# 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+)W(·,·) and +(6+) W(·, ·) spanners over vertex pairs ⊆ V × V on O_(n||^1/3) and O_(n||^1/4) edges for all > 0, which extend previously known unweighted results up to dependence, as well as an all-pairs +4W(·,·) 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 + W(·,·) spanner with O_(n) lightness, and a +(4+) W(·,·) spanner with O_(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.

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