## Minimizing Diameters of Dynamic Trees (1997)

Venue: | In Proc. 24th International Colloquium on Automata, Languages, and Programming (ICALP |

Citations: | 28 - 11 self |

### BibTeX

@INPROCEEDINGS{Alstrup97minimizingdiameters,

author = {Stephen Alstrup and Jacob Holm and Kristian de Lichtenberg and Mikkel Thorup},

title = {Minimizing Diameters of Dynamic Trees},

booktitle = {In Proc. 24th International Colloquium on Automata, Languages, and Programming (ICALP},

year = {1997},

pages = {270--280},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this paper we consider an on--line problem related to minimizing the diameter of a dynamic tree T . A new edge f is added, and our task is to delete the edge e of the induced cycle so as to minimize the diameter of the resulting tree T [ffgnfeg. Starting with a tree with n nodes, we show how each such best swap can be found in worst--case O(log 2 n) time. The problem was raised by Italiano and Ramaswami at ICALP'94 together with a related problem for edge deletions. Italiano and Ramaswami solved both problems in O(n) time per operation. 1 Introduction The diameter of a tree is the length of a longest simple path in the tree and such a path is called a diameter path. The unique midpoint on all diameter paths is called the center, hence the center is the point whose maximal distance to any node is as small as possible. In 1973 Handler [4] showed how one in linear time can compute the diameter (and center) of a tree. However, as pointed out by Rauch [8], too little work has been...

### Citations

153 |
Data structures for on-line updating of minimum spanning trees, with applications
- Frederickson
- 1985
(Show Context)
Citation Context ...ion, where n is the number of nodes in the tree(s) involved. We show this, since to the best of our knowledge, no such algorithm has been presented before. All our results are based on topology trees =-=[3, 2]-=- (the terminology of topology trees is recalled in Section 2). Our algorithm for maintaining the diameter is straightforward, based on a simple observation. Our algorithm for finding a best swap is mu... |

81 | Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees
- Frederickson
- 1997
(Show Context)
Citation Context ...ion, where n is the number of nodes in the tree(s) involved. We show this, since to the best of our knowledge, no such algorithm has been presented before. All our results are based on topology trees =-=[3, 2]-=- (the terminology of topology trees is recalled in Section 2). Our algorithm for maintaining the diameter is straightforward, based on a simple observation. Our algorithm for finding a best swap is mu... |

78 |
Multi-wavelength lightwave networks for computer communication
- Ramaswami
- 1993
(Show Context)
Citation Context .... To the best of our knowledge, the only dynamic algorithms concerning diameters are those given by Italiano and Ramaswami in ICALP'94 [5], motivated by problems in high-speed wide-area networks (see =-=[6, 7]-=- for details). They consider how to minimize the diameter of a dynamic tree T with n nodes and non--negative edge cost. Let f be a new edge which introduce a cycle C in the dynamic tree. Then removing... |

21 |
Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees
- Frederickson
- 1991
(Show Context)
Citation Context ...ion, where n is the number of nodes in the tree(s) involved. We show this, since to the best of our knowledge, no such algorithm has been presented before. All our results are based on topology trees =-=[3, 2]-=- (the terminology of topology trees is recalled in Section 2). Our algorithm for maintaining the diameter is straightforward, based on a simple observation. Our algorithm for finding a best swap is mu... |

6 |
Maintaining spanning trees of small diameter
- Italiano, Ramaswani
- 1994
(Show Context)
Citation Context ... been done to dynamically maintain information about the diameter. To the best of our knowledge, the only dynamic algorithms concerning diameters are those given by Italiano and Ramaswami in ICALP'94 =-=[5]-=-, motivated by problems in high-speed wide-area networks (see [6, 7] for details). They consider how to minimize the diameter of a dynamic tree T with n nodes and non--negative edge cost. Let f be a n... |

5 |
Fully dynamic graph algorithms and their data structures
- Rauch
- 1992
(Show Context)
Citation Context ...e point whose maximal distance to any node is as small as possible. In 1973 Handler [4] showed how one in linear time can compute the diameter (and center) of a tree. However, as pointed out by Rauch =-=[8]-=-, too little work has been done to dynamically maintain information about the diameter. To the best of our knowledge, the only dynamic algorithms concerning diameters are those given by Italiano and R... |

2 |
A simple parallel algorithm for computing the diameters of all vertices in a tree and its application
- Chen
- 1992
(Show Context)
Citation Context ...a dynamic forest under link and cut. The algorithm will be used in the following section. It builds on a generalization of former exploitations of properties of diameters and spanning trees (see e.g. =-=[1, 4, 5]-=-). This generalization, given in the following lemma, makes it possible to construct efficient divide and conquer algorithms. Let T = (V; E) be a tree with n nodes. With each edge e in E is associated... |

2 |
Minimax location of a facility in an undirected tree network. Transportation
- Handler
- 1973
(Show Context)
Citation Context ...th is called a diameter path. The unique midpoint on all diameter paths is called the center, hence the center is the point whose maximal distance to any node is as small as possible. In 1973 Handler =-=[4]-=- showed how one in linear time can compute the diameter (and center) of a tree. However, as pointed out by Rauch [8], too little work has been done to dynamically maintain information about the diamet... |

2 |
Mantaining spanning trees of small diameter. Unpublished revised version of the ICALP paper
- Italiano, Ramaswami
- 1996
(Show Context)
Citation Context .... To the best of our knowledge, the only dynamic algorithms concerning diameters are those given by Italiano and Ramaswami in ICALP'94 [5], motivated by problems in high-speed wide-area networks (see =-=[6, 7]-=- for details). They consider how to minimize the diameter of a dynamic tree T with n nodes and non--negative edge cost. Let f be a new edge which introduce a cycle C in the dynamic tree. Then removing... |