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 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+∈)-spanner of size O(n∈ ^-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+∈ ), 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.