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, and Merritt [Dist. Comp. '02] 的修复利玛的修复利玛证明,在未引导的未加权图表中,任何避免故障边缘的最短路径的替代方法都可以表现为两条原始最短路径的交汇。然而,利玛具有触觉性敏感性:如果一个人为每个节点配对选择一条特定的坎顿最短路径,那么就不再保证一个人能够通过对两条选定的最短路径进行搭配来建造替代路径。它们留下的问题是一个开放的问题,即是否一般有可能使用一条最短路径与这一理想属性连接的方法。我们肯定地解决这个问题的办法是,首次全面建造可恢复性断断层的断层计划。我们随后展示了在断层网络设计中的各种问题的应用。其中包括对子替换路径采用更快的算法,更高效的断层(Exact)距离标签计划、过敏的子距离保护器和美元+4美元的添加器,以及构建这些天体的快速分布式算法。例如我们可恢复性断层断层断层的断层断层的三断断断裂计划几乎的断断断断至不可。