An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+\epsilon)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+\epsilon)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"{\i}vely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $\Omega(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $\Omega(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $\epsilon > 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+\epsilon)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

翻译：许多应用中增生器的基本要求是具备容错性：若从增生器中移除至多$f$个故障点后，剩余非故障点间的$(1+\epsilon)$-增生器仍保持该性质，则称度量空间的$(1+\epsilon)$-增生器为（顶点）$f$容错（$f$-FT）。自20世纪90年代以来，欧几里得与倍增度量空间中的容错增生器得到广泛研究。针对低维欧几里得度量，Czumaj与Zhao在SoCG'03 [CZ03]中证明，通过贪心算法可实现$f$-FT增生器在规模、度与轻量性上的最优保证$O(f n)$、$O(f)$与$O(f^2)$，但该算法朴素运行时间为$O(n^3) \cdot 2^{O(f)}$。能否通过快速构造实现[CZ03]中的最优界这一问题始终悬而未决，其中轻量性参数成为瓶颈。此外，在更广泛的倍增度量空间族中，甚至是否存在轻量性仅依赖于$f$（即便指数级依赖）的$f$-FT增生器也不明确：所有现有构造均具有轻量性$\Omega(\log n)$，因其基于轻量性为$\Omega(\log n)$的层次化网树所导出的网树增生器。本文正面解决了这些长期悬而未决的开放问题。具体而言，我们设计了一种关于所有相关参数（规模、度、轻量性与运行时间）均达最优的$f$-FT增生器构造：对任意$n$点倍增度量、任意$\epsilon > 0$及任意整数$1 \le f \le n-2$，我们的构造在$O(n \log n + f n)$时间内提供规模$O(f n)$、度$O(f)$且轻量性$O(f^2)$的$f$-FT $(1+\epsilon)$-增生器。