Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees [article]

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti
2016 arXiv   pre-print
Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f > 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) σ -approximate single-source shortest-path tree (σ-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 at most by a factor of σ. To this respect, we provide an algorithm that efficiently
more » ... tes an f-EFT (2|F|+1)-ASPT of size O(f n). 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) n. Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in O(f m) time, has size O(fn ^2 n), and is able to report, after the failure of the edge set F, in O(|F|^2 ^2 n) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m ^3 n) time a sensitivity oracle of size O(m ^2 n), that reports in O(k^2 ^2 n) time the (at most 2k) edges either exiting from or entering into the MSF. [...]
arXiv:1601.04169v2 fatcat:l6ytznxbifaaxlneu6r4tpqpfy