Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Abstract

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer \(f \ge 1\), we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) \(\sigma \)-approximate single-source shortest-path tree (\(\sigma \)-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched by a factor of at most \(\sigma \). To this respect, we provide an algorithm that efficiently computes an f-EFT \((2|F|+1)\)-ASPT of size O(fn). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of \(3(f+1)\), plus an additive term of \((f+1) \log n\). Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle, that can be built in \(O(f m\, \alpha (m,n)+fn \log ^3 n)\) time, has size \(O(fn \log ^2 n)\), and in \(O(|F|^2 \log ^2 n)\) time is able to report a \((2|F|+1)\)-approximate distance from the source to any node in \(G-F\). Moreover, our oracle can return a corresponding approximate path in the same amount of time plus the path’s size. The oracle is obtained by tackling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G following a batch of k simultaneous modification (i.e., edge insertions, deletions and weight changes). For this problem, we build in \(O(m \log ^3 n)\) time an oracle of size \(O(m \log ^2 n)\), that reports in \(O(k^2 \log ^2 n)\) time the (at most 2k) edges either exiting from or entering into the MSF. Finally, for any integer \(k \ge 1\), we complement all our results with a lower bound of \(\Omega \left( n^{1+\frac{1}{k}}\right) \) to the size of any f-EFT \(\sigma \)-ASPT with \(f \ge \log n\) and \(\sigma < \frac{3k+1}{k+1}\), that holds if the Erdős’ girth conjecture is true.