An optimal algorithm for computing angle-constrained spanners

Let S be a set of n points in ℝd. A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n logn) time, where t depends only on θ

Fault-tolerant additive weighted geometric spanners

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S, E) is called a t-spanner for the additive weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.d_w(p, q) for a real number t > 1. Here, d_w(p,q) is the additive weighted distance between p and q. For some integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S' \subset S with cardinality at most k, the graph G \ S' is a t-spanner for the points in S \ S'. For any given real number \epsilon > 0, we obtain the following results: - When the points in S belong to Euclidean space R^d, an algorithm to compute a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w). Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between p and q in R^d. - When the points in S belong to a simple polygon P, for the metric space (S, d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here, for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along the shortest path between p and q in P. - When the points in $S$ lie on a terrain T, an algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges.Comment: a few update

Robust Geometric Spanners

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.Comment: 18 pages, 8 figure

An optimal algorithm for computing angle-constrained spanners

Let S be a set of n points in Rd and let t>1 be a real number. A graph G=(S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p andq is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0<θ<π/3, a θ-angle-constrained t-spanner can be computed in O(nlog n) time, where t depends only on θ. For values of θ approaching 0, we havet=1 + O(θ)