Fault-Tolerant Distance Oracles Below the $n \cdot f$ Barrier

2026-03-25

AI Summary

This work addresses the problem of approximate distance queries in graphs under edge failures, circumventing the classic $\Omega(nf)$ space lower bound inherent to fault-tolerant subgraphs. By constructing fault-tolerant distance oracles that are not constrained to subgraph form, the authors introduce a novel framework combining high-connectivity low-diameter graph decomposition with sparse recovery techniques, further extended to streaming and random sketching models. Their main contributions include the first fault-tolerant distance oracle whose space complexity breaks below the $nf$ barrier, achieving stretch $O(\log n \log \log n)$ with $\widetilde{O}(n \sqrt{f})$ space. Moreover, for $n \leq f \leq n^{3/2}$, they obtain an improved construction with stretch 7 and space $\widetilde{O}(n^{3/2} f^{1/3})$.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with $O(k^2 f^{\frac{1}{2}-\frac{1}{2k}} n^{1+\frac{1}{k}} + k f n)$ edges for any odd $k$. For any $k = \tilde{O}(1)$, this bound is essentially optimal for deterministic spanners in part due to a known folklore lower bound that \emph{any} $f$-fault-tolerant spanner requires $Ω(nf)$ edges in the worst case. For $f \geq n$, this $Ω(nf)$ barrier means that any $f$-fault tolerant spanners are trivial in size. Crucially however, this folklore lower bound exploits that the spanner \emph{is itself a subgraph}. It does not rule out distance-reporting data structures that may not be subgraphs. This leads to our central question: can one beat the $n \cdot f$ barrier with fault-tolerant distance oracles? We give a strong affirmative answer to this question. As our first contribution, we construct $f$-fault-tolerant distance oracles with stretch $O(\log(n)\log\log(n))$ that require only $\widetilde{O}(n\sqrt{f})$ bits of space; substantially below the spanner barrier of $n \cdot f$. Beyond this, in the regime $n \leq f \leq n^{3/2}$ we show that by using our new \emph{high-degree, low-diameter} decomposition in combination with tools from sparse recovery, we can even obtain stretch $7$ distance oracles in space $\widetilde{O}(n^{3/2}f^{1/3})$ bits. We also show that our techniques are sufficiently general to yield randomized sketches for fault-tolerant ``oblivious'' spanners and fault-tolerant deterministic distance oracles in bounded-deletion streams, with space below the $nf$ barrier in both settings.

Problem

• fault-tolerant
• distance oracles
• spanner barrier
• space complexity
• graph algorithms

Innovation

• fault-tolerant distance oracles
• sublinear space
• graph spanners
• sparse recovery
• high-degree low-diameter decomposition