4 research outputs found
Relaxed spanners for directed disk graphs
Let be a finite metric space, where is a set of points
and is a distance function defined for these points. Assume that
has a constant doubling dimension and assume that each point
has a disk of radius around it. The disk graph that corresponds
to and is a \emph{directed} graph , whose vertices are
the points of and whose edge set includes a directed edge from to
if . In \cite{PeRo08} we presented an algorithm for
constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where is
the maximal radius . The current paper presents two results. The first
shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for
metrics of constant doubling dimension it is not possible to guarantee a
spanner whose size is independent of . 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+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every , then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for
. Our algorithm is simple and can be implemented efficiently