Near-Optimal Resilient Labeling Schemes

Abstract

Labeling schemes are a prevalent paradigm in various computing settings. In such schemes, an oracle is given an input graph and produces a label for each of its nodes, enabling the labels to be used for various tasks. Fundamental examples in distributed settings include distance labeling schemes, proof labeling schemes, advice schemes, and more. This paper addresses the question of what happens in a labeling scheme if some labels are erased, e.g., due to communication loss with the oracle or hardware errors. We adapt the notion of resilient proof-labeling schemes of Fischer, Oshman, Shamir [OPODIS 2021] and consider resiliency in general labeling schemes. A resilient labeling scheme consists of two parts -- a transformation of any given labeling to a new one, executed by the oracle, and a distributed algorithm in which the nodes can restore their original labels given the new ones, despite some label erasures. Our contribution is a resilient labeling scheme that can handle $F$ such erasures. Given a labeling of $\ell$ bits per node, it produces new labels with multiplicative and additive overheads of $O(1)$ and $O(\log(F))$, respectively. The running time of the distributed reconstruction algorithm is $O(F+(\ell\cdot F)/\log{n})$ in the \textsf{Congest} model. This improves upon what can be deduced from the work of Bick, Kol, and Oshman [SODA 2022], for non-constant values of $F$. It is not hard to show that the running time of our distributed algorithm is optimal, making our construction near-optimal, up to the additive overhead in the label size.