A spanner for the day after

Abstract

\u3cp\u3eWe show how to construct (1 + ε)-spanner over a set P of n points in ℝ\u3csup\u3ed\u3c/sup\u3e that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ, ε ∈ (0, 1), the computed spanner G has O(ε\u3csup\u3e−7d\u3c/sup\u3e log\u3csup\u3e7\u3c/sup\u3e ε\u3csup\u3e−1\u3c/sup\u3e · ϑ\u3csup\u3e−6\u3c/sup\u3en log n(log log n)\u3csup\u3e6\u3c/sup\u3e) 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(n\u3csup\u3e2\u3c/sup\u3e) 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.\u3c/p\u3