Near linear time (1 + ε)-approximation for restricted shortest paths in undirected graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Improved dynamic algorithms for maintaining approximate shortest paths under deletions
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The impact of edge deletions on the number of errors in networks
OPODIS'11 Proceedings of the 15th international conference on Principles of Distributed Systems
Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Fully dynamic randomized algorithms for graph spanners
ACM Transactions on Algorithms (TALG)
Maintaining shortest paths under deletions in weighted directed graphs: [extended abstract]
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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For any fixed $1 \eps 0$ we present a fully dynamic algorithm for maintaining $(2 + \eps)$-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the \emph{expected} amortized update time. \\\indent Our query time $O(\log \log \log n)$. The update time is $\widetilde{O}(mn^{O(1/\sqrt{\log n})}\log(nR))$, where $R$ is the ratio between the heaviest and the lightest edge weight in the graph (so $R = 1$ in unweighted graphs). Unfortunately, the update time does have the drawback of a super-polynomial dependence on $\eps$: it grows as $(3 / \eps)^{\sqrt{\log n / \log(3/\eps)}} = n^{\sqrt{\log(3/\eps) / \log n}}$.\\\indent Our algorithm has a significantly faster update time than any other algorithm with sub-polynomial query time. For exact distances, the state of the art algorithm has an update time of $\widetilde{O}(n^2)$. For approximate distances, the best previous algorithm has a $O(kmn^{1/k})$ update time and returns $(2k - 1)$ stretch paths. Thus, it needs an update time of $O(m\sqrt{n})$ to get close to our approximation, and it has to return $O(\sqrt{\log n})$ approximate distances to match our update time.