Fully dynamic all-pairs shortest paths with worst-case update-time revisited

@article{Abraham2017FullyDA,
  title={Fully dynamic all-pairs shortest paths with worst-case update-time revisited},
  author={Ittai Abraham and Shiri Chechik and Sebastian Krinninger},
  journal={ArXiv},
  year={2017},
  volume={abs/1607.05132}
}
We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter c > 1 has a worst-case update time of O… 

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References

SHOWING 1-10 OF 38 REFERENCES

Dynamic approximate all-pairs shortest paths in undirected graphs

  • Liam RodittyU. Zwick
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
TLDR
Three dynamic algorithms for the approximate all-pairs shortest paths problem in unweighted undirected graphs, including a fully dynamic algorithm with an expected amortized update time of O(mn/t) and worst-case query time ofO(t).

Fully dynamic all pairs shortest paths with real edge weights

  • C. DemetrescuG. Italiano
  • Computer Science, Mathematics
    Proceedings 2001 IEEE International Conference on Cluster Computing
  • 2001
TLDR
This work presents the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with real-valued edge weights, and gives a randomized algorithm with one-sided error which supports updates faster in O(S/spl middot/nlog/sup 3/n) amortized time.

Worst-case update times for fully-dynamic all-pairs shortest paths

TLDR
The first solution to the fully-dynamic all pairs shortest path problem where every update is faster than a recomputation from scratch in Ω(n) time is presented, for a directed graph with arbitrary non-negative edge weights.

Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization

TLDR
An algorithm with a total update time of Ȏ(n5/2) and constant query time that has an additive error of two in addition to the 1 + ϵ multiplicative error is presented, giving the first improved deterministic algorithm since 1981.

A new approach to dynamic all pairs shortest paths

TLDR
A fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in amortized time per update and unit worst-case time per distance query, where n is the number of vertices.

Deterministic decremental single source shortest paths: beyond the o(mn) bound

TLDR
This paper presents the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs, and is faster than all existing randomized algorithms.

Sublinear-time decremental algorithms for single-source reachability and shortest paths on directed graphs

TLDR
A randomized algorithm is obtained which achieves an Õ (mn0.984) expected total update time for SSR and (1 + ε)-approximate SSSP, where Õ(·) hides poly log n and this algorithm serves as a building block for several other dynamic algorithms.

Maintaining all-pairs approximate shortest paths under deletion of edges

TLDR
A hierarchical scheme for efficiently maintaining all-pairs approximate shortest paths in undirected unweighted graphs under deletions of edges and the first update time algorithm based on this scheme is presented.

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

TLDR
The approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph, and is guaranteed to be accurate to within a 1+ $\epsilon$ factor.