The restoration lemma is a classic result by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [PODC '01], which relates the structure of shortest paths in a graph $G$ before and after some edges in the graph fail. Their work shows that, after one edge failure, any replacement shortest path avoiding this failing edge can be partitioned into two pre-failure shortest paths. More generally, this implies an additive tradeoff between fault tolerance and subpath count: for any $f, k$, we can partition any $f$-edge-failure replacement shortest path into $k+1$ subpaths which are each an $(f-k)$-edge-failure replacement shortest path. This generalized result has found applications in routing, graph algorithms, fault tolerant network design, and more. Our main result improves this to a multiplicative tradeoff between fault tolerance and subpath count. We show that for all $f, k$, any $f$-edge-failure replacement path can be partitioned into $O(k)$ subpaths that are each an $(f/k)$-edge-failure replacement path. We also show an asymptotically matching lower bound. In particular, our results imply that the original restoration lemma is exactly tight in the case $k=1$, but can be significantly improved for larger $k$. We also show an extension of this result to weighted input graphs, and we give efficient algorithms that compute path decompositions satisfying our improved restoration lemmas.

翻译：恢复引理是Afek、Bremler-Barr、Kaplan、Cohen和Merritt [PODC '01] 的经典成果，该引理描述了图$G$中某些边故障前后最短路径的结构关系。他们的工作表明，在一条边故障后，任何避开该故障边的替换最短路径可被划分为两条故障前的最短路径。更一般地，这蕴含了容错性与子路径数量之间的可加性权衡：对于任意$f, k$，任何$f$-边故障替换最短路径可被划分为$k+1$条子路径，每条子路径均为$(f-k)$-边故障替换最短路径。这一推广结果已应用于路由、图算法、容错网络设计等领域。我们的主要改进在于将其提升为容错性与子路径数量之间的可乘性权衡。我们证明：对于任意$f, k$，任何$f$-边故障替换路径可被划分为$O(k)$条子路径，每条子路径均为$(f/k)$-边故障替换路径。同时我们给出了渐进匹配的下界。特别地，我们的结果表明原始恢复引理在$k=1$时恰好紧致，但在更大$k$时具有显著改进空间。此外，我们还将该结果推广至带权输入图，并给出了满足改进恢复引理的路径分解高效算法。