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 \emph{f-edge-fault-tolerant} (f-EFT) σ{\em -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 computes an f-EFT (2∣F∣+1)-ASPT of size O(fn). Our structure improves on a previous related construction designed for
\emph{unweighted} graphs, having the same size but guaranteeing a larger
stretch factor of 3(f+1), plus an additive term of (f+1)logn.
Then, we show how to convert our structure into an efficient f-EFT
\emph{single-source distance oracle} (SSDO), that can be built in
O(fm) time, has size O(fnlog2n), and is able to report,
after the failure of the edge set F, in O(∣F∣2log2n) 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 \emph{minimum spanning forest} (MSF) of G after that a
\emph{batch} of k simultaneous edge modifications (i.e., edge insertions,
deletions and weight changes) is performed. For this problem, we build in O(mlog3n) time a \emph{sensitivity} oracle of size O(mlog2n), that
reports in O(k2log2n) time the (at most 2k) edges either exiting from
or entering into the MSF. [...]Comment: 16 pages, 4 figure