(1 + εΒ)-spanner constructions for general graphs

An <italic>(α,Β)</italic>-spanner of a graph <italic>G</italic> is a subgraph <italic>H</italic> such that <italic>d_H(u,w)\le &agr\cdot d_G(u,w)+&Bgr</italic> for every pair of vertices <italic>u,w</italic>, where <italic>d_{G'}(u,w)</italic> denotes the distance between two vertices <italic>u</italic> and <italic>v</italic> in <italic>G'</italic>. It is known that every graph <italic>G</italic> has a polynomially constructible <italic>(2&kgr;-1,0)</italic>-spanner (a.k.a. <italic>multiplicative</italic> <italic>(2&kgr;-1)</italic>-spanner) of size <italic>O(n^{1+1/&kgr})</italic> for every integer <italic>&kgr\ge 1</italic>, and a polynomially constructible <italic>(1,2)</italic>-spanner (a.k.a. <italic>additive</italic> 2-spanner) of size <italic>\tO(n^{3/2})</italic>. This paper explores hybrid spanner constructions (involving both multiplicative and additive factors) for general graphs and shows that the multiplicative factor can be made <italic>arbitrarily close to 1</italic> while keeping the spanner size arbitrarily close to <italic>O(n)</italic>, at the cost of allowing the additive term to be a sufficiently large constant. More formally, we show that for any constant <italic>&egr, &dgr > 0</italic> there exists a constant <italic>&Bgr = &Bgr(&egr, &dgr)</italic> such that for every <italic>n</italic>-vertex graph <italic>G</italic> there is an efficiently constructible <italic>(1+ &egr, &Bgr)</italic>-spanner of size <italic>O(n^{1 + &dgr})<italic>. It follows that for any constant <italic>&egr, &dgr > 0</italic> there exists a constant <italic>&Bgr(&egr, &dgr)</italic> such that for any <italic>n</italic>-vertex graph <italic>G = (V,E)</italic> there exists an efficiently constructible subgraph <italic>(V,H)</italic> with <italic>O(n^{1 +&dgr})</italic> edges such that <italic>d_H(u,w) \le (1 + &egr) d_G(u,w) </italic> for every pair of vertices.