(1+ε, β)-spanner constructions for general graphs

An (α, β)-spanner of a graph G is a subgraph H such that d_H (u, w) ≤ α · d_G (u, w) + β for every pair of vertices u, w, where d_G' (u, w) denotes the distance between two vertices u and v in G'. It is known that every graph G has a polynomially constructible (2k - 1,0)-spanner (a.k.a. multiplicative (2k - 1)-spanner) of size O(n^1+1/k) for every integer k ≥ 1, and a polynomially constructible (1, 2)-spanner (a.k.a. additive 2-spanner) of size Õ(n^3/2). This paper explores hybrid spanner constructions (involving both multiplicative and additive factors) for general graphs and shows that the multiplicative factor can be made arbitrarily close to 1 while keeping the spanner size arbitrarily close to O(n), at the cost of allowing the additive term to be a sufficiently large constant. More formally, we show that for any constant ε, δ > 0 there exists a constant β = β(ε, δ) such that for every n-vertex graph G there is an efficiently constructible (1+ε,β)-spanner of size O(n^1+δ). It follows that for any constant ε, δ > 0 and graph G, there exists a spanning subgraph H with O(n^1+δ) edges which behaves like a multiplicative (1 + ε)-spanner for "sufficiently distant" pairs of vertices.