# 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}
}
• Published 18 July 2016
• Computer Science, Mathematics
• ArXiv
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

• Computer Science
45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
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

• Computer Science, Mathematics
Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
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

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

• Computer Science
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
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

• Computer Science, Mathematics
STOC '03
• 2003
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

• Computer Science
STOC
• 2016
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

• Computer Science
STOC
• 2014
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

• Computer Science
SODA '03
• 2003
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

• Computer Science
Algorithmica
• 1998
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.