Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: La, An, Le, Hung
Institution: University of Massachusetts Amherst
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: HDR TRIPODS: Institute for Integrated Data Science: A Transdisciplinary Approach to Understanding Fundamental Trade-offs and Theoretical Foundations, NSF-BSF: Small: AF: Towards a Unified Theory of Spanners, G5987536167, G6671297155, CAREER: Geometric Techniques for Topological Graph Algorithms, G8616775632
OpenAlex ID: W4402704300