Deterministic Fault-Tolerant Connectivity Labeling Scheme

TL;DR

This paper introduces a deterministic fault-tolerant connectivity labeling scheme with small labels and polynomial construction time, solving an open problem and advancing fault-tolerant graph algorithms.

Contribution

It develops a deterministic graph sketch technique and a new graph sparsification method, enabling efficient fault-tolerant connectivity labels and related applications.

Findings

Achieves $O(f^2 ext{polylog}(n))$-bit label size.

Provides the first deterministic fault-tolerant approximate distance labeling scheme.

Improves deterministic fault-tolerant compact routing.

Abstract

The \emph{$f$-fault-tolerant connectivity labeling} ($f$-FTC labeling) is a scheme of assigning each vertex and edge with a small-size label such that one can determine the connectivity of two vertices $s$ and $t$ under the presence of at most $f$ faulty edges only from the labels of $s$, $t$, and the faulty edges. This paper presents a new deterministic $f$-FTC labeling scheme attaining $O(f^2 \mathrm{polylog}(n))$-bit label size and a polynomial construction time, which settles the open problem left by Dory and Parter [PODC'21]. The key ingredient of our construction is to develop a deterministic counterpart of the graph sketch technique by Ahn, Guha, and McGreger [SODA'12], via some natural connection with the theory of error-correcting codes. This technique removes one major obstacle in de-randomizing the Dory-Parter scheme. The whole scheme is obtained by combining this technique with a new deterministic graph sparsification algorithm derived from the seminal $\epsilon$-net theory, which is also of independent interest. As byproducts, our result deduces the first deterministic fault-tolerant approximate distance labeling scheme with a non-trivial performance guarantee and an improved deterministic fault-tolerant compact routing. The authors believe that our new technique is potentially useful in the future exploration of more efficient FTC labeling schemes and other related applications based on graph sketches.