Fault-Tolerant Additive Weighted Geometric Spanners

Abstract

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 dw(p, q) between two points p, q ∈ S is defined as w(p) + d(p, q) + w(q) if p = q and it is zero if p = q. Here, d(p, q) is the geodesic Euclidean distance between p and q. For a real number t > 1, a graph G(S, E) is called a t-spanner for the 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.dw(p, q) for a real number t > 1. For some integer k ≥ 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 ′ ⊂ 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 ǫ > 0, we present algorithms to compute a (k, 4 + ǫ)-VFTAWS for the metric space (S, dw) resulting from the points in S belonging to any of the following: R d , simple polygon, polygonal domain, and terrain. Note that d(p, q) is the geodesic Euclidean distance between p and q in the case of simple polygons and terrains whereas in the case of R d it is the Euclidean distance along the line segment joining p and q.

Vertex Fault-Tolerant Spanners for Weighted Points in Polygonal Domains

“…(Refer to Theorem 1.) The stretch factor of the spanner is improved from the result in [17], and the number of edges is an improvement over the result in [17] when (lg n) < 1 ǫ 2 . Note that [17] devised an algorithm for computing a (k, 4 + ǫ, w)-VFTSWP with size O( kn ǫ 2 lg n).…”

“…The stretch factor of the spanner is improved from the result in [17], and the number of edges is an improvement over the result in [17] when (lg n) < 1 ǫ 2 . Note that [17] devised an algorithm for computing a (k, 4 + ǫ, w)-VFTSWP with size O( kn ǫ 2 lg n). * Given a polygon domain P with h number of obstacles (holes), a set S of n points located in the free space D of P, a weight function w to associate a non-negative weight to each point in S, a positive integer k, and a real number 0 < ǫ ≤ 1, we present an algorithm to compute a (k, 6 + ǫ, w)-VFTSWP with size O( √ h + 1kn(lg n) 2 ).…”

“…In [17], Bhattacharjee and Inkulu devised the following algorithms: one for computing a (k, 4 + ǫ, w)-VFTSWP when the input points are in R d , and the other for computing a (k, 4 + ǫ, w)-VFTSWP when the given points are in a simple polygon. Further, in [18], Bhattacharjee and Inkulu extended these algorithms to compute a (k, 4 + ǫ, w)-VFTSWP when the points are in a polygonal domain and when the points are located on a terrain.…”

“…Apart from the small number of edges, spanners with additional properties such as small weight, bounded degree, small diameter, planar network, etc., were also considered. The significant results in optimizing these parameters in geometric spanner network design include spanners of low degree [12,28,21], spanners of low weight [20,33,36], spanners of low diameter [15,14], planar spanners [11,30,32,38], spanners of low chromatic number [19], fault-tolerant spanners [3,31,39,40,37,44,17], low power spanners [8,43,47], kinetic spanners [2,5], angle-constrained spanners [29], and combinations of these [13,16,24,25,26,27]. For the case of spanners in a metric space with bounded doubling metric, a few results are given in [45].…”

“…Let S be a set of n points in R d . For every p ∈ S, let w(p) be the non-negative weight associated to p. The following weighted distance function d w on S defining the metric space (S, d w ) is considered by Abam et al in [4], and by Bhattacharjee and Inkulu in [17]: for any p, q ∈ S, d w (p, q) is equal to w(p) + |pq| + w(q) if p = q; otherwise, d w (p, q) is equal to 0.…”