$(α, β)$ Spanners and Hybrid Spanners with Nearly Tight Bounds

2026-09-30

AI Summary

This study addresses the limitation that existing graph spanner construction methods fail to achieve theoretical lower bounds in both multiplicative stretch and edge count. By integrating graph-theoretic algorithm design, combinatorial analysis, and asymptotic complexity optimization techniques, this work proposes a novel spanner construction framework. It achieves near-optimal $(\alpha, \beta)$-spanners and hybrid spanners with $O(n^{1+1/k}+kn)$ edges. Furthermore, the proposed approach reduces the multiplicative stretch from $O(k^\varepsilon)$ to $O(\log k)$ and eliminates the $k^2$ factor in hybrid spanners, approaching optimal bounds under the Erdős girth conjecture. These results significantly reduce edge complexity while enhancing distance approximation accuracy, comprehensively improving upon prior work presented at SODA'20.

Abstract

For an $n$-vertex undirected, unweighted graph $G=(V,E)$ and a positive integer $k$, we present new spanner constructions with $O_k(n^{1+1/k})$ edges that achieve nearly optimal guarantees for all distances $d\le k$. Specifically, we construct a spanner $H\subseteq G$ with $O(n^{1+1/k}+(k+d\log d)n)$ edges, ensuring that any pair at original distance at most $d$ satisfies $\mathrm{dist}_H(u,v)\le 2k+O(d\log d)$. Equivalently, the multiplicative stretch for pairs at distance $d$ is $2k/d+O(\log d)$. In particular, setting $d=k/\log k$ yields an $(O(\log k),O(k))$-spanner with $O(n^{1+1/k}+kn)$ edges. For comparison, Ben-Levy and Parter (SODA'20) obtained, for every fixed $\varepsilon>0$ and sufficiently large $k$, an $(O(k^\varepsilon),O_\varepsilon(k))$-spanner with $O_{\varepsilon,k}(n^{1+1/k})$ edges. Our result improves the multiplicative stretch from $O(k^\varepsilon)$ to $O(\log k)$ while keeping the additive term linear in $k$, bringing us closer to the goal of $(O(1),O(k))$-spanners. Furthermore, Ben-Levy and Parter obtained multiplicative stretch $O_\varepsilon(k/d)$ for distances $d\le k^{1-\varepsilon}$, for every fixed $\varepsilon>0$, and an explicit bound of $7k/d$ for $d\le\sqrt{k}/2$. We achieve $2k/d+O(\log d)$, which is $(2+o(1))k/d$ whenever $d=o(k/\log k)$. Our second result is an improved construction of $k$-hybrid spanners, which guarantee stretch $2k-1$ for adjacent pairs and $k$ for non-adjacent pairs. Parter's original construction uses $O(k^2 n^{1+1/k})$ edges; we achieve the same guarantees with $O(n^{1+1/k}+kn)$ edges, removing the $k^2$ factor from the $n^{1+1/k}$ term. For every fixed $k$, our edge bound is optimal up to a constant factor under Erdős' girth conjecture.

Problem

• spanners
• hybrid spanners
• stretch factor
• graph sparsification
• distance approximation

Innovation

• spanner construction
• hybrid spanners
• multiplicative stretch
• additive stretch
• graph sparsification