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 = c W , 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 + c W ( 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 + ε W ( · , · ) 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.