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.

翻译：Afek、Bremler-Barr、Kaplan、Cohen与Merritt [Dist. Comp. '02]提出的修复引理证明：在无向无权图中，任何绕过故障边的替换最短路径均可表示为两条原始最短路径的串联。然而该引理对断链策略具有敏感性：若为每对节点选择特定的规范最短路径，则无法保证能通过串联两条选定最短路径构建替换路径。他们遗留了一个开放问题：是否普遍存在具有这种理想属性的最短路径断链方法。我们通过首次构建可恢复断链方案的一般性构造，给出肯定性回答。随后展示该方案在容错网络设计各类问题中的应用，包括：更快速的子集替换路径算法、更高效的容错（精确）距离标记方案、容错子集距离保持器及稀疏性优化的$+4$加法扳手，以及构建这些对象的快速分布式算法。例如，作为可恢复断链方案的直接推论，我们首次实现了针对三个故障具有鲁棒性的稀疏容错距离保持器的非平凡分布式构造。