Fault-Tolerant Labeling and Compact Routing Schemes

TL;DR

This paper introduces new fault-tolerant labeling and routing schemes for general graphs, achieving nearly optimal label lengths and improved routing performance under faults, advancing the robustness and efficiency of network algorithms.

Contribution

It proposes two independent fault-tolerant connectivity labeling schemes with nearly optimal label lengths for general graphs, filling a significant research gap.

Findings

A randomized FT labeling scheme with $O(f+ ext{log} n)$ bits, optimal for $f=O( ext{log} n)$.

A randomized FT labeling scheme with $O( ext{log}^3 n)$ bits, independent of $f$.

A new routing scheme with sub-linear FT labeling and routing tables, improving previous bounds.

Abstract

The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of the vertices $s$ and $t$, and at most $f$ failing edges $F$, one can determine if $s$ and $t$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an $n$-vertex graph and a fault bound $f$ are: -- There is a randomized FT connectivity labeling scheme with a label length of $O(f+\log n)$ bits, hence optimal for $f=O(\log n)$. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG '11]. -- There is a randomized FT connectivity labeling scheme with a label length of $O(\log^3 n)$ bits (independent of the number of faults $f$). This scheme is based on the notion of linear sketches of [Ahn et al., SODA '12]. -- For $k\geq 1$, there is a randomized routing scheme that routes a message from $s$ to $t$ in the presence of a set $F$ of faulty edges, with stretch $O(|F|^2 k)$ and routing tables of size $\tilde{O}(f^3 n^{1/k})$. This significantly improves over the state-of-the-art bounds by [Chechik, ICALP '11], providing the first scheme with sub-linear FT labeling and routing schemes for general graphs.