Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the $Ω(\log n)$ Lightness Barrier

TL;DR

The paper resolves long-standing open questions on fault-tolerant $(1+\epsilon)$-spanners in doubling metrics by breaking the $\Omega(\log n)$ lightness barrier. It introduces a light net-forest (LNF) and leverages surrogate sets on a net-tree framework to construct an $f$-FT $(1+\epsilon)$-spanner with optimal size $O(fn)$, degree $O_{\epsilon,d}(f)$, and lightness $O(\epsilon^{-O(d)} f^2)$, in time $O_{\epsilon,d}(n \log n + fn)$. The approach carefully augments a base light spanner with cross edges and surrogate-based bipartite connections, ensuring fault tolerance while maintaining low weight via a detailed edge decomposition and weight analysis. Beyond doubling metrics, the method yields improvements for Euclidean spaces and advances practical, scalable construction of robust geometric networks with provable guarantees.

Abstract

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 ($f$-FT) 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 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which naïvely runs in $O(n^3) \cdot 2^{O(f)}$ time. 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. 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 $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+ε)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

Figures (5)

• Figure 1: A path from $u$ to $v$ in $H \setminus F$. Every purple node has $f + 1$ surrogates. We replace some edges and some subpaths of the shortest path from $u$ to $v$ in $G$ by the cross edges between nodes having $f + 1$ surrogates. It might be misleading that $\delta(\tilde{u}_2, \tilde{u}_6)$ is much smaller than $\delta(u_2, u_6)$. Indeed, $\delta(\tilde{u_2}, \tilde{u_6})$ is a good approximation of the length of $(u_2, u_3, u_4, u_5, u_6)$.
• Figure 2: An example of $A_{i_1}, A_{i_2}$ and $A_{i_3}$. Here, $A_{i_1}, A_{i_2}$ and $A_{i_3}$ are disjoint. For each $k' \in \{1, 2, 3\}$, $A_{i_{k'}} \subseteq \mathbf{B}(v_{i_{k}}, 4r_{i_{k}})$ for all $k > k'$.
• Figure 3: An illustration of $(x_k, y_k), (w_k, z_k)$ for $i \in [1, 3]$. Here, it might be misleading that $u_{a_k}$ is far from $x_{k + 1}$ and $y_k$. In fact, $u_{a_k}$ is in both $\mathbf{B}(x_{k + 1}, 16r_{k + 1})$ (the blue ball) and $\mathbf{B}(y_k, 16r_k)$ (the yellow ball). Furthermore, the higher ball in $\{\mathbf{B}(x_{k + 1}, 16r_{k + 1}), \mathbf{B}(y_k, 16r_k)\}$ contains the other.
• Figure 4: An illustration of two cases. The blue, orange edges and the cyan and yellow balls have the same meaning as those in \ref{['fig:lm13-1']}. The purple node is the $\mathrm{LCA}$ of $u_{a_s}$ and the white node $\tilde{v}$ is the ancestor of $v = u_{a_{s + 1}}$ at the same level. The cross edge between $t$ and $\tilde{v}$ is not in $E^*$ for the first case $(a)$. In figure $(b)$, $(t, \tilde{v}) \in E^*$ and hence there is an edge of $H - F$ in the bipartite connection between $S(t)$ and $S(\tilde{v})$.
• Figure 5: An example of a $P$-stair-jump with three cross edges. Note that $u_{a_1} = u$. Each pink node $y_k$ is a descendant of a purple node $x_{k + 1}$. When we construct a $P$-stair-jump later in the proof of \ref{['lm:detour-exist']}, every pink node is incomplete and every purple node is complete.

Theorems & Definitions (76)

• Theorem 1
• Lemma 1: Packing bound
• Definition 1: Greedy Net Tree
• Claim 1
• proof : Proof of \ref{['clm:leaf-dist']}
• Definition 2: Surrogate Sets
• Definition 3: Bipartite Connection
• Theorem 2
• Lemma 2
• proof