Let $H$ be an edge-weighted graph, and let $G$ be a subgraph of $H$. We say that $G$ is an $f$-fault-tolerant $t$-spanner for $H$, if the following is true for any subset $F$ of at most $f$ edges of $G$: For any two vertices $p$ and $q$, the shortest-path distance between $p$ and $q$ in the graph $G \setminus F$ is at most $t$ times the shortest-path distance between $p$ and $q$ in the graph $H \setminus F$. Recently, Bodwin, Haeupler, and Parter generalized this notion to the case when $F$ can be any set of edges in $G$, as long as the maximum degree of $F$ is at most $f$. They gave constructions for general graphs $H$. We first consider the case when $H$ is a complete graph whose vertex set is an arbitrary metric space. We show that if this metric space contains a $t$-spanner with $m$ edges, then it also contains a graph $G$ with $O(fm)$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor $O(ft)$. Next, we consider the case when $H$ is a complete graph whose vertex set is a metric space that admits a well-separated pair decomposition. We show that, if the metric space has such a decomposition of size $m$, then it contains a graph with at most $(2f+1)^2 m$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor at most $1+\varepsilon$, for any given $\varepsilon > 0$. For example, if the vertex set is a set of $n$ points in $\mathbb{R}^d$ ($d$ being a constant) or a set of $n$ points in a metric space of bounded doubling dimension, then the spanner has $O(f^2 n)$ edges. Finally, for the case when $H$ is a complete graph on $n$ points in $\mathbb{R}^d$, we show how natural variants of the Yao- and $\Theta$-graphs lead to graphs with $O(fn)$ edges, that are resilient to edge faults of maximum degree $f$ and have stretch factor at most $1+\varepsilon$, for any given $\varepsilon > 0$.

翻译：设 $H$ 为一边加权图，$G$ 为 $H$ 的子图。若对 $G$ 中任意边子集 $F$（$|F| \leq f$）均满足以下性质：对于任意两个顶点 $p$ 和 $q$，图 $G \setminus F$ 中 $p$ 与 $q$ 的最短路径距离不超过图 $H \setminus F$ 中 $p$ 与 $q$ 最短路径距离的 $t$ 倍，则称 $G$ 是 $H$ 的 $f$ 容错 $t$ 生成器。最近，Bodwin、Haeupler 和 Parter 将这一概念推广至 $F$ 可为 $G$ 中任意边集的情形，仅要求 $F$ 的最大度不超过 $f$，并给出了针对一般图 $H$ 的构造方法。我们首先考虑 $H$ 为完全图且其顶点集构成任意度量空间的情形。证明若该度量空间存在含 $m$ 条边的 $t$ 生成器，则必存在含 $O(fm)$ 条边的图 $G$，该图对最大度为 $f$ 的边故障具有鲁棒性，且其拉伸因子为 $O(ft)$。其次，考虑 $H$ 为完全图且其顶点集为允许良分离对分解的度量空间。证明若该度量空间存在规模为 $m$ 的此类分解，则存在边数不超过 $(2f+1)^2 m$ 的图，该图对最大度为 $f$ 的边故障具有鲁棒性，且对任意给定 $\varepsilon > 0$，其拉伸因子不超过 $1+\varepsilon$。例如，若顶点集为 $\mathbb{R}^d$（$d$ 为常数）中的 $n$ 个点集或有界倍增维度度量空间中的 $n$ 个点集，则该生成器具有 $O(f^2 n)$ 条边。最后，针对 $H$ 为 $\mathbb{R}^d$ 中 $n$ 个点构成的完全图的情形，我们证明 Yao 图和 $\Theta$ 图的自然变体可构造出含 $O(fn)$ 条边的图，该图对最大度为 $f$ 的边故障具有鲁棒性，且对任意给定 $\varepsilon > 0$，其拉伸因子不超过 $1+\varepsilon$。