A linear algorithm for the pos/neg-weighted 1-median problem on a cactus

Burkhard, R. E.; Krarup, Jakob

doi:10.1007/bf02684332

Cited by 77 publications

(54 citation statements)

References 6 publications

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“…The second is the subset of N CG-nodes, where an N CG-node is a node not included in any cycle. The remaining nodes, if any, are referred to as H nodes, or hinges [5].…”

Section: Model and Definitionsmentioning

confidence: 99%

“…A maximal subtree is a subtree for which the subset of N CG-and H nodes defining it cannot be extended. A graft is a maximal subtree with no two H nodes belonging to the same cycle [5]. For convenience we will view each graft of a cactus graph as a block.…”

Section: Model and Definitionsmentioning

confidence: 99%

“…A k-cactus graph is a cactus graph with each block containing at most k edges. It is not difficult to see that a cactus graph consists of blocks, where each block is either a cycle or a graft [5], and the blocks are glued by H nodes [5].…”

Section: Model and Definitionsmentioning

confidence: 99%

“…Two nodes of T CG are connected by an edge if and only if one of them represents an H node and the other represents a block containing this H node [2,5,6]. Figure 4 shows an example for constructing a tree from a cactus graph.…”

Section: Model and Definitionsmentioning

confidence: 99%

“…Lan and Wang [28] showed that the median problem in 4-cactus graphs can be solved as efficiently as on trees. Burkard and Krarup [5] presented a linear time algorithm for the median problem in a cactus graph with positive and negative node weights. Recently, Hatzl [21] presented a linear time algorithm for the median problem on wheel graphs and cactus graphs, where a wheel graph is a graph consisting of a cycle of order p − 1 and an additional node that is connected by an edge to each of the cycle nodes.…”

Section: Related Workmentioning

confidence: 99%

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Centdian Computation in Cactus Graphs

Ben‐Moshe¹,

Dvir²,

Segal³

et al. 2012

JGAA

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This paper focuses on the centdian problem in a cactus network where a cactus network is a connected undirected graph, and any two simple cycles in the graph have at most one node in common. The cactus network has important applications for wireless sensor networks when a tree topology might not be applicable and for extensions to the ring architecture. The centdian criterion represents a convex combination of two QoS requirements: transport and delay. To the best of our knowledge, no efficient algorithm has yet been developed for constructing a centdian node in a cactus graph, either sequential or distributed. We first investigate the properties of the centdian node in a cycle graph, and then explore the behavior of the centdian node in a cactus graph. Finally, we present new efficient sequential and distributed algorithms for finding all centdian nodes in a cycle graph and a cactus graph.

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“…The second is the subset of N CG-nodes, where an N CG-node is a node not included in any cycle. The remaining nodes, if any, are referred to as H nodes, or hinges [5].…”

Section: Model and Definitionsmentioning

confidence: 99%

Section: Model and Definitionsmentioning

confidence: 99%

Section: Model and Definitionsmentioning

confidence: 99%

Section: Model and Definitionsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 3 more Smart Citations

Centdian Computation in Cactus Graphs

Ben‐Moshe¹,

Dvir²,

Segal³

et al. 2012

JGAA

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Robust location problems with pos/neg weights on a tree

Burkard

Dollani

2001

Networks

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In this paper, we consider different aspects of robust 1‐median problems on a tree network with uncertain or dynamically changing edge lengths and vertex weights which can also take negative values. The dynamic nature of a parameter is modeled by a linear function of time. A linear algorithm is designed for the absolute dynamic robust 1‐median problem on a tree. The dynamic robust deviation 1‐median problem on a tree with n vertices is solved in O(n2 α(n) log n) time, where α(n) is the inverse Ackermann function. Examples show that both problems do not possess the vertex optimality property. The uncertainty is modeled by given intervals, in which each parameter can take a value randomly. The absolute robust 1‐median problem with interval data, where vertex weights might also be negative, can be solved in linear time. The corresponding deviation problem can be solved in O(n2) time. © 2001 John Wiley & Sons, Inc.

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Minimizing the makespan in multiserver network restoration problems

Averbakh

2017

Networks

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Suppose that a destroyed network needs to be restored by a number of servers (construction crews) that are initially located at some nodes of the network (depots). Each server can restore one unit of length of the network per unit of time. When several servers are simultaneously working at the same point, their restoration speeds combine additively. The servers can travel within the already restored part of the network with infinite speed, which means that travel times are negligible with respect to construction times. It is required to minimize the time when each node becomes connected to at least one of the depots. We show that the problem is strongly NP‐hard on general networks, and present fast polynomial algorithms for trees and cactus networks which are connected networks where each node and each edge belong to at most one cycle. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 60–68 2017

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A linear algorithm for the pos/neg-weighted 1-median problem on a cactus

Cited by 77 publications

References 6 publications

Centdian Computation in Cactus Graphs

Centdian Computation in Cactus Graphs

Robust location problems with pos/neg weights on a tree

Minimizing the makespan in multiserver network restoration problems

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