Sparsification-a technique for speeding up dynamic graph algorithms

@article{Eppstein1992SparsificationaTF,
  title={Sparsification-a technique for speeding up dynamic graph algorithms},
  author={David Eppstein and Zvi Galil and Giuseppe F. Italiano and Amnon Nissenzweig},
  journal={Proceedings., 33rd Annual Symposium on Foundations of Computer Science},
  year={1992},
  pages={60-69}
}
The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and graph 2-edge-connectivity, in time O(n/sup 1/2/log(m/n)) per change; 3-edge-connectivity, in time O(n/sup 2/3/) per change; 4-edge-connectivity, in time O(n alpha (n)) per change; k-edge-connectivity, in time O(n log n) per change; bipartiteness, 2-vertex-connectivity, and 3-vertex-connectivity, in… 
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References

SHOWING 1-10 OF 22 REFERENCES
Fully dynamic algorithms for edge connectivity problems
TLDR
Algorithms to test at any time whether two vertices belong to the same 2-edge-connected component of a connected graph, and how to insert and delete an edge in 0(m213) time in the worst case, where m is the current number of edges in the graph.
Fully dynamic biconnectivity in graphs
  • M. Rauch
  • Computer Science
    Proceedings., 33rd Annual Symposium on Foundations of Computer Science
  • 1992
TLDR
The author presents an algorithm for maintaining the bi-connected components of a graph during a sequence of edge insertions and deletions, which is the first sublinear algorithm for this problem.
A matroid approach to finding edge connectivity and packing arborescences
  • H. Gabow
  • Computer Science, Mathematics
    STOC '91
  • 1991
TLDR
An algorithm that finds k edge-disjoint arborescences on a directed graph in time O(kmn + k3n2)2 is presented, based on two theorems of Edmonds that link these two problems and show how they can be solved.
On-line maintenance of the four-connected components of a graph
TLDR
The authors present a static data structure that answers k-connectivity queries for k<or=4 that supports queries and updates in time O( alpha (l,n)) amortized, and an efficient algorithm for testing whether graph G is four-connected that runs in O(n alpha (n, n)+m) time using O( n+m) space.
Efficient algorithms for finding minimum spanning trees in undirected and directed graphs
TLDR
This paper uses F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs and can be extended to allow a degree constraint at one vertex.
Efficient Algorithms for Graphic Matroid Intersection and Parity (Extended Abstract)
TLDR
Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network.
Algorithms for parallel k-vertex connectivity and sparse certificates
TLDR
It is shown that sparse certificates for undirected graphs can be computed by executing k breadth first searches in sequence, and sequential algorithms for finding (undirected) sparse certificates ‘(on-line”, and for finding - are given.
Maintenance of Triconnected Components of Graphs (Extended Abstract)
In this paper, optimal algorithms and data structures are presented to maintain the triconnected components of a general graph, under insertions of edges in the graph. At any moment, the data
Finding thek smallest spanning trees
TLDR
It is shown that the best spanning trees for a set of points in the plane can be computed in timeO(min(k2n+n logn,k2+kn log log(n/k))).
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