Geodesic spanners for points on a polyhedral terrain

Let S be a set S of n points on a polyhedral terrain T in R3, and let " > 0 be a xed constant. We prove that S admits a (2 + ")-spanner with O(n log n) edges with respect to the geodesic distance. This is the rst spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in Rd admits an additively weighted (2 + ")-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + "), and almost matches the lower bound

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

Geodesic Spanners for Points on a Polyhedral Terrain

Let S be a set of n points on a polyhedral terrain \scrT in \BbbR 3 , and let \varepsilon > 0 be a fixed constant. We prove that S admits a (2 + \varepsilon )-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in \BbbR d admits an additively weighted (2 + \varepsilon )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + \varepsilon ) and almost matches the lower bound

Geodesic spanners for points on a polyhedral terrain

Let S be a set S of n points on a polyhedral terrain T in R3, and let > 0 be a xed constant. We prove that S admits a (2 + )-spanner with O(n log n) edges with respect to the geodesic distance. This is the rst spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in Rd admits an additively weighted (2 + )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ), and almost matches the lower bound

Geodesic spanners for points on a polyhedral terrain

Let S be a set S of n points on a polyhedral terrain T in ℝ3, and let ∊ > 0 be a fixed constant. We prove that S admits a (2 + ∊)-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in ℝd admits an additively weighted (2 + ∊)-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ∊), and almost matches the lower bound

Geodesic spanners for points on a polyhedral terrain

Let S be a set S of n points on a polyhedral terrain T in ℝ3, and let ∊ > 0 be a fixed constant. We prove that S admits a (2 + ∊)-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in ℝd admits an additively weighted (2 + ∊)-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ∊), and almost matches the lower bound

Geodesic Spanners for Points on a Polyhedral Terrain

Let S be a set of n points on a polyhedral terrain \scrT in \BbbR 3 , and let \varepsilon > 0 be a fixed constant. We prove that S admits a (2 + \varepsilon )-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in \BbbR d admits an additively weighted (2 + \varepsilon )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + \varepsilon ) and almost matches the lower bound