  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.
