Fault-Tolerant Additive Weighted Geometric Spanners

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 6= q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S,E) is called a t-spanner for the additive 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. Here, dw(p, q) is the additive weighted distance between p and q. 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 obtain the following results: – When the points in S belong to Euclidean space R, an algorithm to compute a (k, (2 + ǫ))VFTAWS with O(kn) edges for the metric space (S, dw). Here, for any two points p, q ∈ S, d(p, q) is the Euclidean distance between p and q in R. – When the points in S belong to a simple polygon P , for the metric space (S, dw), one algorithm to compute a geodesic (k, (2+ ǫ))-VFTAWS with O( kn ǫ2 lg n) edges and another algorithm to compute a geodesic (k, ( √ 10 + ǫ))-VFTAWS with O(kn(lg n)) edges. Here, for any two points p, q ∈ S, d(p, q) is the geodesic Euclidean distance along the shortest path between p and q in P .