{(eta)}(u,v)le alphacdot d_G(u,v)$. 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 $ ilde{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 $(alpha,eta)$-hopset of size $O(n^{1+1/k})$ must have $alphacdot etageOmega(k)$.

Problem

• Efficient Spanners
• Hopsets Construction
• Optimal Graph Algorithms

Innovation

• unified algorithm
• graph spanners
• hopsets optimality