Let (V,δ) be a finite metric space, where V is a set of n points and δ is a distance function defined for these points. Assume that (V,δ) has a doubling dimension d and assume that each point p∈V has a disk of radius r(p) around it. The disk graph that corresponds to V and r(⋅) is a directed graph I(V,E,r), whose vertices are the points of V and whose edge set includes a directed edge from p to q if δ(p,q)≤r(p). In Peleg and Roditty (Proc. 7th Int. Conf. on Ad‐Hoc Networks and Wireless (AdHoc‐NOW), pp. 622–633, 2008) we presented an algorithm for constructing a (1+ϵ)‐spanner of size O(nϵ
−d
logM), where M is the maximal radius r(p). The current paper presents two results. The first shows that the spanner of Peleg and Roditty (in Proc. 7th Int. Conf. on Ad‐Hoc Networks and Wireless (AdHoc‐NOW), pp. 622–633, 2008) is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of M. The second result shows that by slightly relaxing the requirements and allowing a small augmentation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r
1+ϵ
), where r
1+ϵ
(p)=(1+ϵ)⋅r(p) for every p∈V, then it is possible to get a (1+ϵ)‐spanner of size O(n/ϵ
d
) for I(V,E,r). Our algorithm is simple and can be implemented efficiently.
