On Additive Spanners in Weighted Graphs with Local Error

Abstract

An additive $+\beta$ spanner of a graph G is a subgraph which preserves distances up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners.

For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error $\beta = cW$, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error $+cW(s,t)$ for each vertex pair (s, t), where W(s, t) is the maximum edge weight along the shortest s–t path in G. These include pairwise $+(2+\varepsilon )W(\cdot ,\cdot )$ and $+(6+\varepsilon ) W(\cdot , \cdot )$ spanners over vertex pairs $\mathcal {P}\subseteq V \times V$ on $O_{\varepsilon }(n|\mathcal {P}|^{1/3})$ and $O_{\varepsilon }(n|\mathcal {P}|^{1/4})$ edges for all $\varepsilon > 0$, which extend previously known unweighted results up to $\varepsilon$ dependence, as well as an all-pairs $+4W(\cdot ,\cdot )$ spanner on $\widetilde{O}(n^{7/5})$ edges.

Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a $+\varepsilon W(\cdot ,\cdot )$ spanner with $O_{\varepsilon }(n)$ lightness, and a $+(4+\varepsilon ) W(\cdot ,\cdot )$ spanner with $O_{\varepsilon }(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.