# Sparsification—a technique for speeding up dynamic graph algorithms

@article{Eppstein1997SparsificationaTF, title={Sparsification—a technique for speeding up dynamic graph algorithms}, author={David Eppstein and Zvi Galil and Giuseppe F. Italiano and Amnon Nissenzweig}, journal={J. ACM}, year={1997}, volume={44}, pages={669-696} }

We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time<italic>O</italic>(<italic>n</italic><supscrpt>1/2</supscrpt>) per change; 3-edge connectivity, in time <italic>O</italic>(<italic>n</italic><supscrpt>2/3</supscrpt>) per change; 4-edge connectivity, in time <italic>O</italic>(<italic>n</italic…

## 261 Citations

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.

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

- Computer ScienceJACM
- 2001

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…

Fully dynamic planarity testing with applications

- MathematicsJACM
- 1999

The first algorithm for this problem with sub-linear running time is affirmatively answered, and it affirmatively answers a question posed in Epstein et al.

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…

An Experimental Study of Polylogarithmic, Fully Dynamic, Connectivity Algorithms

- Computer ScienceJEAL
- 2001

An experimental study of different variants of the fully-dynamicconnectivity algorithm of Holm, de Lichtenberg, and Thorup (STOC'98) sheds light upon similarities and differences betweent the two algorithms.

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.

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].

Maintaining dynamic graph properties deterministically

- Computer Science
- 2001

The algorithms match the previous best randomized bounds, and improve substantially over the best deterministic bounds, for maintaining several properties on undirected graphs subject to edge insertions and deletions in polylogarithmic time per operation.

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.

Vertex Sparsifiers for Hyperedge Connectivity

- Computer Science, MathematicsArXiv
- 2022

This paper constructs a sparsiﬁer whose size matches the state-of-the-art for normal graphs and studies a natural extension for c -hyperedge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for c-edge st-connectivity.

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