Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $\varepsilon,\psi \in (0,1)$ be parameters. Here, we consider the task of constructing a $(1+\varepsilon)$-spanner for $P$, where every edge might fail (independently) with probability $1-\psi$. For example, for $\psi=0.1$, about $90\%$ of the edges of the graph fail. Nevertheless, we show how to construct a spanner that survives such a catastrophe with near linear number of edges. The measure of reliability of the graph constructed is how many pairs of vertices lose $(1+\varepsilon)$-connectivity. Surprisingly, despite the spanner constructed being of near linear size, the number of failed pairs is close to the number of failed pairs if the underlying graph was a clique. Specifically, we show how to construct such an exact dependable spanner in one dimension of size $O(\tfrac{n}{\psi} \log n)$, which is optimal. Next, we build an $(1+\varepsilon)$-spanners for a set $P \subseteq \mathbb{R}^d$ of $n$ points, of size $O( C n \log n )$, where $C \approx 1/\bigl(\varepsilon^{d} \psi^{4/3}\bigr)$. Surprisingly, these new spanners also have the property that almost all pairs of vertices have a $\leq 4$-hop paths between them realizing this short path.

翻译：设$P$为$\mathbb{R}^d$中$n$个点的集合，$\varepsilon,\psi \in (0,1)$为参数。本文研究为$P$构建$(1+\varepsilon)$-生成图的任务，其中每条边以$1-\psi$的概率独立失效。例如，当$\psi=0.1$时，图中约$90\%$的边会失效。尽管如此，我们展示了如何构建一个在如此灾难性情况下仍能存活的生成图，其边数接近线性。所构建图的可靠性度量是失去$(1+\varepsilon)$-连通性的顶点对数量。令人惊讶的是，尽管所构建的生成图规模接近线性，其失效顶点对的数量却接近底层图为完全图时的失效对数。具体而言，我们在一维情况下构建了规模为$O(\tfrac{n}{\psi} \log n)$的精确可靠生成图，该规模是最优的。进一步地，我们为$\mathbb{R}^d$中$n$个点的集合$P$构建了规模为$O( C n \log n )$的$(1+\varepsilon)$-生成图，其中$C \approx 1/\bigl(\varepsilon^{d} \psi^{4/3}\bigr)$。值得注意的是，这些新型生成图还具有以下特性：几乎所有顶点对之间都存在实现该短路径的$\leq 4$跳路径。