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

Let G be an n -node and m -edge positively real-weighted undirected graph. For any given integer $$f \ge 1$$ f ≥ 1 , we study the problem of designing a sparse f -edge-fault-tolerant ( f -EFT) $$\sigma $$ σ -approximate single-source shortest-path tree ( $$\sigma $$ 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 by a factor of at most $$\sigma $$ σ . To this respect, we provide an algorithm that efficiently computes an f -EFT $$(2|F|+1)$$ ( 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 $$3(f+1)$$ 3 ( f + 1 ) , plus an additive term of $$(f+1) \log n$$ ( 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 $$O(f m\, \alpha (m,n)+fn \log ^3 n)$$ O ( f m α ( m , n ) + f n log 3 n ) time, has size $$O(fn \log ^2 n)$$ O ( f n log 2 n ) , and in $$O(|F|^2 \log ^2 n)$$ O ( | F | 2 log 2 n ) time is able to report a $$(2|F|+1)$$ ( 2 | F | + 1 ) -approximate distance from the source to any node in $$G-F$$ 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 $$O(m \log ^3 n)$$ O ( m log 3 n ) time an oracle of size $$O(m \log ^2 n)$$ O ( m log 2 n ) , that reports in $$O(k^2 \log ^2 n)$$ O ( k 2 log 2 n ) time the (at most 2 k ) edges either exiting from or entering into the MSF. Finally, for any integer $$k \ge 1$$ k ≥ 1 , we complement all our results with a lower bound of $$\Omega \left( n^{1+\frac{1}{k}}\right) $$ Ω n 1 + 1 k to the size of any f -EFT $$\sigma $$ σ -ASPT with $$f \ge \log n$$ f ≥ log n and $$\sigma < \frac{3k+1}{k+1}$$ σ < 3 k + 1 k + 1 , that holds if the Erdős’ girth conjecture is true.