# Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the Ω(log n) Lightness Barrier

An essential requirement of spanners in many applications is to be fault-tolerant: a (1+ϵ)-spanner of a metric space is called (vertex) f-fault-tolerant (f-FT) if it remains a (1+ϵ)-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 naïvely runs in O(n^3) · 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 Ω(log n) since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness Ω(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 ϵ > 0, and any integer 1 ≤ f ≤ n-2, our construction provides, within time O(n log n + f n), an f-FT (1+ϵ)-spanner with size O(f n), degree O(f) and lightness O(f^2).

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