6 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
Efficiently Navigating a Random Delaunay Triangulation
International audiencePlanar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. Whilst many algorithms have been proposed, very little theoretical analysis is available for the properties of the paths generated or the computational resources required to generate them. In this work, we propose and analyse a new planar navigation algorithm for the Delaunay triangulation. We then demonstrate a number of strong theoretical guarantees for the algorithm when it is applied to a random set of points in a convex region
Efficient Delaunay-based Localized Routing for Wireless Sensor Networks
Consider a wireless sensor network consisting of n wireless sensors randomly distributed in a twodimensional plane. In this paper, we show that with high probability we can locally find a path for any pair of sensors such that the length of the path is no more than a constant factor of the minimum. By assuming each sensor knows its position, our new routing method decides where to forward the message purely based on the position of current node, its neighbors, and the positions of the source and the target. Our method is based on a novel structure called localized Delaunay triangulation [1] and a geometric routing method [2] that guarantees that the distance traveled by the packets is no more than a small constant factor of the minimum when the Delaunay triangulation of sensor nodes are known. Our experiments show that the delivery rates of existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed topologies, and our localized routing protocol based on Delaunay triangulation works well in practice. We also conducted extensive simulations of another localized routing protocol, face routing [13]. The path found by this protocol is also reasonably good compared with previous one although it cannot guarantee a constant approximation on the length of the path traveled theoretically. Copyright c â—‹ 2000 John Wiley & Sons, Ltd. key words: Wireless sensor networks, Delaunay triangulation, localized routing, power efficiency