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 d H ∖ 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 ≥2 n , 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 ( k 2 n ) edges and maximum degree O ( k 2 ).