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

An (α,β)-spanner of a graph G is a subgraph H such that dist _H ≤ α · distt _G(u, ω) + β for every pair of vertices u, ω, where dist _G′ (u, ω) denotes the distance between two vertices u and v in G′. It is known that every graph G has a polynomially constructible (2κ -1,0)-spanner (also known as multiplicative (2κ - 1)-spanner) of size O(n ^1-1/κ) for every integer κ ≥ 1, and a polynomially constructible (1, 2)-spanner (also known as additive 2-spanner) of size Ō(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 ∈,λ > 0 there exists a constant β= β(ε, λ) such that for every n-vertex graph G there is an efficiently constructible (1 + ε, β)-spanner of size O(n ^1+λ).