Additive Spanners: A Simple Construction

Abstract

We consider additive spanners of unweighted undirected graphs. Let G be a graph and H a subgraph of G. The most naïve way to construct an additive k-spanner of G is the following: As long as H is not an additive k-spanner repeat: Find a pair (u,v) ∈ H that violates the spanner-condition and a shortest path from u to v in G. Add the edges of this path to H.

We show that, with a very simple initial graph H, this naïve method gives additive 6- and 2-spanners of sizes matching the best known upper bounds. For additive 2-spanners we start with H = ∅ and end with O(n ^3/2) edges in the spanner. For additive 6-spanners we start with H containing $\lfloor n^{1/3} \rfloor$ arbitrary edges incident to each node and end with a spanner of size O(n ^4/3).