Sparse Fault-Tolerant Spanners for Doubling Metrics with Bounded Hop-Diameter or Degree

We study fault-tolerant spanners in doubling metrics. A subgraph H for a metric space X is called a k-vertex-fault-tolerant t-spanner ((k,t)-VFTS or simply k-VFTS), if for any subset S⊆X with |S|≤k, it holds that dH∖S(x, y)≤t ·d(x, y), for any pair of x, y∈X∖S. For any doubling metric, we give a basic construction of k-VFTS with stretch arbitrarily close to 1 that has optimal O(kn) edges. In addition, we also consider bounded hop-diameter, which is studied in the context of fault-tolerance for the first time even for Euclidean spanners. We provide a construction of k-VFTS with bounded hop-diameter: for m≥2n, we can reduce the hop-diameter of the above k-VFTS to O(α(m, n)) by adding O(km) edges, where α is a functional inverse of the Ackermann's function. Finally, we construct a fault-tolerant single-sink spanner with bounded maximum degree, and use it to reduce the maximum degree of our basic k-VFTS. As a result, we get a k-VFTS with O(k2n) edges and maximum degree O(k2).