Vertex Fault-Tolerant Geometric Spanners for Weighted Points
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Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer k ≥1, and a real number ϵ> 0, we present algorithms for computing a spanner network G(S, E) for the metric space (S, d_w) induced by the weighted points in S. The weighted distance function d_w on the set S of points is defined as follows: for any p, q ∈S, d_w(p, q) is equal to w(p) + d_π(p, q) + w(q) if p q, otherwise, d_w(p, q) is 0. Here, d_π(p, q) is the Euclidean distance between p and q if points in S are in ℝ^d, otherwise, it is the geodesic (Euclidean) distance between p and q. The following are our results: (1) When the weighted points in S are located in ℝ^d, we compute a k-vertex fault-tolerant (4+ϵ)-spanner network of size O(k n). (2) When the weighted points in S are located in the relative interior of the free space of a polygonal domain P, we detail an algorithm to compute a k-vertex fault-tolerant (4+ϵ)-spanner network with O(kn√(h+1)/ϵ^2 n) edges. Here, h is the number of simple polygonal holes in P. (3) When the weighted points in S are located on a polyhedral terrain T, we propose an algorithm to compute a k-vertex fault-tolerant (4+ϵ)-spanner network, and the number of edges in this network is O(kn/ϵ^2 n).
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