I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previous best lower bound of 1.4161 for the worst-case dilation of plane spanners. (II) For every n ≥ 13, there exists an n-element point set S such that the degree 3 dilation of S equals 1 + √ 3 = 2.7321 . . . in the domain of plane geometric spanners. In the same domain, we show that for every n ≥ 6...

Tractability of multivariate problems has become nowadays a popular research subject. Polynomial tractability means that the solution of a d-variate problem can be solved to within ε with polynomial cost in ε−1 and d. Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems. By an unweighted problem we mea...

Consider a network linking the points of a rate-1 Poisson point process on the plane. Write ! (s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function ! (s), and on the analogous “worst-case” function !(s) where t...

We introduce a new geometric spanner whose construction is based on a generalization of the known Stable Roommates problem. The Stable Roommates Spanner combines the most desirable properties of geometric spanners: a natural definition, small degree, linear number of edges, strong (1 + ε)-spanner for every > 0, and an efficient construction algorithm. It is an improvement over the well-known Ya...

We present a distributed algorithm for creation of geometric spanners in a wireless sensor network. Given any connected network, we show that the algorithm terminates inO(1) time, irrespective of network size. Our algorithm uses an underlying clustering algorithm as a foundation for creating spanners and only relies on the periodic heartbeat messages associated with cluster maintenance for the ...

The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them. A path has stretch t if its weight is at most t times the distance between its end points. We consider a weighted undirected graph G = (V, E) and present algorithms that compute paths with stretch 2 5 t 5 log n. We present a q((m + k)n('+')lt) time randomized algorithm that finds paths bet...

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find (1+ ε)-approximate farthest neighbors in time O(logn) per query in D-dimensional Euclidean space for any constants D and ε. A...