A Constant-Approximation Distance Labeling Scheme under Polynomially Many Edge Failures

Abstract: A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$ in $G \setminus F$ by reading only the labels of $F \cup {p,q}$. For any $k$, we present a deterministic polynomial-time scheme with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults $f$, resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in-$f$ approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just $f = Θ(\log n)$ faults, all previous oracles either have super-linear query time, linear-in-$f$ approximation [CLPR12], or exponentially worse $2^{{\rm poly}(k)}$ approximation dependency in $k$ [HLS24].

• The paper establishes the first constant-approximation distance labeling scheme with an O(k^4) factor that is independent of the number of edge failures.
• It employs a deterministic polynomial-time algorithm using nested length-constrained expander hierarchies to efficiently encode local graph structures.
• The approach advances fault-tolerant oracles by significantly reducing query work and ensuring scalable performance even under polynomially many faults.

Constant-Approximation Fault-Tolerant Distance Labeling under Polynomial Edge Failures

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Introduction and Problem Context

The paper "A Constant-Approximation Distance Labeling Scheme under Polynomially Many Edge Failures" (2604.01829) addresses the longstanding open problem of fault-tolerant distance labeling in undirected weighted graphs subject to edge failures. Given a graph $G$ of $n$ vertices, the challenge is to design a distributed data structure that assigns compact labels to vertices and edges so that, for any set $F$ of up to $f$ failed edges, the approximate distance between vertices $p$ and $q$ in $G \setminus F$ can be determined solely by reading the labels of $F \cup \{p,q\}$.

A core issue in prior work, notably that of Dory and Parter, was the dependence of the stretch (approximation factor) on $f$: existing schemes attained only $O(kf)$ approximation for parameter $n$0, with stretch scaling linearly in the number of faults. The paper achieves—for the first time—a constant approximation ratio that is independent of $n$1, even for polynomially many failures.

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Main Contributions

Constant-Approximation Labeling Scheme

The authors provide a deterministic polynomial-time scheme for general (weighted undirected) graphs with the following properties:

• Approximation Ratio: $n$2 for parameter $n$3, independent of the number of failures $n$4
• Label Size: $n$5 bits per label
• Stretch Independence: The scheme tolerates $n$6 faults, maintaining a constant approximation
• Query Efficiency: Decoding distances requires only $n$7 work
• Polynomial Construction: The labeling is computed deterministically in polynomial time

This conclusively resolves the question of achieving stretch independent of $n$8, previously posed as an open problem in the literature.

Improved Distance Sensitivity Oracles

The scheme strictly improves prior sensitivity oracles for the case of $n$9 faults and beyond, where prior oracles either suffered from super-linear query time, linear-in-$F$0 approximation, or an exponential-in-$F$1 dependency. The authors describe a fault-tolerant oracle with:

• Approximation: $F$2
• Space: $F$3
• Query Time: $F$4
• Polynomial Construction

They also introduce a two-stage scenario: precompile the oracle for failures in $F$5 time, then answer distance queries in $F$6 time with slightly degraded $F$7 approximation.

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Technical Methodology

Expander Hierarchies and Length-Constrained Expanders

A pivotal innovation is the use of length-constrained expander hierarchies as opposed to standard expanders. The authors generalize and strengthen techniques from prior connectivity labeling (using classic expanders) and dynamic oracles (using length-constrained expanders) to the distributed labeling regime:

• Expander Hierarchy Construction: The hierarchy consists of nested, length-constrained expanders, enabling the preservation of distance information across hierarchical decomposition.
• Heavy-Light Component Analysis: Edge failures break cluster trees into connected components. The analysis distinguishes between heavy components (with high degree under node-weighting) and light components, enabling efficient local discovery and avoiding linear-in-$F$8 stretch loss.

Distributed and Scalable Distance Recovery

The labeling scheme relies on local neighborhood covers, shortest-path trees associated with clusters, and explicit expander cuts at multiple scales. The nestedness property of the expander hierarchy is essential for a polynomial, rather than exponential, dependency in $F$9. The labels encode enough local structure so that only labels of endpoints and failed edges are needed to reconstruct an approximate distance, leveraging recursive discovery among heavy components but limiting additive losses to within constant stretch bounds.

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Numerical Bounds and Contradictory Claims

• First Constant-Approximation for Polynomial Faults: Prior schemes (e.g., Dory and Parter) admitted only $f$0 approximation. Here, the approximation ratio is provably $f$1 and does not increase with $f$2.
• Label and Oracle Size: The polynomial dependency on $f$3 in label/oracle size is traded for constant approximation. All previous schemes with constant approximation required exponential space or were restricted to small $f$4.
• Non-adaptive Decoding: The scheme is a distributed labeling solution—only local labels (query endpoints plus failed edges) are read, with no adaptive global access to the structure.

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Implications and Theoretical Significance

Practical Impact

This result transforms the scalability of distributed distance recovery in reliability-critical networks, e.g., in precomputed routing, distributed sensor networks, and road networks under disruptions. The independence from $f$5 in stretch enables robust distance estimates even under extensive faults, which is vital for applications requiring operational guarantees.

Theoretical Innovations

The nested length-constrained expander hierarchy is a conceptual advancement. It refines previously centralized notions of expanders, enabling recursive local discovery and efficient global reconstruction. This methodology is expected to impact future work on robust network algorithms, scalable distributed oracles, and metric embeddings under failure.

Future Directions

• Hierarchical Expansion in Directed and Weighted Regimes: The generic construction of length-constrained nested hierarchies may be extensible to directed graphs, multi-commodity flow, and more general network reliability settings.
• Label Size Optimization: Reducing the polynomial dependency on $f$6 or improving the $f$7 approximation to $f$8 with exponential construction time may yield further improvements for practical deployments.
• Sensitivity Oracles in Dynamic Streams: The compiled oracle approach could fuse with dynamic graph data structures for fast reconfiguration under adversarial failure events.

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Conclusion

The paper establishes the first constant-approximation fault-tolerant distance labeling scheme that remains effective under any polynomial number of edge failures, with label sizes and query complexity strictly sub-linear in $f$9, and approximation ratio independent of $p$0 (2604.01829). This resolves a critical gap in prior art, advances the state of the art for sensitivity oracles, and brings new structural theory in length-constrained expanders to distributed labeling algorithms. The implications span both reliable distributed computation and the combinatorial foundations of robust graph representations.