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.

翻译：$f$容错连通标签（$f$-FTC标签）是一种为每个顶点和边分配小尺寸标签的方案，使得仅通过顶点$s$、$t$及至多$f$条故障边的标签，即可判定在存在故障边时两顶点$s$与$t$的连通性。本文提出一种新的确定性$f$-FTC标签方案，其标签大小为$O(f^2 \mathrm{polylog}(n))$比特，且构造时间为多项式复杂度，解决了Dory与Parter [PODC'21] 遗留的开放问题。我们构造的核心要素在于，通过纠错码理论的自然关联，为Ahn、Guha与McGreger [SODA'12] 提出的图草图技术开发确定性对偶版本。该技术消除了Dory-Parter方案去随机化的主要障碍之一。完整方案通过将该技术与源自经典$\epsilon$-网理论的新型确定性图稀疏化算法相结合而获得，后者本身也具有独立研究价值。作为副产品，我们的结果推导出首个具有非平凡性能保证的确定性容错近似距离标签方案，并改进了确定性容错紧凑路由。作者认为，我们的新技术在未来探索更高效的FTC标签方案及其他基于图草图的相关应用中具有潜在价值。