On Additive Spanners in Weighted Graphs with Local Error

An additive + β spanner of a graph G is a subgraph which preserves distances up to an additive + β 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 β= 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 + ε) W(·, · ) and + (6 + ε) W(·, · ) spanners over vertex pairs P⊆ V× V on O_ε(n| P|^{1 / 3}) and O_ε(n| P|^{1 / 4}) edges for all ε> 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs + 4 W(·, · ) spanner on 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 + εW(·, · ) spanner with O_ε(n) lightness, and a + (4 + ε) W(·, · ) spanner with O_ε(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.