Fault-Tolerant Labeling and Compact Routing Schemes

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 a vertex pair s and t, and the labels of 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 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(log3 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 a given stretch parameter 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 (unknown to s) over a path of length O(|F|2 k)⋅G \ F (s,t). The routing labels have Õ (f) bits, the messages have Õ (f3) bits, and each routing table has only Õ (f3 n1/k) bits1. The results also hold for weighted graphs with positive polynomial weights. 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.