Nearly Optimal Fault Tolerant Distance Oracle

TL;DR

This work presents an exact distance oracle that tolerates up to $f$ edge faults in undirected weighted graphs with weights in $[1..W]$, returning the shortest path from $s$ to $t$ avoiding a fault set $F$ of size $f$. It introduces Jump Sequence and $(f+1)$-decomposable path representations to enable efficient, fault-robust path recovery, and proves correctness via a layered inductive construction coupled with specialized maximisers. The resulting oracle uses $O(f^4 n^2 \log^2(nW))$ space and supports queries in time $O\big(c^{(f+1)^2} f^{8(f+1)^2} \log^{2(f+1)^2}(nW)\big)$, returning the path itself in a decomposed form, making it near-optimal for constant $f$. Compared to previous work, the approach achieves favorable space and query-time trade-offs with polylogarithmic dependence on $n$ and $W$, at the cost of a large preprocessing time, and applies to integral edge weights without requiring negative weights. The results significantly advance exact fault-tolerant distance querying and practical fault-resilient path computation in networks.

Abstract

We present an $f$-fault tolerant distance oracle for an undirected weighted graph where each edge has an integral weight from $[1 \dots W]$. Given a set $F$ of $f$ edges, as well as a source node $s$ and a destination node $t$, our oracle returns the \emph{shortest path} from $s$ to $t$ avoiding $F$ in $O((cf \log (nW))^{O(f^2)})$ time, where $c > 1$ is a constant. The space complexity of our oracle is $O(f^4n^2\log^2 (nW))$. For a constant $f$, our oracle is nearly optimal both in terms of space and time (barring some logarithmic factor).

Figures (4)

• Figure 1: $X$ is the set of green vertices and $Y$ is the set of blue vertices. $x$ satisfies \ref{['item:3']} and $y$ satisfies \ref{['item:4']}.
• Figure 2: $X$ is the set of green vertices and $Y$ is the set of blue vertices. $x$ satisfies \ref{['item:3']} and $y$ satisfies \ref{['item:4']}.
• Figure 3: A jump from $x_{i-1}$ using $u$.
• Figure 6: Here the blue path is $P_i$. $su$ is intact from failure $F$ and is in the first segment of $R$. So, the detour of $R$ starts somewhere in the dashed path from $u$ to $a_k$. Or the detour starts on $P_k$. This contradicts our assumption in \ref{['item:setting']} that the detour starts on $P_i$.

Theorems & Definitions (32)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 3.1
• Definition 3.2
• Lemma 5.1
• Lemma 5.2
• proof
• Lemma 5.3