New weighted additive spanners
An La, Hung Le
TL;DR
The paper addresses constructing additive spanners for weighted graphs with constant additive stretch while maintaining sparsity; it achieves a +6W_{ ext{max}} spanner with \\tilde{O}(n^{4/3}) edges by adapting Woodruff's unweighted +6 spanner to the weighted setting using d-light initialization and a tailored analysis. It introduces MECSP and Weak CSSSP as subroutines to speed up path computations, enabling fast constructions and enabling a \\tilde{O}(n^{2})-time algorithm for a +(6+\\varepsilon)W_{ ext{max}} spanner with edges \\tilde{O}(n^{4/3}/\\varepsilon). The work also extends to a +2W-subsetwise spanner with \\tilde{O}(n\\sqrt{|S|}) edges and provides additional fast constructions for +4W_{ ext{max}} and related variants, advancing the transfer of unweighted spanner techniques to weighted graphs and reducing dependencies on W_{ ext{max}}. Overall, the results yield near-optimal sparsity and practical time bounds for weighted additive spanners, with potential local-stretch extensions and broader applicability to distance-preserving graph compressions.
Abstract
Ahmed, Bodwin, Sahneh, Kobourov, and Spence (WG 2020) introduced additive spanners for weighted graphs and constructed (i) a $+2W_{\max}$ spanner with $O(n^{3/2})$ edges and (ii) a $+4W_{\max}$ spanner with $\tilde{O}(n^{7/5})$ edges, and (iii) a $+8W_{\max}$ spanner with $O(n^{4/3})$ edges, for any weighted graph with $n$ vertices. Here $W_{\max} = \max_{e\in E}w(e)$ is the maximum edge weight in the graph. Their results for $+2W_{\max}$, $+4W_{\max}$, and $+8W_{\max}$ match the state-of-the-art bounds for the unweighted counterparts where $W_{\max} = 1$. They left open the question of constructing a $+6W_{\max}$ spanner with $O(n^{4/3})$ edges. Elkin, Gitlitz, and Neiman (DISC 2021) made significant progress on this problem by showing that there exists a $+(6+ε)W_{\max}$ spanner with $O(n^{4/3}/ε)$ edges for any fixed constant $ε> 0$. Indeed, their result is stronger as the additive stretch is local: the stretch for any pair $u,v$ is $+(6+ε)W_{uv}$ where $W_{uv}$ is the maximum weight edge on the shortest path from $u$ to $v$. In this work, we resolve the problem posted by Ahmed et al. (WG 2020) up to a poly-logarithmic factor in the number of edges: We construct a $+6W_{\max}$ spanner with $\tilde{O}(n^{4/3})$ edges. We extend the construction for $+6$-spanners of Woodruff (ICALP 2010), and our main contribution is an analysis tailoring to the weighted setting. The stretch of our spanner could also be made local, in the sense of Elkin, Gitlitz, and Neiman (DISC 2021). We also study the fast constructions of additive spanners with $+6W_{\max}$ and $+4W_{\max}$ stretches. We obtain, among other things, an algorithm for constructing a $+(6+ε)W_{\max}$ spanner of $\tilde{O}(\frac{n^{4/3}}ε)$ edges in $\tilde{O}(n^2)$ time.