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.
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Bhattacharjee, S., Inkulu, R. (2019). Fault-Tolerant Additive Weighted Geometric Spanners. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_3
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