Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist.\ Comp.\ '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible. We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and $+4$ additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults.