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-F T) 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 90 s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees O(f n), O(f) and O(f2) on the size, degree and lightness of f-FT spanners can be achieved via a greedy algorithm, which naïvely runs in O(n3) · 2O(f) time.1 An earlier construction, by Levcopoulos et al. [LNS98] from STOC'98, has a faster running time of O(n log n)+n 2O(f), but has a slack of 2Ω(f) in all the three involved parameters. 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: Any construction (other than [CZ03]) has lightness either 2Ω(f) or Ω(log n). 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 ≤ log f n≤ + nfn-), an2, our construction provides,-spanner with within time O(n size O (fn), degree O(f) and lightness O(f2). To break the Ω (log n) lightness barrier, we introduce a new geometric object - the light net-forest. Like the net-tree, the light net-forest is induced by a hierarchy of nets. However, to ensure small lightness, the light net-forest is inherently less 'well-connected' than the net-tree, which, in turn, makes the task of achieving fault-tolerance significantly more challenging. Further, to achieve the optimal degree (and size) together with optimal lightness, and to do so within the optimal running time - we overcome several highly nontrivial technical challenges.