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spanner的一个基本要求是容错性：若度量空间的一个(1+ϵ)-spanner在移除至多f个故障点后仍能保持(1+ϵ)-spanner（针对非故障点），则称其为（顶点）f-容错（f-FT）。自90年代以来，欧几里得与双倍度量空间中的容错spanner已得到广泛研究。对于低维欧几里得度量，Czumaj与Zhao在SoCG'03 [CZ03]中证明，通过贪心算法可实现f-FT spanner在规模、度与轻量性上的最优保证O(f n)、O(f)与O(f²)，但该算法朴素运行时间为O(n³)·2^{O(f)}。能否通过快速构造实现[CZ03]的最优界这一问题始终未解，其中轻量性参数成为瓶颈。此外，在更广泛的双倍度量空间族中，甚至是否存在轻量性仅取决于f（即使指数级）的f-FT spanner也不明确：现有全部构造的轻量性均为Ω(log n)，因其基于轻量性为Ω(log n)的分层网树所诱导的网树spanner。本文以肯定方式解决了这些长期悬而未决的问题。具体而言，我们设计了一种对所有相关参数（规模、度、轻量性与运行时间）均最优的f-FT spanner构造：对任意n点双倍度量空间、任意ϵ>0及任意整数1≤f≤n-2，该构造在O(n log n + f n)时间内提供规模O(f n)、度O(f)且轻量性O(f²)的f-FT (1+ϵ)-spanner。