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 $\mathbb{R}^d$ 空间中包含 $n$ 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，我们证明了如何构建一个在如此大规模的边失效情况下仍能保持连通性的生成图，且其边数接近线性。所构建图的可靠性度量是：有多少对顶点会失去 $(1+\varepsilon)$-连通性。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于底层图为完全图时的失效顶点对数量。具体而言，我们展示了如何在一维空间中构建精确的可靠生成图，其规模为 $O(\tfrac{n}{\psi} \log n)$，且该规模是最优的。进一步地，我们为 $\mathbb{R}^d$ 空间中包含 $n$ 个点的集合 $P$ 构建了 $(1+\varepsilon)$-生成图，其规模为 $O( C n \log n )$，其中 $C \approx 1/\bigl(\varepsilon^{d} \psi^{4/3}\bigr)$。值得注意的是，这些新构建的生成图还具有以下特性：几乎所有顶点对之间都存在 $\leq 4$ 跳的路径来实现这一短路径。