Fault-Tolerant Labeling and Compact Routing Schemes

Abstract.

The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as improved FT routing schemes. For a given [Formula: see text]-vertex graph [Formula: see text] and a bound [Formula: see text] on the number of faults, an [Formula: see text]-FT connectivity labeling scheme is a distributed data structure that assigns to each of the graph edges and vertices a short label, such that given the labels of a vertex pair [Formula: see text] and [Formula: see text], and the labels of at most [Formula: see text] failing edges [Formula: see text], one can determine if [Formula: see text] and [Formula: see text] are connected in [Formula: see text]. 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 [Formula: see text]-vertex graph and a fault bound [Formula: see text] are (1) There is a randomized FT connectivity labeling scheme with a label length of [Formula: see text] bits, hence optimal for [Formula: see text]. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG ’11]. (2) There is a randomized FT connectivity labeling scheme with a label length of [Formula: see text] bits (independent of the number of faults [Formula: see text]). This scheme is based on the notion of linear sketches of [Ahn et al., SODA ’12]. (3) For a given stretch parameter [Formula: see text], there is a randomized routing scheme that routes a message from [Formula: see text] to [Formula: see text] in the presence of a set [Formula: see text] of faulty edges (unknown to [Formula: see text]) over a path of length [Formula: see text]. The routing labels have [Formula: see text] bits, the header size is [Formula: see text] bits, and each routing table has only [Formula: see text] bits. (Throughout the paper, we use the notation [Formula: see text] to hide poly-logarithmic in [Formula: see text] terms.) 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 sublinear FT labeling and routing schemes for general graphs.