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 + ǫ)-spanner of size O(n/ǫd 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...

• Type: Conference
• Year: 2010
• Where: STACS
• Authors: David Peleg, Liam Roditty
• Keywords: Constant Doubling Dimension, Disk Graph, STACS 2010, Theoretical Computer Science, finite Metric Space