Vertex FaultTolerant Geometric Spanners for Weighted Points
Abstract
Given a set S of n points, a weight function w to associate a nonnegative weight to each point in S, a positive integer k \ge 1, and a real number \epsilon > 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 \in S, d_w(p, q) is equal to w(p) + d_\pi(p, q) + w(q) if p \ne q, otherwise, d_w(p, q) is 0. Here, d_\pi(p, q) is the Euclidean distance between p and q if points in S are in \mathbb{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 \mathbb{R}^d, we compute a kvertex faulttolerant (4+\epsilon)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 \cal P, we detail an algorithm to compute a kvertex faulttolerant (4+\epsilon)spanner network with O(\frac{kn\sqrt{h+1}}{\epsilon^2} \lg{n}) edges. Here, h is the number of simple polygonal holes in \cal P. (3) When the weighted points in S are located on a polyhedral terrain \cal T, we propose an algorithm to compute a kvertex faulttolerant (4+\epsilon)spanner network, and the number of edges in this network is O(\frac{kn}{\epsilon^2} \lg{n}).
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.03354
 Bibcode:
 2020arXiv201103354B
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 preliminary version accepted to COCOON 2019. arXiv admin note: substantial text overlap with arXiv:1709.01061