(1 + ε )-SPANNER CONSTRUCTIONS FOR GENERAL GRAPHS.

An (α,β)-spanner of a graph G is a subgraph H such that distH(u,w) ≤ α disttG (u, w) + β for every pair of vertices u, w, where distG', (u, w) 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(n1+1/κ) for every integer it κ ≥ 1, and a polynomially constructible (1,2)-spanner (also known as additive 2-spanner) of size O(n&frac32;). 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, 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(n1+λ). [ABSTRACT FROM AUTHOR]