Efficient Parallel Algorithms for Optimally Locating a Path and a Tree of a Specified Length in a Weighted Tree Network

Wang, Biing-Feng

doi:10.1006/jagm.1999.1020

Cited by 26 publications

(15 citation statements)

References 17 publications

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“…(This uniformity assumption simplifies the combinatorial complexity of the model significantly. For example, the unweighted path center with a length constraint is solved in O(n) time in [35], while the best algorithm known for the weighted version has O(n log n) complexity [33,Section 3.2].) Studying only this unweighted model without a length constraint on the selected path, Averbakh and Berman [3] proved that there are at most n − 1 nondominated outcomes, and gave an O(n log n) algorithm to generate all of them.…”

Section: Introductionmentioning

confidence: 99%

The bi‐criteria doubly weighted center‐median path problem on a tree

Puerto

Rodríguez‐Chía

Tamir³

et al. 2006

Networks

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Given a tree network T with n nodes, let PL be the subset of all discrete paths whose length is bounded above by a prespeciﬁed value L. We consider the location of a path-shaped facility P ∈ PL, where customers are represented by the nodes of the tree. We use a bi-criteria model to represent the total transportation cost of the customers to the facility. Each node is associated with a pair of nonnegative weights: the center-weightand the median-weight. In this doubly weighted model, a path P is assigned a pair of values (MAX (P ), SUM (P )),which are, respectively, the maximum center-weighted distance and the sum of the median-weighted distances from P to the nodes of the tree. Viewing PL and the planar set {(MAX (P), SUM (P )) : P ∈ PL} as the decisionspace and the bi-criteria or outcome space respectively,we focus on ﬁnding all the nondominated points of the bi-criteria space. We prove that there are at most 2n non-dominated outcomes, even though the total number of efﬁcient paths can be Ω(n2), and they can all be generated in O(n log n) optimal time. We apply this result to solve the cent-dian model, whose objective is a convex combination of the weighted center and weighted median functions. We also solve the restricted models, where the goal is to minimize one of the two functions MAX or SUM, subject to an upper bound on the other one, both withand without a constraint on the length of the path. All these problems are solved in linear time, once the set of non dominated outcomes has been obtained, which in turn, results in an overall complexity of O(n log n). The latter bounds improve upon the best known results by a factor of O(log n).Ministerio de Ciencia y Tecnologí

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Section: Introductionmentioning

confidence: 99%

The bi‐criteria doubly weighted center‐median path problem on a tree

Puerto

Rodríguez‐Chía

Tamir³

et al. 2006

Networks

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“…13 16: end if 17: end if 18: until the size of T is no more than 270. 19: Solve the problem on T with a constant size. end…”

Section: Algorithm 1 Continuoustree-center(t = (V E) S L)mentioning

confidence: 99%

“…All the results mentioned above [8,12,16,19] considered the tree network to be edge-weighted but not vertex-weighted. In recent papers, Tamir et al [17,18] proposed O(n log n)-time algorithms to solve the conditional problems in tree networks where the vertices and the edges are weighted.…”

Section: Introductionmentioning

confidence: 98%

“…However, in recent years there has been growing interest in studying the location of connected structures (referred to as extensive facilities) [7,8,12,[15][16][17][18][19][20]. These studies were motivated by concrete decision problems related to routing and network design [13,18].…”

Section: Introductionmentioning

confidence: 99%

“…In the case when there are no existing facilities located in the networks, Hedetmiemi et al [8] proposed optimal algorithms for locating a path-shaped facility without length constraint that take advantage of a canonical recursive representation of a tree. Wang [19] proposed an optimal parallel algorithm for locating a path-shaped facility of a specified length in a weighted tree network. For locating a tree-shaped facility, Shioura and Shigeno [16] showed that these problems in tree networks are related to the bottleneck knapsack problems, and presented linear-time algorithms by using this relationship.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Algorithms for the Path/Tree-Shaped Facility Location Problems in Trees

2007

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Extensive facility location models in networks deal with the location of special types of subgraphs such as paths or trees and can be considered as extensions of classical facility location models. In this paper we consider the problem of locating a path-shaped or tree-shaped (extensive) facility in trees, under the condition that existing facilities are already located. We introduce a parametric-pruning method to solve the conditional discrete/continuous extensive weighted 1-center location problems in trees in linear time. This improves the recent results of O(n log n) by Tamir et al. (J. Algebra 56:50-75, 2005).

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Minimax regret path location on trees

Puerto

Ricca

Scozzari³

2011

Networks

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This work studies the problem of finding optimal paths with respect to the center, median and centdian objective functions, on networks with uncertain vertex weights that are given as intervals. Our approach looks for minimax regret paths which minimize the worst-case opportunity loss in the corresponding objective function. These problems are NP-hard on general graphs, therefore we study them on trees. We show that a discrete optimal path always exists for each of them, and provide polynomial time solution algorithms.

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Efficient Parallel Algorithms for Optimally Locating a Path and a Tree of a Specified Length in a Weighted Tree Network

Cited by 26 publications

References 17 publications

The bi‐criteria doubly weighted center‐median path problem on a tree

The bi‐criteria doubly weighted center‐median path problem on a tree

Optimal Algorithms for the Path/Tree-Shaped Facility Location Problems in Trees

Minimax regret path location on trees

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