### Abstract

We show how to construct (1+ε)-spanner over a set P of n points in Rd that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ,ε∈(0,1), the computed spanner G has O(ε−cϑ−6nlogn(loglogn)6) edges, where c=O(d). Furthermore, for any k, and any deleted set B⊆P of k points, the residual graph G∖B is (1+ε)-spanner for all the points of P except for (1+ϑ)k of them. No previous constructions, beyond the trivial clique with O(n2) edges, were known such that only a tiny additional fraction (i.e., ϑ) lose their distance preserving connectivity.

Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion.

Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion.

Original language | English |
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Publisher | Cornell university |

Number of pages | 21 |

Publication status | Published - 2018 |

### Keywords

- Computational Geometry