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

Bernhard Haeupler, Yaowei Long, Antti Roeyskoe, Thatchaphol Saranurak

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].

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 with vertices so that, for any edge set of size , one can approximate the distance between and in by reading only the labels of . For any , we present a deterministic polynomial-time scheme with approximation and label size. This is the first scheme to achieve a constant approximation while handling any number of edge faults , resolving the open problem posed by Dory and Parter [DP21]. All previous schemes provided only a linear-in- approximation [DP21, LPS25]. Our labeling scheme directly improves the state of the art in the simpler setting of distance sensitivity oracles. Even for just faults, all previous oracles either have super-linear query time, linear-in- approximation [CLPR12], or exponentially worse approximation dependency in [HLS24].

Paper Structure

This paper contains 54 sections, 39 theorems, 59 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Distance Labeling Schemes: Well-Understood.
  3. Fault-Tolerant Distance Labeling Schemes.
  4. Main Result: Constant-Approximate FT Labeling Scheme
  5. Consequence: Improved Sensitivity Oracles
  6. Previous Oracles: Few Faults or Inefficient.
  7. The Exceptions.
  8. Our Technique
  9. Expander Hierarchies.
  10. Our Technical Contribution: Nested Length-Constrained Expander Hierarchies.
  11. Other Related Results
  12. Overview
  13. The Expander Case
  14. Previous $O(kf)$-Approximate Oracles.
  15. The Key Property of Length-Constrained Expanders.
  16. ...and 39 more sections

Key Result

Theorem 1.1

For every $k \ge 1$, there is a fault-tolerant distance labeling scheme for an undirected $n$-vertex graph with edge lengths in $[1, \mathrm{poly}(n)]$ undergoing $f$ edge faults, with $O(k^{4})$ approximation and $\tilde{O}(f^{4}n^{1/k})$ label size. The approximate distance can be decoded from the

Figures (7)

  • Figure 1: A cluster $S$ and its tree $T_S$ (solid tree edges vs. dotted non-tree edges). The two red tree edges failing breaks the cluster into the three very light blue, dotted components.
  • Figure 2: A witness shortest $(p, q)$-path $P$ in $G \setminus F$, contained in a cluster $S_P$ of $\mathcal{N}_d$ with components $\Gamma_1, \Gamma_2$, and $\Gamma_3$. The $C_d$-edges on $P$ are drawn thick, and the subpaths between the cut edges are labeled $\check{P}_1$ to $\check{P}_5$. Suppose that all of the components $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are $A_d$-light. Then, each of the $C_d$-edges on $P$ is discovered, and each of the subpaths $\check{P}_1$ to $\check{P}_5$ corresponds to a recursive scenario 1.
  • Figure 3: Suppose that instead the component $\Gamma_2$ is $A_d$-heavy, and the components $\Gamma_1$ and $\Gamma_3$ are $A_d$-light. Then, the subpaths $\check{P}_1$ and $\check{P}_5$ (blue) correspond to recursive scenario 1, and the subpaths $\check{P}_2$ and $\check{P}_4$ (red) correspond to recursive scenario 2. The subpath $\check{P}_3$ does not correspond to any recursive scenario, as we jump directly from the first undiscovered $C_d$-edge on $P$ to the last.
  • Figure 4: If instead the components $\Gamma_2$ and $\Gamma_3$ were both $A_d$-heavy, and the component $\Gamma_1$ was $A_d$-light, only the subpath $\check{P}_1$ (blue) corresponds to recursive scenario 1 and the subpath $\check{P}_2$ (red) to recursive scenario 2. Since $q$ itself appears in an $A_d$-heavy component $\Gamma_3$, we can directly jump from $\pi(\Gamma_2)$ to $\pi(\Gamma_3)$ and then take the type-2 edge from $\pi(\Gamma_3)$ to $q$.
  • Figure 5: A tree and its Euler tour rooted at $a$. The subtrees of $b$, $e$ and $g$ are highlighted in the tour and the tree. Note how the intervals of $e$ and $g$ are disjoint as neither is an ancestor of the other, and both are contained in the interval of their ancestor $b$.
  • ...and 2 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Corollary 1.2
  • lemma 2.1: The key property
  • lemma 2.2: The general key property
  • theorem 3.1
  • theorem 3.2: Routing Characterization of Length-Constrained Expanders OrigLCED22
  • definition 3.3: Clustering
  • definition 3.4: Neighborhood Cover
  • lemma 3.5: Constructive Neighborhood Cover NHCover98
  • lemma 3.6
  • ...and 60 more