We show how to construct a \((1+\varepsilon )\)-spanner over a set \({P}\) of n points in \({\mathbb {R}}^d\) that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters \({\vartheta },\varepsilon \in (0,1)\), the computed spanner \({G}\) has

edges. Furthermore, for anyk, and any deleted set \({{B}}\subseteq {P}\) of k points, the residual graph \({G}\setminus {{B}}\) is a \((1+\varepsilon )\)-spanner for all the points of \({P}\) except for \((1+{\vartheta })k\) of them. No previous constructions, beyond the trivial clique with \({{\mathcal {O}}}(n^2)\) edges, were known with this resilience property (i.e., only a tiny additional fraction of vertices, \(\vartheta |B|\), 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.

我们展示了如何在\（{\ mathbb {R}} ^ d \）中具有n个点的一组\（{P} \）上构造一个\（（1+ \ varepsilon）\）-扳手，该扳手对节点的灾难性故障。具体来说，对于指定参数\（{\ vartheta}，\ varepsilon \ in（0,1）\），计算出的扳手\（{G} \）具有

边缘。此外，对于任何ķ，和任何删除集\（{{B}} \ subseteq {P} \）的ķ分，剩余图\（{G} \ setminus {{B}} \）是一个\（（ 1+ \ varepsilon）\） -spanner对于所有的点\（{P} \）除了\（（1 + {\ vartheta}）K \）它们。除了具有\（{{\ mathcal {O}}}（n ^ 2）\）边的琐碎小集团之外，没有其他构造具有此弹性属性（即，顶点的一小部分只是\（\ vartheta） | B | \），失去了保持连接的距离）。我们的构造首先通过在一个维度上解决确切的问题，然后在更高维度上显示出令人惊讶的简单而优雅的构造来工作，该构造以黑盒子的方式使用一维构造。