From Hierarchical Partitions to Hierarchical Covers: Optimal Fault-Tolerant Spanners for Doubling Metrics

Abstract: In this paper we devise an optimal construction of fault-tolerant spanners for doubling metrics. Specifically, for any $n$-point doubling metric, any $\eps > 0$, and any integer $0 \le k \le n-2$, our construction provides a $k$-fault-tolerant $(1+\eps)$-spanner with optimal degree $O(k)$ within optimal time $O(n \log n + k n)$. 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 $O(k² + k \log n) \cdot \omega(MST)$, while preserving the optimal guarantees on the degree $O(k)$ and the running time $O(n \log n + k n)$. 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. Our spanner construction is based on a novel \emph{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 \emph{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.