## Maintaining Minimum Spanning Trees in Dynamic Graphs (1997)

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Venue: | IN PROC. 24TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING (ICALP |

Citations: | 26 - 2 self |

### BibTeX

@INPROCEEDINGS{Henzinger97maintainingminimum,

author = {Monika Rauch Henzinger and Valerie King},

title = {Maintaining Minimum Spanning Trees in Dynamic Graphs},

booktitle = {IN PROC. 24TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING (ICALP},

year = {1997},

pages = {594--604},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

We present the first fully dynamic algorithm for maintaining a minimum spanning tree in time o( # n) per operation. To be precise, the algorithm uses O(n 1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic. An immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and, bipartiteness in amortized time O(n 1/3 log n) per update, with O(1) worst case time per query.

### Citations

296 |
A data structure for dynamic trees
- Sleator, Tarjan
- 1983
(Show Context)
Citation Context ...O(1) time. In 1992, Eppstein et. al. [3, 4] improved the update time to O( √ n) using the sparsification technique. If only edge insertions are allowed, the Sleator-Tarjan dynamic tree data structure =-=[13]-=- maintains the minimum spanning forest in time O(log n) per insertion or query. If only edge deletions are allowed (“deletions-only”), then no algorithm faster than the �( √ n) fully dynamic algorithm... |

222 |
Introduction to Algorithms
- Corman, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...sing a constant number of splits and joins on the corresponding B-trees. Also we can test two vertices of the forest to determine whether they are in the same tree in time O(log d n). See for example =-=[1, 11]-=- for operations on B-trees. If d = n # , for # a positive constant, then the join and split operations take time O(d) and the test operation takes time O(1). We refer to the B-trees used to store ET-s... |

150 |
Data structures for on-line updating of minimum spanning trees, with applications
- Frederickson
- 1985
(Show Context)
Citation Context ... reconnects F if e exists or return null if e does not exist. In addition, the data structure permits the following type of query: connected(u,v): Determine if vertices u and v are connected. In 1985 =-=[7]-=-, Fredrickson introduced a data structure known as topology trees for the fully dynamic minimum spanning tree problem with a worst case cost of O( # m) per update His data structure permitted connecti... |

145 |
Data Structures and Algorithms 1: Sorting and Searching. EATCS,Monographs on Theoretical Computer Science
- Mehlhorn
- 1984
(Show Context)
Citation Context ...sing a constant number of splits and joins on the corresponding B-trees. Also we can test two vertices of the forest to determine whether they are in the same tree in time O(log d n). See for example =-=[1, 11]-=- for operations on B-trees. If d = n # , for # a positive constant, then the join and split operations take time O(d) and the test operation takes time O(1). We refer to the B-trees used to store ET-s... |

123 |
Sparsification – a technique for speeding up dynamic graph algorithms
- Eppstein, Galil, et al.
- 1997
(Show Context)
Citation Context ...r the fully dynamic minimum spanning tree problem with a worst case cost of O( # m) per update His data structure permitted connectivity queries to be answered in O(1) time. In 1992, Eppstein et. al. =-=[3, 4]-=- improved the update time to O( # n) using the sparsification technique. If only edge insertions are allowed, the Sleator-Tarjan dynamic tree data structure [13] maintains the minimum spanning forest ... |

74 | Balanced matroids
- Feder, Mihail
- 1992
(Show Context)
Citation Context ...lowed, however, much faster times are achievable [9, 10]. Additionally, improvements can be achieved in the following static problems (see [4, 3]): randomly sampling spanning forests of a given graph =-=[6]-=-; finding a color-constrained minimum spanning tree [8]. The paper is structured as follows: In Section 2 we give a deletions-only minimum spanning tree algorithm. In Section 3, we show how to use a s... |

49 | Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph”, Algorithmica 7 - Nagamochi, Ibaraki - 1992 |

47 | Randomized dynamic graph algorithms with polylogarithmic time per operation
- Henzinger, King
- 1995
(Show Context)
Citation Context ...n that the fully dynamic connectivity problem, i.e., the restricted problem where all edge costs are the same, can be solved in amortized time O(log 2 n) per update and O(logn) per connectivity query =-=[9, 10]-=-. However, this approach could not be extended to arbitrary edge 2 weights, leaving the question open as to whether the fully dynamic minimum spanning tree problem can be solved in time o( # n). In th... |

39 | Dynamic Euclidean minimum spanning trees and extrema of binary functions
- Eppstein
- 1995
(Show Context)
Citation Context ...r the following problems: connectivity, bipartiteness, k-edge witness, maximal spanning forest decomposition, and Euclidean minimum spanning tree. See [9] for all but the last reduction; see Eppstein =-=[2]-=- for the last reduction. For these problems, the new algorithm achieves an O(n 1/6 / log n) factor improvement over the previously best deterministic running time. If randomization is allowed, however... |

29 |
An on-line edge-deletion problem
- Shiloach, Even
- 1981
(Show Context)
Citation Context ... following: Claim 2.1 O( P w(T 1 )) summed over all smaller components T 1 which split from a tree T on any given level during all Replace operations is O(w(T ) log n). The proof of the claim follows =-=[5]-=-. The first time a smaller component T 1 of a tree T is searched, it can have weight no greater than w(T )/2. Between two successive times that |L f (i) (v)| contributes to the weight of a smaller com... |

28 | Improved sparsification
- Eppstein, Galil, et al.
- 1993
(Show Context)
Citation Context ...r the fully dynamic minimum spanning tree problem with a worst case cost of O( # m) per update His data structure permitted connectivity queries to be answered in O(1) time. In 1992, Eppstein et. al. =-=[3, 4]-=- improved the update time to O( # n) using the sparsification technique. If only edge insertions are allowed, the Sleator-Tarjan dynamic tree data structure [13] maintains the minimum spanning forest ... |

17 |
Tarjan, "A data structure for dynamic trees
- Sleator, E
- 1983
(Show Context)
Citation Context ...O(1) time. In 1992, Eppstein et. al. [3, 4] improved the update time to O( # n) using the sparsification technique. If only edge insertions are allowed, the Sleator-Tarjan dynamic tree data structure =-=[13] maintains-=- the minimum spanning forest in time O(log n) per insertion or query. If only edge deletions are allowed ("deletions-only"), then no algorithm faster than the #( # n) fully dynamic algorithm... |

10 | Improved Sampling with Applications to Dynamic Graph Algorithms
- Henzinger, Thorup
- 1996
(Show Context)
Citation Context ...n that the fully dynamic connectivity problem, i.e., the restricted problem where all edge costs are the same, can be solved in amortized time O(log 2 n) per update and O(logn) per connectivity query =-=[9, 10]-=-. However, this approach could not be extended to arbitrary edge 2 weights, leaving the question open as to whether the fully dynamic minimum spanning tree problem can be solved in time o( # n). In th... |

6 |
Srinivas, “Algorithms and data structure for an expanded family of matroid intersection problems
- Frederickson, A
- 1989
(Show Context)
Citation Context ...]. Additionally, improvements can be achieved in the following static problems (see [4, 3]): randomly sampling spanning forests of a given graph [6]; finding a color-constrained minimum spanning tree =-=[8]-=-. The paper is structured as follows: In Section 2 we give a deletions-only minimum spanning tree algorithm. In Section 3, we show how to use a sequence of deletions-only data structures to create a f... |

3 |
Introduction to Algorithms.MITPress
- Corman, Leiserson, et al.
- 1989
(Show Context)
Citation Context ...sing a constant number of splits and joins on the corresponding B-trees. Also we can test two vertices of the forest to determine whether they are in the same tree in time O(log d n). See for example =-=[1, 11]-=- for operations on B-trees. If d = n α ,for αa positive constant, then the join and split operations take time O(d) and the test operation takes time O(1). We refer to the B-trees used to store ET-seq... |

2 |
Data Structures and Network
- Tarjan
- 1983
(Show Context)
Citation Context .... The correctness of the algorithm follows easily from the invariants. We use the well-known fact that an edge is in the minimum spanning tree iff it is not the heaviest edge in any cycle ("red r=-=ule" [14]-=-). We also note that every edge in the composite data structure is an edge in G. Let e be an edge of the MST which is deleted. Let e # be the correct replacement edge. Consider the state of the compos... |

1 |
Srinivas, "Algorithms and data structures for an expanded family of matroid intersection problems
- Frederickson, A
- 1989
(Show Context)
Citation Context ...]. Additionally, improvements can be achieved in the following static problems (see [4, 3]): randomly sampling spanning forests of a given graph [6]; finding a color-constrained minimum spanning tree =-=[8]-=-. The paper is structured as follows: In Section 2 we give a deletions-only minimum spanning tree algorithm. In Section 3, we show how to use a sequence of deletions-only data structures to create a f... |

1 | Data Structures and Network Flow, SIAM - Tarjan - 1983 |