## Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures (1998)

### Cached

### Download Links

- [www.mat.uniroma2.it]
- [www.emis.de]
- [www.cs.brown.edu]
- [www.mat.uniroma2.it]
- [www.emis.math.ca]
- [www.emis.de]
- [jgaa.info]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of Graph Algorithms and Applications |

Citations: | 14 - 6 self |

### BibTeX

@ARTICLE{Nardelli98findingall,

author = {Enrico Nardelli and Guido Proietti},

title = {Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures},

journal = {Journal of Graph Algorithms and Applications},

year = {1998},

volume = {5},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the all-best-swaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where |V | = n and |E | = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1

### Citations

1130 |
Graph Theory
- Harary
- 1969
(Show Context)
Citation Context ...f a real new MDST. Finally, in Section 5, we present conclusions and list some open problems. 2 Solving the ABS problem 2.1 Problem statement For basic definitions of graph terminologies, we refer to =-=[6]-=-. Let T =(V,ET )be a spanning tree of a 2-edge connected, undirected graph G =(V,E), of n nodes and m edges, and with a nonnegative real length |e| associated with each edge e ∈ E. Let D(T )=〈d1,d2,..... |

150 |
Data structures for on-line updating of minimum spanning trees, with applications
- Frederickson
- 1985
(Show Context)
Citation Context ...(n) time per update, using O(m) space and preprocessing time. For the edge deletion case, that is of interest for this paper, the authors make use of a topology tree and a 2-dimensional topology tree =-=[3]-=-, augmented with some extra information. This requires O(m) time and space for initialization, and allows to compute the length of the diameter of any tree obtained as a consequence of a swap in O(n) ... |

109 |
Leeuwen. Worst-case analysis of set union algorithms
- Tarjan, van
(Show Context)
Citation Context ...ves the ABS problem for a MDST in O(n √ m) time and O(m) space, thus strictly improving previous bounds for m = o(n 2 ). This can be done by adapting the well-known path halving compression technique =-=[10]-=- for answering in O(1) amortized time (over a total of Θ(n √ m) queries) the following query: GivenarootedtreeT and a pair of nodes y and v in T such that v belongs to the subtree of T rooted at y, wh... |

69 | Design of survivable networks
- Grötschel, Monma, et al.
- 1995
(Show Context)
Citation Context ...idual network components fail. In the past few years, the ability of a network to survive a failure (its survivability) has been studied intensely (an excellent survey paper on survivable networks is =-=[5]-=-). From the practical side, this has largely been a consequence of the replacement of metal wire meshes by fiber optic networks: Their extremely high bandwidth makes it economically attractive to make... |

54 | Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time
- Dixon, Rauch, et al.
- 1992
(Show Context)
Citation Context ...– for every edge in the network. In the past, the ABS problem has been solved both when the network is a MST and a SPT. In the first case, the fastest solution known to date is an O(m) time algorithm =-=[2]-=-, while in the second case, an O(n 2 ) time algorithm has been presented in [9]. However, in several applications, the used spanning tree is neither an MST nor a SPT. Rather, many network architecture... |

54 | Dynamic graph algorithms
- Eppstein, Galil, et al.
- 1999
(Show Context)
Citation Context ...has been solved in [7]. This problem belongs to the family of problems that is concerned with updating the solution of a graph problem after dynamic changes of the graph; for a recent survey, consult =-=[2]-=-. The approach in [7] is more general than what is needed for solving the ABS problem. Hence, if we use it for solving the ABS problem, we spend O(n2 )timeandO(m+n)space and preprocessing time. We get... |

28 | Minimizing diameters of dynamic trees
- Alstrup, Holm, et al.
- 1997
(Show Context)
Citation Context ... it for solving the ABS problem, we spend O(n 2 ) time and O(m) space and preprocessing time. We get these bounds by computing a best swap in O(n) time for each deleted edge. Recently, Alstrup et al. =-=[1]-=- improved the runtime for computing a best swap in an incremental context (i.e., when no deletions are allowed) to O(log 2 n). In the same paper, the authors claim that it is possible to maintain the ... |

27 |
The k most vital arcs in the shortest path problem
- Malik, Mittal, et al.
- 1989
(Show Context)
Citation Context ...case, the problem of finding an edge in the tree whose removal results in the largest increase of the distance between the source node r and a given node s has been solved efficiently by Malik et al. =-=[9]-=-, who gave an O(m+n log n) time algorithm. Recently, Nardelli et al. [10] defined a different parameter for measuring the vitality of an edge along a shortest path, looking for the edge whose removal ... |

24 | On the minimum diameter spanning tree problem
- Hassin, Tamir
- 1995
(Show Context)
Citation Context ...|/|D(TG−e)|, where, as usual, e ′ is a best swap edge for e. We start by observing that the MDST T must contain at least one path of length D(T ) which is a shortest path between its two ending nodes =-=[7]-=-. W.l.o.g., let us assume that such a path is 〈d1,...,dk〉, and therefore 〈dc,...,d1〉 and 〈dc,...,dk〉 are shortest paths. The following holds: Theorem 4.1 For any edge e =(x, y) ∈ ET , we have σe ≤ 5/2... |

21 |
Ambivalent data structures for dynamic 2-edge connectivity and k-smallest spanning trees
- Frederickson
- 1991
(Show Context)
Citation Context ...t dc = di+1. Let � T denote a source directed tree obtained by rooting T in dc and orienting the edges towards the leaves. Following [8], we maintain a topology tree and a 2-dimensional topology tree =-=[3, 4]-=-, augmented with some extra-information, to efficiently retrieve only O( √ m) selected edges among the O(m) replacement edges, whenever an edge e in T is deleted. In fact, among the selected edges, a ... |

19 | Swapping a failing edge of a single source shortest paths tree is good and fast. Algorithmica 35 - Nardelli, Proietti, et al. - 2003 |

12 | Efficient algorithms for finding the most vital edge of a minimum spanning tree - Iwano, Katoh - 1993 |

6 |
Maintaining spanning trees of small diameter
- Italiano, Ramaswani
- 1994
(Show Context)
Citation Context ... and the ABS problem for a MDST is defined accordingly. In this paper we present an efficient solution precisely for this latter problem. Our approach makes use of some of the techniques presented in =-=[8]-=-, in which the authors consider the related problem of computing a best swap edge in a fully dynamic context, where the original MDST evolves due to repeated insertions and deletions of edges in the g... |

3 |
Finding the detour-critical edge of a shortest path between two nodes
- Nardelli, Proietti, et al.
- 1998
(Show Context)
Citation Context ...n the largest increase of the distance between the source node r and a given node s has been solved efficiently by Malik et al. [9], who gave an O(m+n log n) time algorithm. Recently, Nardelli et al. =-=[10]-=- defined a different parameter for measuring the vitality of an edge along a shortest path, looking for the edge whose removal will result in the worst detour to reach the destination node, and they s... |

1 |
de Lichtenberg and M. Thorup, Minimizing diameters of dynamic trees
- Alstrup, Holm, et al.
- 1997
(Show Context)
Citation Context ...use it for solving the ABS problem, we spend O(n2 )timeandO(m+n)space and preprocessing time. We get these bounds by computing a best swap in O(n) time for each deleted edge. Recently, Alstrup et al. =-=[1]-=- improved the runtime for computing a best swap in an incremental context (i.e., when no deletions are allowed) to O(log 2 n). By using some of the results in [7], their approach can be adapted to sol... |