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

@article{Henzinger2013DynamicAA, title={Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization}, author={Monika Henzinger and Sebastian Krinninger and Danupon Nanongkai}, journal={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, year={2013}, pages={538-547} }

We study dynamic (1 + ϵ)-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected n-node m-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of Ȏ(mn) and constant query time by Roditty and Zwick (FOCS 2004). The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach (JACM 1981); it has a total update time of O(mn2) and constant query time. We improve these results as…

## 73 Citations

### A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths

- Computer ScienceSODA
- 2014

The first approximation algorithm whose total update time is faster than O(mn) for all values of m is obtained, and is the first that breaks through the quadratic time barrier.

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

- Computer ScienceSTOC
- 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.

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

- Computer Science, MathematicsSODA
- 2017

This work revisits the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph and gives worst-case guarantees on the time needed to process a single update, and shows that randomization along with a more direct approach can provide better bounds.

### Fast Deterministic Fully Dynamic Distance Approximation

- Computer ScienceArXiv
- 2021

Improved dynamic algorithms that, given an unweighted and undirected graph G = ( V, E ) undergoing edge insertions and deletions, maintain (1 + (cid:15) )-approximations of the st -distance between a given pair of nodes s and t are obtained.

### Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

- Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

This paper develops an efficient (1 + ε) -approximation algorithm for this query using fast matrix multiplication and obtains the first dynamic APSP algorithm with subquadratic update time and sublinear query time.

### Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

- Computer ScienceSTOC
- 2017

This work presents two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions and provides the first polynomial improvement over the long-standing O(√n) bound for such type of algorithms.

### Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time

- Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

A (1 + ε)-approximation algorithm with O(m1+o(1) total update time, thus obtaining near-linear time for the weighted case, where the edge weights are integers from 1 to W and the only prior work on weighted graphs in o(mn log W) time is the O(mn0.986 log W)-time algorithm by Henzinger, Krinninger, and Nanongkai (STOC 2014).

### New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

- Computer Science, Mathematics
- 2022

Two new approximate decremental APSP algorithms are provided, one for weighted and one for unweighted graphs, and they both improve over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016].

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

- Computer ScienceSTOC
- 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.

### A New Deterministic Algorithm for Dynamic Set Cover

- Computer Science, Mathematics2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

This work presents a deterministic dynamic algorithm for maintaining a (1+ε)f-approximate minimum cost set cover with O(f log(Cn)/ε^2) amortized update time, when the input set system is undergoing element insertions and deletions, via the primal-dual approach.

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