A spanner for the day after

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(ε^−7d log⁷ ε⁻¹ · ϑ⁻⁶n log n(log log n)⁶) edges. 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.