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 $d_w(p, q)$ between two points $p,q \in S$ is defined as $w(p) + d(p, q) + w(q)$ if $p \ne q$ 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.$d_w(p, q)$ for a real number $t > 1$. For some integer $k \ge 1$, 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 $S' \subset S$ with cardinality at most k, the graph $G \setminus S'$ is a t-spanner for the points in $S \setminus S'$. For any given real number $\epsilon > 0$, we present algorithms to compute a $(k, 4+\epsilon )$-VFTAWS for the metric space $(S, d_w)$ resulting from the points in S belonging to either $\mathbb {R}^d$ or located in the given simple polygon. Note that d(p, q) is the geodesic Euclidean distance between p and q in the case of simple polygons whereas in the case of $\mathbb {R}^d$ it is the Euclidean distance along the line segment joining p and q.

R. Inkulu—This research is supported in part by NBHM grant 248(17)2014-R&D-II/1049.