### Fault-Tolerant Labeling and Compact Routing Schemes [article]

Michal Dory, Merav Parter
2021 arXiv   pre-print
The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given n-vertex graph G and a bound f on the number of faults, an f-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of the vertices s and t, and at most f failing edges F, one can determine if s and t are connected in G ∖ F. The primary complexity measure is the length
more » ... of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an n-vertex graph and a fault bound f are: – There is a randomized FT connectivity labeling scheme with a label length of O(f+log n) bits, hence optimal for f=O(log n). This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG '11]. – There is a randomized FT connectivity labeling scheme with a label length of O(log^3 n) bits (independent of the number of faults f). This scheme is based on the notion of linear sketches of [Ahn et al., SODA '12]. – For k≥ 1, there is a randomized routing scheme that routes a message from s to t in the presence of a set F of faulty edges, with stretch O(|F|^2 k) and routing tables of size Õ(f^3 n^1/k). This significantly improves over the state-of-the-art bounds by [Chechik, ICALP '11], providing the first scheme with sub-linear FT labeling and routing schemes for general graphs.