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Vertex Fault-Tolerant Geometric Spanners for Weighted Points
arXiv - CS - Computational Geometry Pub Date : 2020-11-05 , DOI: arxiv-2011.03354
Sukanya Bhattacharjee, R. Inkulu

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 \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 k-vertex fault-tolerant (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 k-vertex fault-tolerant (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 k-vertex fault-tolerant (4+\epsilon)-spanner network, and the number of edges in this network is O(\frac{kn}{\epsilon^2} \lg{n}).

中文翻译:

加权点的顶点容错几何扳手

给定一个由 n 个点组成的集合 S、一个将非负权重与 S 中的每个点相关联的权重函数 w、一个正整数 k\ge 1 和一个实数 \epsilon > 0,我们提出了计算扳手网络的算法G(S, E) 用于由 S 中的加权点引起的度量空间 (S, d_w)。 点集 S 上的加权距离函数 d_w 定义如下:对于任何 p, q \in S, d_w( p, q) 等于 w(p) + d_\pi(p, q) + w(q) 如果 p \ne q,否则,d_w(p, q) 为 0。这里,d_\pi(p, q) 是 p 和 q 之间的欧几里德距离,如果 S 中的点在 \mathbb{R}^d 中,否则,它是 p 和 q 之间的测地线(欧几里德)距离。以下是我们的结果: (1) 当 S 中的加权点位于 \mathbb{R}^d 时,我们计算大小为 O(kn) 的 k-顶点容错 (4+\epsilon)-spanner 网络. (2) 当 S 中的加权点位于多边形域 \cal P 的自由空间的相对内部时,我们详细说明了计算 k 顶点容错 (4+\epsilon)-spanner 网络的算法,其具有O(\frac{kn\sqrt{h+1}}{\epsilon^2} \lg{n}) 边。这里,h 是 \cal P 中简单多边形孔的数量。 (3) 当 S 中的加权点位于多面体地形 \cal T 上时,我们提出了一种算法来计算 k 顶点容错 (4+ \epsilon)-spanner 网络,该网络的边数为 O(\frac{kn}{\epsilon^2} \lg{n})。
更新日期:2020-11-09
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