Õptimal Fault-Tolerant Labeling for Reachability and Approximate Distances in Directed Planar Graphs

TL;DR

This work introduces near-constant-size fault-tolerant labeling schemes for directed weighted planar graphs that, given labels for vertices s, t, and f, produce a (1+ε)-approximate s-to-t distance in G\setminus{f} in time \tilde{O}(1). Building on Thorup's separator-based decomposition, the authors develop a modular labeling framework that reduces fault-tolerant reachability to a sequence of 'Find the First' subproblems, extends these constructs to approximate distances using a 3-layered, scale-aware approach, and introduces the Good Cross–Bad Cross technique to control additive error propagation. The resulting schemes achieve label sizes of $\tilde{O}(1)$ for reachability and $\tilde{O}(\mathrm{poly}(1/\varepsilon))$ in the approximate-distance setting, with query times of $\tilde{O}(1)$, matching the best-known non-faulty bounds up to polylogarithmic factors. This work closes a gap between fault-tolerant reachability/oracle bounds and labeling schemes in directed planar graphs and offers techniques potentially extendable to multiple failures and related graph families.

Abstract

We present a labeling scheme that assigns labels of size $\tilde O(1)$ to the vertices of a directed weighted planar graph $G$, such that for any fixed $\varepsilon>0$ from the labels of any three vertices $s$, $t$ and $f$ one can determine in $\tilde O(1)$ time a $(1+\varepsilon)$-approximation of the $s$-to-$t$ distance in the graph $G\setminus\{f\}$. For approximate distance queries, prior to our work, no efficient solution existed, not even in the centralized oracle setting. Even for the easier case of reachability, $\tilde O(1)$ queries were known only with a centralized oracle of size $\tilde O(n)$ [SODA 21].

Figures (24)

• Figure 1: If $R$ is a (blue) path from $s$ to $t$ that crosses $P$ (at $a'$), then there exists an (orange) path from $s$ to $t$ that goes through $a=\mathsf{first_{}}(s,P)$, then from $a$ to $a'$ along $P$ and finally from $a'$ to $t$ along $R$).
• Figure 2: An lustration of the case $f\notin P$, a path from $s$ to $b=\mathsf{first_{G\setminus \{f\}}}(s,P)$. The blue subpath is from $s$ to $a\in P'$, and the gray subpath is from $a$ to $b \in P$.
• Figure 3: An lustration of the case $f\in P$.
• Figure 4: A naive attempt to use the reachability strategy for approximate distances may find a path which is much longer than the shortest path, hence fails to approximate the distance.
• Figure 5: An illustration of $G_1$. We are interested in $P_2$ to $P_1$ paths (like the blue path) that may start with some edges of $P_2$ and then continue with a subpath which is internally disjoint from $P$. Formally, this is achieved by removing all in-going edges to $P_2$ and all out-going edges from $P_1$ (the removed edges are displayed in gray in the figure).

Theorems & Definitions (100)

• Theorem 4.1
• Lemma 4.1
• Lemma 4.1
• proof : Proof of \ref{['thm:reach']}
• Lemma 4.1
• Lemma 4.1
• proof : Proof of \ref{['lem:stfopreach']}
• proof : Proof of \ref{['lem:atfopreach']}.
• Lemma 4.1
• Lemma 4.1