# Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity

@article{Holm2001PolylogarithmicDF, title={Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity}, author={Jacob Holm and Kristian de Lichtenberg and Mikkel Thorup}, journal={J. ACM}, year={2001}, volume={48}, pages={723-760} }

Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity. Assuming that we start with no edges in a graph with <i>n</i> vertices, the amortized operation costs are <i>O</i>(log<sup>2</sup> <i>n</i>) for connectivity, <i>O</i>(log<sup>4</sup> <i>n</i>) for minimum spanning forest, 2-edge connectivity, and <i>O</i>(log<sup>5</sup> <i>n</i>) biconnectivity.

## 368 Citations

### Faster dynamic matchings and vertex connectivity

- Computer ScienceSODA '07
- 2007

We present first fully dynamic subquadratic algorithms for: computing maximum matching size, computing maximum bipartite matching weight, computing maximum number of vertex disjoint <i>s, t</i> paths…

### Simple Deterministic Algorithms for Fully Dynamic Maximal Matching

- Computer ScienceTALG
- 2015

The first deterministic fully dynamic algorithm that outperforms the trivial one is shown and is a 3/2-approximate maximum cardinality matching (MCM), which addresses an open question of Onak and Rubinfeld [2010].

### Fully-dynamic min-cut

- Computer ScienceSTOC '01
- 2001

It is shown that the algorithm can maintain up to polylogarithmic edge connectivity for a fully-dynamic graph in <italic>\tilde O(\sqrt{n})</italic) time per edge insertion or deletion, which matches the best time bound for 1-edge connectivity.

### Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O(n2 barrier

- Computer ScienceJACM
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An algorithm for directed acyclic graphs that breaks through the O(n)(2) barrier on the single-operation complexity of fully dynamic transitive closure, where <i>n</i> is the number of edges in the graph.

### Dynamic Graph Algorithms with Applications

- Computer ScienceSWAT
- 2000

Amortized fully-dynamic polylogarithmic algorithms for connectivity, minimum spanning trees (MST), 2-edge- and biconnectivity, and output sensitive algorithms for dynamic shortest paths have been applied successfully to speed up local search algorithms for improving routing on the internet.

### Worst-case Deterministic Fully-Dynamic Planar 2-vertex Connectivity

- Computer Science, MathematicsArXiv
- 2022

It is shown that the best for fully-dynamic biconnectivity was an amortised O (log 3 n ) for general graphs, and algorithms with worst-case polylogarithmic update times were known only in the partially dynamic (insertion-only or deletion-only) setting.

### Optimal Decremental Connectivity in Non-Sparse Graphs

- Computer ScienceArXiv
- 2021

A dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions that is Monte-Carlo randomized and can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time.

### Fully dynamic randomized algorithms for graph spanners

- Computer ScienceTALG
- 2012

This work presents fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments designed for undirected unweighted graphs and use randomization in a crucial manner.

### Fully-Dynamic Min-Cut*

- Computer ScienceComb.
- 2007

It is shown that this algorithm can maintain up to polylogarithmic edge connectivity for a fully-dynamic graph and immediately gets a sampling based expected factor (1+o(1))) approximation to general edge connectivity.

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- Computer ScienceSTOC '05
- 2005

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.

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