## Swapping a failing edge of a single source shortest paths tree is good and fast (1999)

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Venue: | Algorithmica |

Citations: | 19 - 7 self |

### BibTeX

@ARTICLE{Nardelli99swappinga,

author = {Enrico Nardelli and Guido Proietti and Peter Widmayer},

title = {Swapping a failing edge of a single source shortest paths tree is good and fast},

journal = {Algorithmica},

year = {1999},

volume = {35},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Let G = (V, E) be a 2-edge connected, undirected and nonnegatively weighted graph, and let S(r) be a single source shortest paths tree (SPT) of G rooted at r ∈ V. Whenever an edge e in S(r) fails, we are interested in reconnecting the nodes now disconnected from the root by means of a single edge e ′ crossing the cut created by the removal of e. Such an edge e ′ is named a swap edge for e. Let Se/e ′(r) be the swap tree (no longer an SPT, in general) obtained by swapping e with e ′ , and let Se be the set of all possible swap trees with respect to e. Let F be a function defined over Se that expresses some feature of a swap tree, such as the average length of a path from the root r to all the nodes below edge e, or the maximum length, or one of many others. A best swap edge for e with respect to F is a swap edge f such that F(Se/f (r)) is minimum. In this paper we present efficient algorithms for the problem of finding a best swap edge, for each edge e of S(r), with respect to several objectives. Our work is motivated by a scenario in which individual connections in a communication network suffer transient failures. As a consequence of an edge failure, the shortest paths to all the nodes below the failed edge might completely change, and it might be desirable to avoid an expensive switch to a new SPT, because the failure is only temporary. As an aside, what we get is not even far from a new SPT: our analysis shows that the trees obtained from the swapping have features very similar to those of the corresponding SPTs rebuilt from scratch. Key Words. Network survivability, Single source shortest paths tree, Swap algorithms. 1. Introduction. Survivability

### Citations

1414 |
Network Flows: Theory, Algorithms and Application
- Ahuja, Magnati, et al.
- 1993
(Show Context)
Citation Context ...Te/f is at most 5/2 times the diameter of T ′ , the MDST of G − e. However, many network architectures are based on a single source shortest paths tree (SPT) rooted at a given node r, sayS(r) =(V,ES) =-=[1]-=-. This is especially true for centralized network, where there exists a privileged node broadcasting messages to all the other nodes. In this case, a best swap policy for a failing edge is not unique ... |

1137 |
Graph Theory
- Harary
- 1971
(Show Context)
Citation Context ...shortest paths trees, by extending our preliminary results presented in [10]. 1.1. Basic Definitions. We first recall some of the basic graph terminology that we use in what follows; for details, see =-=[5]-=-. Let G = (V, E) be an undirected graph, where V is the set of nodes and E ⊆ V × V is the set of edges, with a nonnegative real length |e| associated with each edge e ∈ E. Let n and m denote the numbe... |

577 |
Fibonacci heaps and their uses in improved network optimization algorithms
- Fredman, Tarjan
- 1987
(Show Context)
Citation Context ...on for this problem is known to be asymptotically better than recomputing n times an SPT [3]. Since the fastest algorithm for computing an SPT in a comparison-based model requires O(m + n log n) time =-=[2]-=-, it follows that recomputing from scratch S ′ e (r) for each failing edge e ∈ ES results in an O(mn + n2 log n) time algorithm. Although it might appear unfair to compare S ′ e (r) and Se/f (r), sinc... |

329 |
Fast Algorithms for Finding Nearest Common Ancestors
- Harel, Tarjan
(Show Context)
Citation Context ...e ′ = (u ′ ,v ′ ), let ze ′ be the nearest common ancestor in S(r) of u ′ and v ′ . Recall that the nearest common ancestors of all nontree edges can be computed in O(m · α(m, n)) time and O(n) space =-=[6]-=-. Since de/e ′(u,v)= d(r, u′ ) + d(r,v ′ ) +|e ′ |−|e|−2d(r, ze ′), we can compute in O(1) time de/e ′(u,v), and thus we spend O(m) time to select a best swap edge. Hence, it follows that we spend a t... |

254 |
Efficiency of a Good But Not Linear Set Union Algorithm
- Tarjan
(Show Context)
Citation Context ...st swap edge, i.e., a total O(n · m) time to solve the problem. We now show how the above time bound can be improved substantially. Let α(m, n) denote the functional inverse of the Ackermann function =-=[11]-=-. The following can be proved: THEOREM 3.1. There exists a swap algorithm solving the {r,v}-Problem in O(m · α(m, n)) time and O(m) space. PROOF. A swap algorithm needs to process all the edges in Ce,... |

87 |
Applications of path compression on balanced trees
- Tarjan
- 1979
(Show Context)
Citation Context ...rk is a minimum spanning tree of a given weighted graph, then the optimum might be to choose a swap edge of minimum weight. For minimum spanning trees, this question has been studied before [1], [8], =-=[13]-=-. In this paper we study the corresponding question for shortest paths trees, by extending our preliminary results presented in [10]. 1.1. Basic Definitions. We first recall some of the basic graph te... |

69 | Design of survivable networks
- Grötschel, Monma, et al.
- 1995
(Show Context)
Citation Context ...lity of the network to remain operational even if individual network components (such as a link or even a node) fail. In the past few years, several survivability problems have been studied intensely =-=[4]-=-. 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 ... the network is a minimum spanning tree of a given weighted graph, then the optimum might be to choose a swap edge of minimum weight. For minimum spanning trees, this question has been studied before =-=[1]-=-, [8], [13]. In this paper we study the corresponding question for shortest paths trees, by extending our preliminary results presented in [10]. 1.1. Basic Definitions. We first recall some of the bas... |

29 |
Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees
- Tarjan
- 1982
(Show Context)
Citation Context ...cle with e. Therefore, we can build a transmuter having as source nodes all the edges belonging to S(r) and as sink nodes all the nontree edges. This can be done in O(m · α(m, n)) time and O(m) space =-=[14]-=-. To associate e with its best swap edge, it remains to establish the value that hass62 E. Nardelli, G. Proietti, and P. Widmayer to be given to a sink node. Since we have to minimize the value of de/... |

24 | Fully dynamic output bounded single source shortest path problem, in:SODA
- Frigioni, Marchetti-Spaccamela, et al.
- 1996
(Show Context)
Citation Context ...g for each edge e ∈ ES from scratch an SPT S ′ e (r) of G − e. It is fair to compare with a recomputation from scratch, since no dynamic solution for this problem is known to be asymptotically better =-=[3]-=-. Furthermore, we compare Se/f (r) and S ′ e (r), on the basis of those features captured by the functions that have been considered for the swapping. More precisely, if a swap tree is obtained starti... |

18 | The complexity of finding most vital arcs and nodes
- Bar-Noy, Khuller, et al.
- 1995
(Show Context)
Citation Context ...urvivability problem is that of maintaining the shortest path between two specified nodes in the network under the assumption that at most k arcs or nodes along the original shortest path are removed =-=[2]-=-. In the extreme, a network might be designed as a spanning tree of some underlying graph of all possible links. A sparse network, however, is more likely to react catastrophically to failures, especi... |

14 | Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures
- Nardelli, Proietti, et al.
(Show Context)
Citation Context ...the diameter of the new spanning tree as low as possible (for practical motivations, see [7]). The problem of finding a best swap edge for each edge in T can be solved in O(n √ m) time and O(m) space =-=[9]-=-. Notice that in this case Te/f does not coincide with the MDST T ′ of G − e. However, it can be shown that the diameter of e Te/f is at most 5/2 times the diameter of T ′ e . Therefore, Te/f is funct... |

12 |
Efficient algorithms for finding the most vital edge of a minimum spanning tree
- Iwano, Katoh
- 1993
(Show Context)
Citation Context ...network is a minimum spanning tree of a given weighted graph, then the optimum might be to choose a swap edge of minimum weight. For minimum spanning trees, this question has been studied before [1], =-=[8]-=-, [13]. In this paper we study the corresponding question for shortest paths trees, by extending our preliminary results presented in [10]. 1.1. Basic Definitions. We first recall some of the basic gr... |

10 |
Complexity of monotone networks for computing conjunctions
- Tarjan
- 1978
(Show Context)
Citation Context ...red. Clearly, it also remains to establish whether the algorithms we have proposed are optimal. All the algorithms making use of a transmuter will need time �(m · α(m, n)), the size of the transmuter =-=[12]-=-; to improve beyond that, a different approach must be used. Finally, swap problems in other network topologies (e.g., bipartite networks, spanners, etc.) deserve further study. Acknowledgments. The a... |

6 |
Maintaining spanning trees of small diameter
- Italiano, Ramaswani
- 1994
(Show Context)
Citation Context ... the natural function to be defined is the diameter of Te/e ′, and a best swap edge is a swap edge which makes the diameter of the new spanning tree as low as possible (for practical motivations, see =-=[7]-=-). The problem of finding a best swap edge for each edge in T can be solved in O(n √ m) time and O(m) space [9]. Notice that in this case Te/f does not coincide with the MDST T ′ of G − e. However, it... |

1 |
Proietti and P.Widmayer, Finding all the best swaps of a minimum diameter spanning tree under transient edge failures
- Nardelli, G
(Show Context)
Citation Context ...est swap is a swap edge which makes the diameter of the new spanning tree as low as possible. The problem of finding a best swap for each edge in T has been solved by Nardelli et al. in O(n √ m) time =-=[7]-=-. They also showed that the diameter of Te/f is at most 5/2 times the diameter of T ′ , the MDST of G − e. However, many network architectures are based on a single source shortest paths tree (SPT) ro... |