# Fault-Tolerant Additive Weighted Geometric Spanners

• Sukanya Bhattacharjee
• R. Inkulu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

## 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(pq) is the (geodesic) Euclidean distance between p and q. For a real number $$t > 1$$, a graph G(SE) 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 (kt)-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(pq) 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.

## Keywords

Computational geometry Geometric spanners Approximation algorithms

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