通过不可靠边构建可靠支撑图
Dependable Spanners via Unreliable Edges
AI总结：
研究边以高概率独立失效时如何构建可靠的 $(1+\varepsilon)$-spanner，给出一维最优构造及高维近线性构造，并证明其连通失效对数接近完全图且多数点对路径不超过4跳。
摘要：
令 $P$ 为 $\mathbb{R}^d$ 中 $n$ 个点的集合，并令 $\varepsilon,ψ\in (0,1)$ 为参数。这里，我们考虑为 $P$ 构建一个 $(1+\varepsilon)$-spanner（支撑图）的任务，其中每条边都可能以概率 $1-ψ$ 独立失效。例如，当 $ψ=0.1$ 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能够在这种灾难中幸存、且边数接近线性的支撑图。所构建图的可靠性度量是有多少对顶点失去 $(1+\varepsilon)$-连通性。令人惊讶的是，尽管所构建的支撑图规模接近线性，失效点对的数量却接近于底层图为完全图（clique）时的失效点对数量。具体而言，我们展示了如何在一维中构建这样一个规模为 $O(\tfrac{n}ψ \log n)$ 的精确可靠支撑图，该规模是最优的。接着，我们为 $\mathbb{R}^d$ 中包含 $n$ 个点的集合 $P$ 构建一个 $(1+\varepsilon)$-spanner，其规模为 $O( C n \log n )$，其中 $C \approx 1/\bigl(\varepsilon^{d} ψ^{4/3}\bigr)$。令人惊讶的是，这些新的支撑图还具有如下性质：几乎所有顶点对之间都存在跳数 $\leq 4$ 的路径来实现这条短路径。
英文摘要：
Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $\varepsilon,ψ\in (0,1)$ be parameters. Here, we consider the task of constructing a $(1+\varepsilon)$-spanner for $P$, where every edge might fail (independently) with probability $1-ψ$. For example, for $ψ=0.1$, 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 $O(\tfrac{n}ψ \log n)$, 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 $C \approx 1/\bigl(\varepsilon^{d} ψ^{4/3}\bigr)$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
