Deterministic Fault-Tolerant Connectivity Labeling Scheme with Adaptive Query Processing Time

The $f$-fault-toleratant 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. 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. An interesting byproduct of our result is that one can obtain an improved randomized $f$-FTC labeling scheme attaining adaptive query processing time (i.e., the processing time does not depend on $f$, but only on the actual number of faulty edges). This scheme is easily obtained from our deterministic scheme just by replacing the graph sparsification part with the conventional random edge sampling.