Relaxed spanners for directed disk graphs

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 constant 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 [8] we presented an algorithm for constructing a (1 + ∈)-spanner of size O(n/∈^d log M), where M is the maximal radius r(p). The current paper presents two results. The first shows that the spanner of [8] 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 perturbation 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+∈), where r_1+∈(p) = (1 + ∈) ·r(p) for every p ∈ V, then it is possible to get a (1 + ∈)-spanner of size O(n/∈^d) for I(V,E,r). Our algorithm is simple and can be implemented efficiently.