Deterministic fault-tolerant connectivity labeling scheme

Abstract The 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 $$O(f^2 \textrm{polylog}(n))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mtext>polylog</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -bit label size and a polynomial construction time, which settles the open problem left by Dory and Parter (in: Proceedings of the 2021 ACM symposium on principles of distributed computing (PODC), pp 445–455, 2021). The key ingredient of our construction is to develop a deterministic counterpart of the graph sketch technique by Ahn et al. (in: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI symposium on principles of database systems (PODS), pp 5–14, 2012), 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 $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> - 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.