$(1 + \epsilon,\beta)$-Spanner Constructions for General Graphs

• Authors: Michael Elkin;David Peleg
• Venue: SIAM Journal on Computing
• Year: 2004

An {\em $(\alpha,\beta)$-spanner} of a graph G is a subgraph H such that $\mathit{dist}_H(u,w)\le \alpha\cdot \mathit{dist}t_G(u,w)+\beta$ 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 $(2\kappa-1,0)$-spanner (also known as multiplicative $(2\kappa-1)$-spanner) of size $O(n^{1+1/\kappa})$ for every integer $\kappa\ge 1$, and a polynomially constructible (1,2)-spanner (also known as additive 2-spanner) of size ${\tilde O}(n^{3/2})$. 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 $\epsilon, \lambda 0$ there exists a constant $\beta = \beta(\epsilon, \lambda)$ such that for every $n$-vertex graph G there is an efficiently constructible $(1+ \epsilon, \beta)$-spanner of size $O(n^{1 + \lambda})$.