Vertex Fault-Tolerant Geometric Spanners for Weighted Points

Abstract

Given a set [Formula: see text] of [Formula: see text] points, a weight function [Formula: see text] to associate a non-negative weight to each point in [Formula: see text], a positive integer [Formula: see text], and a real number [Formula: see text], we present algorithms for computing a spanner network [Formula: see text] for the metric space [Formula: see text] induced by the weighted points in [Formula: see text]. The weighted distance function [Formula: see text] on the set [Formula: see text] of points is defined as follows: for any [Formula: see text], [Formula: see text] is equal to [Formula: see text] if [Formula: see text], otherwise, [Formula: see text] is [Formula: see text]. Here, [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text] if points in [Formula: see text] are in [Formula: see text], otherwise, it is the geodesic (Euclidean) distance between [Formula: see text] and [Formula: see text]. The following are our results: (1) When the weighted points in [Formula: see text] are located in [Formula: see text], we compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network of size [Formula: see text]. (2) When the weighted points in [Formula: see text] are located in the relative interior of the free space of a polygonal domain [Formula: see text], we detail an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network with [Formula: see text] edges. Here, [Formula: see text] is the number of simple polygonal holes in [Formula: see text]. (3) When the weighted points in [Formula: see text] are located on a polyhedral terrain [Formula: see text], we propose an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network, and the number of edges in this network is [Formula: see text].