We show how to construct (1+\varepsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters \vartheta,\varepsilon \in (0,1), the computed spanner G has O (n \bigl( \varepsilon^{-c} \vartheta^{-6} \log^{c\bigr) edges, where c= O(d). Furthermore, for any k, and any deleted set B \subseteq P of k points, the residual graph G \setminus B is (1+\varepsilon)-spanner for all the points of P except for (1+\vartheta)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., \vartheta) lose their distance preserving connectivity  .
We show how to construct (1+\varepsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters \vartheta,\varepsilon \in (0,1), the computed spanner G has O \bigl( \varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6\bigr) edges, where c= O(d). Furthermore, for any k, and any deleted set B \subseteq P of k points, the residual graph G \setminus B is (1+\varepsilon)-spanner for all the points of P except for (1+\vartheta)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., \vartheta) 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 .
D
(n
R
\log^{c
n \log n (\log\log n)^6
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. 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