The inverse 1-median problem on a tree and on a path

Galavii, Mohammadreza

doi:10.1016/j.endm.2010.05.157

Cited by 36 publications

(14 citation statements)

References 4 publications

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“…The authors derived a combinatorial approach which solves the problem in O(n log n) time for unit cost and under the assumption that the prespecified point that should become a 1-median does not coincide with a given point in the plane. Galavii [9] showed in his Ph.D. thesis that the 1-median on a path with pos/neg weights lies in one of the vertices with positive weights or lies in one of the end points of the path. This property allows the inverse 1-median problem to be solved on a path with negative weights in O(n) time.…”

Section: Introductionmentioning

confidence: 99%

“…They proposed a greedy-like O(n log n) time algorithm for the inverse 1-median problem with vertex weight modifications on tree networks. Hatzl [13] as well as Galavii [9] showed later that this problem can actually be solved in O(n) time. Moreover, Burkard et al [3] proved that the inverse 1-median problem on the plane under Manhattan (or Chebyshev) norm can be solved in O(n log n) time.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

Alizadeh

Burkard

2010

Networks

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Abstract. In an inverse network absolute (or vertex) 1-center location problem the parameters of a given network, like edge lengths or vertex weights, have to be modified at minimum total cost such that a prespecified vertex s becomes an absolute (or a vertex) 1-center of the network. In this article, the inverse absolute and vertex 1-center location problems on unweighted trees with n + 1 vertices are considered where the edge lengths can be changed within certain bounds. For solving these problems a fast method is developed for reducing the height of one tree and increasing the height of a second tree under minimum cost until the heights of both trees become equal. Using this result a combinatorial O(n 2 ) time algorithm is stated for the inverse absolute 1-center location problem in which no topology change occurs. If topology changes are allowed, an O(n 2 r) time algorithm solves the problem where r, r < n, is the compressed depth of the tree network T rooted in s. Finally, the inverse vertex 1-center problem with edge length modifications is solved on T . If all edge lengths remain positive this problem can be solved within the improved O(n 2 ) time complexity by balancing the height of two trees. In the general case one gets the improved O(n 2 r v ) time complexity where the parameter r v is bounded by n.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

Alizadeh

Burkard

2010

Networks

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“…For the inverse 1-median problem on trees, Burkard et al [6] and Galavii [9] modeled the problem as a linear knapsack problem and thus solved this problem in linear time. Burkard et al [7] investigated the inverse 1-median problem on a cycle and solved the problem in O(n 2 ) time by exploring the concavity of linear constraints.…”

Section: Introductionmentioning

confidence: 99%

A model for the inverse 1-median problem on trees under uncertain costs

Nguyen

Linh

2016

OpMath

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Abstract.We consider the problem of justifying vertex weights of a tree under uncertain costs so that a prespecified vertex become optimal and the total cost should be optimal in the uncertainty scenario. We propose a model which delivers the information about the optimal cost which respect to each confidence level α ∈ [0, 1]. To obtain this goal, we first define an uncertain variable with respect to the minimum cost in each confidence level. If all costs are independently linear distributed, we present the inverse distribution function of this uncertain variable in O(n 2 log n) time, where n is the number of vertices in the tree.

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“…The authors derived a combinatorial approach which solves the problem in O(n log n) time for unit cost and under the assumption that the prespecified point that should become a 1-median does not coincide with a given point in the plane. Galavii [11] showed in his Ph.D. thesis that the 1-median on a path with pos/neg weights lies in one of the vertices with positive weights or lies in one of the end points of the path. This property allows to solve the inverse 1-median problem on a path with negative weights in O(n) time.…”

mentioning

confidence: 99%

“…They proposed a greedy-like O(n log n) time algorithm for the inverse 1-median problem with vertex weight modification on tree networks. Hatzl [17] as well as Galavii [11] showed later that this problem can actually be solved in O(n) time. Moreover, Burkard et al [4] proved that the inverse 1-median problem on the plane under Manhattan (or Chebyshev) norm can be solved in O(n log n) time.…”

mentioning

confidence: 99%

Inverse 1-center location problems with edge length augmentation on trees

2009

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This paper considers the inverse 1-center location problem with edge length augmentation on a tree network T with n + 1 vertices. The goal is to increase the edge lengths at minimum total cost subject to given modification bounds such that a prespecified vertex s becomes an absolute 1-center under the new edge lengths. Using a set of suitably extended AVL-search trees we develop a combinatorial algorithm which solves the inverse 1-center location problem with edge length augmentation in O(n log n) time. Moreover, it is shown that the problem can be solved in O(n) time if all the cost coefficients are equal.

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The inverse 1-median problem on a tree and on a path

Cited by 36 publications

References 4 publications

Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

Combinatorial algorithms for inverse absolute and vertex 1‐center location problems on trees

A model for the inverse 1-median problem on trees under uncertain costs

Inverse 1-center location problems with edge length augmentation on trees

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