New weighted additive spanners

TL;DR

This paper constructs a near-optimal weighted additive spanner with a stretch of +6W_max and about O(n^{4/3}) edges, improving previous bounds and extending unweighted spanner techniques to weighted graphs.

Contribution

It provides a construction of a +6W_max weighted additive spanner with near-optimal size, resolving an open problem and extending unweighted spanner methods to weighted graphs.

Findings

Constructed a +6W_max spanner with O(n^{4/3}) edges.

Extended unweighted +6-spanner techniques to weighted graphs.

Developed fast algorithms for constructing +6W_max and +4W_max spanners.

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 $+(6+\epsilon)W_{\max}$ spanner with $O(n^{4/3}/\epsilon)$ edges for any fixed constant $\epsilon > 0$. Indeed, their result is stronger as the additive stretch is local: the stretch for any pair $u,v$ is $+(6+\epsilon)W_{uv}$ 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 $+(6+\epsilon)W_{\max}$ spanner of $\tilde{O}(\frac{n^{4/3}}{\epsilon})$ edges in $\tilde{O}(n^2)$ time.