From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics

A (1+ε)-spanner for a doubling metric (X, δ) is a subgraph H of the complete graph corresponding to (X, δ), which preserves all pairwise distances to within a factor of 1 + ε. A natural requirement from a spanner, which is essential for many applications (mainly in distributed systems or wireless networks), is to be robust against vertex and edge failures -- so that even when some vertices and edges in the network fail, we still have a (1 + ε)-spanner for what remains. The spanner H is called a k-fault-tolerant (1 + ε)-spanner, for 1 ≤ k ≤ n -- 2, if for any F ⊆ X with |F| ≤ k, the graph H -- F (obtained by removing from H the vertices of F and their incident edges) is a (1 + ε)-spanner for X -- F. In this paper we devise an optimal construction of fault-tolerant spanners for doubling metrics: For any n-point doubling metric, any ε > 0, and any integer 1 ≤ k ≤ n -- 2, our construction provides a k-fault-tolerant (1+ε)-spanner with optimal degree O(k) within optimal time O(n log n + kn). We then strengthen this result to provide near-optimal (up to a factor of log k) guarantees on the diameter and weight of our spanners, namely, diameter O(log n) and weight O(k2 + k log n) · ω(MST), while preserving the optimal guarantees on the degree O(k) and the runtime O(n log n + kn). Our result settles several fundamental open questions in this area, culminating a long line of research that started with the STOC'95 paper of Arya et al. and the STOC'98 paper of Levcopoulos et al. On the way to this result we develop a new technique for constructing spanners in doubling metrics. In particular, our spanner construction is based on a novel hierarchical cover of the metric, whereas most previous constructions of spanners for doubling and Euclidean metrics (such as the net-tree spanner) are based on hierarchical partitions of the metric. We demonstrate the power of hierarchical covers in the context of geometric spanners by improving the state-of-the-art results in this area.