Additive Spanners in Nearly Quadratic Time

We consider the problem of efficiently finding an additive C-spanner of an undirected unweighted graph G, that is, a subgraph H so that for all pairs of vertices u,v, δ H (u,v) ≤ δ G (u,v) + C, where δ denotes shortest path distance. It is known that for every graph G, one can find an additive 6-spanner with O(n 4/3) edges in O(mn 2/3) time. It is unknown if there exists a constant C and an additive C-spanner with o(n 4/3) edges. Moreover, for C ≤ 5 all known constructions require Ω(n 3/2) edges.

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