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

Abstract: For an $n$-vertex undirected, unweighted graph $G=(V,E)$ and a positive integer $k$, we present new spanner constructions with $O_k(n<sup>{1+1/k})$ 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 $\mathrm{dist}<em>H(u,v)\le 2k+O(d\log d)$. 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 $O(n<sup>{1+1/k}+kn)$ edges. For comparison, Ben-Levy and Parter (SODA'20) obtained, for every fixed $\varepsilon&gt;0$ and sufficiently large $k$, an $(O(k^{\varepsilon),O}</em>\varepsilon(k))$-spanner with $O_{\varepsilon,k}(n<sup>{1+1/k})$ edges. Our result improves the multiplicative stretch from $O(k<sup>\varepsilon)$ 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 $d\le k<sup>{1-\varepsilon}$, for every fixed $\varepsilon&gt;0$, 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 $O(k² n<sup>{1+1/k})$ edges; we achieve the same guarantees with $O(n<sup>{1+1/k}+kn)$ edges, removing the $k<sup>2$ factor from the $n<sup>{1+1/k}$ term. For every fixed $k$, our edge bound is optimal up to a constant factor under Erdős' girth conjecture.