Finding the most vital node of a shortest path

Nardelli, Enrico; Proietti, Guido; Widmayer, Peter

doi:10.1016/s0304-3975(02)00438-3

Cited by 87 publications

(52 citation statements)

References 13 publications

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“…We have shown that, surprisingly, such a data structure can be stored using nearly the same space required to store a single distance matrix, while still supporting queries in constant time. Our oracle can easily be constructed in O(mn 2 +n 3 log n) time, matching the preprocessing time of all pairs variants of similar problems such as most vital node detection [18] and Vickrey pricing [11] in networks; while these algorithms require Θ(n 3 ) space, our oracle requires only O(n 2 log n) space. Furthermore, we have shown that by using O(n 2.5 ) space we can improve construction time to O(mn 1.5 + n 2.5 log n).…”

Section: Discussionmentioning

confidence: 99%

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Oracles for Distances Avoiding a Failed Node or Link

Demetrescu¹,

et al. 2008

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Abstract. We consider the problem of preprocessing an edge-weighted directed graph G to answer queries that ask for the length and first hop of a shortest path from any given vertex x to any given vertex y avoiding any given vertex or edge. As a natural application, this problem models routing in networks subject to node or link failures. We describe a deterministic oracle with constant query time for this problem that uses O(n 2 log n) space, where n is the number of vertices in G. The construction time for our oracle is O(mn 2 + n 3 log n). However, if one is willing to settle for Θ(n 2.5 ) space, we can improve the preprocessing time to O(mn 1.5 + n 2.5 log n) while maintaining the constant query time. Our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path.

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Section: Discussionmentioning

confidence: 99%

“…Of a similar flavor is the most vital node (or arc) problem [1,2,4,18], which is the problem of identifying the vertex (or edge) on a given shortest path, whose removal results in the longest replacement path.…”

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confidence: 99%

Oracles for Distances Avoiding a Failed Node or Link

Demetrescu¹,

et al. 2008

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“…In this respect, a first result was obtained in [16], where an O(m + n log n) time algorithm was presented for the problem of finding a most vital node of a shortest path (i.e., a node whose removal induces a longest replacement shortest path between s and t). As the authors pointed out, such an algorithm can be used to implement a truthful mechanism for solving the shortest path problem in a communication network in which nodes are owned by selfish agents.…”

Section: Previous Resultsmentioning

confidence: 99%

Nearly Linear Time Minimum Spanning Tree Maintenance for Transient Node Failures

2004

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Abstract. Given a 2-node connected, real weighted, and undirected graph G = (V, E), with n nodes and m edges, and given a minimum spanning tree (MST) T = (V, E T ) of G, we study the problem of finding, for every node v ∈ V , a set of replacement edges which can be used for constructing an MST of G − v (i.e., the graph G deprived of v and all its incident edges). We show that this problem can be solved on a pointer machine in O(m · α(m, n)) time and O(m) space, where α is the functional inverse of Ackermann's function. Our solution improves over the previously best known O(min{m · α(n, n), m + n log n}) time bound, and allows us to close the gap existing with the fastest solution for the edge-removal version of the problem (i.e., that of finding, for every edge e ∈ E T , a replacement edge which can be used for constructing an MST of G − e = (V, E\{e})). Our algorithm finds immediate application in maintaining MST-based communication networks undergoing temporary node failures. Moreover, in a distributed environment in which nodes are managed by selfish agents, it can be used to design an efficient, truthful mechanism for building an MST.

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“…In [CG84,SBvL87], it is proved that for any sequence of edge deletions that do not disconnect the graph, the diameter D of any unweighted graph turns to be less than D( + 1). Our work is also related to the computation of the most vital node of a shortest path [NPW03], that is the node whose removal results in the largest increase of the distance for a given pair of source/target, and the Vickrey pricing of edges [HS01].…”

Section: Related Workmentioning

confidence: 99%

The Impact of Edge Deletions on the Number of Errors in Networks

Glacet

Hanusse

Ilcinkas

2011

Lecture Notes in Computer Science

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Abstract. In this paper, we deal with an error model in distributed networks. For a target t, every node is assumed to give an advice, ie.to point to a neighbour that take closer to the destination. Any node giving a bad advice is called a liar . Starting from a situation without any liar, we study the impact of topology changes on the number of liars. More precisely, we establish a relationship between the number of liars and the number of distance changes after one edge deletion. Whenever deleted edges are chosen uniformly at random, for any graph with n nodes, m edges and diameter D, we prove that the expected number of liars and distance changes is O( 2 Dn m ) in the resulting graph. The result is tight for = 1. For some specific topologies, we give more precise bounds.

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Finding the most vital node of a shortest path

Cited by 87 publications

References 13 publications

Oracles for Distances Avoiding a Failed Node or Link

Oracles for Distances Avoiding a Failed Node or Link

Nearly Linear Time Minimum Spanning Tree Maintenance for Transient Node Failures

The Impact of Edge Deletions on the Number of Errors in Networks

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