14 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
Asynchronous Local Construction of Bounded-Degree Network Topologies Using Only Neighborhood Information
We consider ad-hoc networks consisting of wireless nodes that are located
on the plane. Any two given nodes are called neighbors if they are located
within a certain distance (communication range) from one another. A given node
can be directly connected to any one of its neighbors and picks its connections
according to a unique topology control algorithm that is available at every
node. Given that each node knows only the indices (unique identification
numbers) of its one- and two-hop neighbors, we identify an algorithm that
preserves connectivity and can operate without the need of any synchronization
among nodes. Moreover, the algorithm results in a sparse graph with at most
edges and a maximum node degree of . Existing algorithms with the same
promises further require neighbor distance and/or direction information at each
node. We also evaluate the performance of our algorithm for random networks. In
this case, our algorithm provides an asymptotically connected network with
edges with a degree less than or equal to for fraction
of the nodes. We also introduce another asynchronous connectivity-preserving
algorithm that can provide an upper bound as well as a lower bound on node
degrees.Comment: To appear in IEEE Transactions on Communication
A Framework for Computing the Greedy Spanner
The highest quality geometric spanner (e.g. in terms of edge count, both in theory and in practice) known to be computable in polynomial time is the greedy spanner. The state-of-the-art in computing this spanner are a O(n^2 log n) time, O(n^2) space algorithm and a O(n^2 log^2 n) time, O(n) space algorithm, as well as the `improved greedy' algorithm, taking O(n^3 log n) time in the worst case and O(n^2) space but being faster in practice thanks to a caching strategy. We identify why this caching strategy gives speedups in practice. We formalize this into a framework and give a general efficiency lemma. From this we obtain many new time bounds, both on old algorithms and on new algorithms we introduce in this paper. Interestingly, our bounds are in terms of the well-separated pair decomposition, a data structure not actually computed by the caching algorithms. Specifically, we show that the `improved greedy' algorithm has a O(n^2 log n log Phi) running time (where Phi is the spread of the point set) and a variation has a O(n^2 log^2 n) running time. We give a variation of the linear space state-of-the-art algorithm and an entirely new algorithm with a O(n^2 log n log Phi) running time, both of which improve its space usage by a factor O(1/(t-1)). We present experimental results comparing all the above algorithms. The experiments show that - when using low t - our new algorithm is up to 200 times more space efficient than the existing linear space algorithm, while being comparable in running time and much easier to implement