A Spanner for the Day After
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We show how to construct (1+ε)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ,ε∈ (0,1), the computed spanner G has O(n ε^-cϑ^-6^c n ) 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(n^2) edges, were known such that only a tiny additional fraction (i.e., ϑ) lose their distance preserving connectivity.
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