# New Results on Fault Tolerant Geometric Spanners

## Abstract

We investigate the problem of constructing spanners for a given set of points that are tolerant for edge/vertex faults. Let ℝ^{ d } be a set of *n* points and let k be an integer number. A *k*-edge/vertex fault tolerant spanner for S has the property that after the deletion of k arbitrary edges/vertices each pair of points in the remaining graph is still connected by a short path. Recently it was shown that for each set *S* of *n* points there exists a *k*-edge/vertex fault tolerant spannerwith *O*(*k* ^{2} *n*) edges which can be constructed in *O*(*n* log *n* + *k* ^{2} *n*) time. Furthermore, it was shown that for each set *S* of *n* points there exists a *k*-edge/vertex fault tolerant spannerwhose degree is bouned by *O*(*c* ^{k+1}) for some constant *c*.

Our first contribution is a construction of a *k*-vertex fault tolerant spanner with O(kn) edges which is a tight bound. The computation takes *O*(*n*log^{d−1}n + *kn* log log *n*) time. Then we show that the same *k*-vertex fault tolerant spanner is also *k*-edge fault tolerant. Thereafter, we construct a *k*-vertex fault tolerant spanner with *O*(*k* ^{2} *n*) edges whose degree is bounded by *O*(*k* ^{2}). Finally, we give a more natural but stronger definition of *k*-edge fault tolerance which not necessarily can be satisfied if one allows only simple edges between the points of *S*. We investigate the question whether Steiner points help.We answer this question affirmatively and prove Θ(*kn*) bounds on the number of Steiner points and on the number of edges in such spanners.

## Keywords

Minimum Span Tree Computational Geometry Real Constant Steiner Point Simplicial Cone## Preview

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