A Unified Framework for Hopsets and Spanners

TL;DR

The paper introduces a unified framework that simultaneously constructs and analyzes hopsets, spanners, and emulators via a generalized TZ-style algorithm. By parameterizing a blend of linear-TZ and exponential-TZ strategies and using rounding functions, it achieves all state-of-the-art regimes across multiplicative, additive, and hybrid stretches, while removing the $\log\Lambda$ factor from sizes in many cases. A key theoretical result is a lower bound showing $\alpha\cdot\beta = \Omega(k)$ for $O(n^{1+1/k})$-size hopsets, which matches the BP20 upper bounds in the considered regime. The work also provides a comprehensive toolkit for converting hopsets into spanners and emulators, including non-simultaneous and simultaneous variants, with tight size and stretch guarantees and detailed analysis via Jumping Lemmas. Overall, the framework unifies prior approaches, simplifies constructions, and sharpens both upper and lower bounds for core graph-structure tasks.

Abstract

Given an undirected graph $G=(V,E)$, an {\em $(α,β)$-spanner} $H=(V,E')$ is a subgraph that approximately preserves distances; for every $u,v\in V$, $d_H(u,v)\le α\cdot d_G(u,v)+β$. An $(α,β)$-hopset is a graph $H=(V,E")$, so that adding its edges to $G$ guarantees every pair has an $α$-approximate shortest path that has at most $β$ edges (hops), that is, $d_G(u,v)\le d_{G\cup H}^{(β)}(u,v)\le α\cdot d_G(u,v)$. 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 $α$. 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 $(O(k^ε),O(k^{1-ε}))$-hopset of size $\tilde{O}(n^{1+1/k})$ was shown for any $n$-vertex graph and parameter $0<ε<1$, and they asked whether this result is best possible. We resolve this open problem, showing that any $(α,β)$-hopset of size $O(n^{1+1/k})$ must have $α\cdot β\geΩ(k)$.

Figures (3)

• Figure 1: $P_{u,v}$ is marked by green edges. Every edge of the hopset $H$ is colored in blue, and the shortest path of $G$ connecting its ends is marked by a dashed line. The edges of the path $P\subseteq G\cup H$ are marked by a thick line, and we assume by negation that its detours are disjoint from $P_{u,v}$. Then, since also $P$ doesn't use the edge $(a,b)$ of $P_{u,v}$, there must be a simple cycle with size $\leq|P_{u,v}|+|\hat{P}|$.
• Figure 2: The potential path between $u$ and $u'$. Notice that $d(u,p_{i-1}(u))\leq r_{i-1}$ and $d(u,p_{f^{-1}(i-1)}(u))\leq r_{f^{-1}(i-1)}$, since $score(u)=i$.
• Figure 3: The graph $G(k,f,\alpha,n)$. $P$ contains $n^{\frac{1}{2k}}$ vertices, each of them is the $0$'th layer of a copy of $T(k,f,\alpha,n)$.

Theorems & Definitions (69)

• Theorem 1: LPS88
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof
• Definition 1
• Lemma 3
• proof