A constructive algorithm for realizing a distance matrix

Varone, Sacha

doi:10.1016/j.ejor.2005.02.071

Cited by 13 publications

(10 citation statements)

References 31 publications

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“…In [9] and [19] the authors work with any admissible α satisfying the condition of Theorem 1, but, here, we are interested in considering the maximal value α for which D i (α) is still a distance matrix. Moreover, we want to work for each index i of compaction, with i = 1, .…”

Section: Preliminariesmentioning

confidence: 99%

“…Once the compaction vector a D of D is computed, we consider the following matrix Remark 5. Notice that now we omit the expression "with respect to a D " and we do not refer to a specific index i, since we are considering the compaction process along all the indices and only with respect to the corresponding values of the compaction vector; while in the standard definition of compaction matrix the computation is taken only with respect to an index and a given value α ( [19]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Using and generalizing the definitions of compaction and reduction processes contained in [19] (and basing, especially compaction, on the results in [9]), we establish a recursive method to "contract" a distance matrix D, until we end with a matrix where contractions are no more possible. The analysis of this final matrix by Theorem 15, and Propositions 10 and 13, permits to establish if it is realizable, or not, by an n−cycle or a tree.…”

Section: Introductionmentioning

confidence: 99%

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Realization of distance matrices by graphs of genus 1

Bocci,

Capresi

2020

Preprint

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Given a distance matrix D, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of D by a weighted graph. To this end, we first give a general result on realization by n−cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Realization of distance matrices by graphs of genus 1

Bocci,

Capresi

2020

Preprint

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“…As already mentioned, all these applications have in common the notion of a graph realization of a distance matrix [1]. Over the years several authors studied the characteristics of the distance matrix and its graph and tree realization [23,35,39,52,53,54,56,57].…”

Section: Introductionmentioning

confidence: 99%

Compact mixed integer linear programming models to the minimum weighted tree reconstruction problem

Fortz

Oliveira

Requejo

2017

European Journal of Operational Research

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The Minimum Weighted Tree Reconstruction (MWTR) problem consists of finding a minimum length weighted tree connecting a set of terminal nodes in such a way that the length of the path between each pair of terminal nodes is greater than or equal to a given distance between the considered pair of terminal nodes. This problem has applications in several areas, namely, the inference of phylogenetic trees, the modeling of traffic networks and the analysis of internet infrastructures. In this paper, we investigate the MWTR problem and we present two compact mixed-integer linear programming models to solve the problem. Computational results using two different sets of instances, one from the phylogenetic area and another from the telecommunications area, show that the best of the two models is able to solve instances of the problem having up to 15 terminal nodes.

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“…The actual construction of optimal realizations is a rather difficult problem even for metric spaces with a small number of points (Koolen and Lesser, 2009), (Sturmfels and Yu, 2004). A constructive algorithm was given in (Varone, 2006) which works well in many cases. As a finite metric space is itself a complete weighted graph, the question amounts to minimizing the total length of the ``connecting threads" between the nodes.…”

Section: Introductionmentioning

confidence: 99%

Optimal Embeddings of Finite Metric Spaces Into Graphs

Çelik¹

2015

BTDB

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We consider the embedding of a finite metric space into a weighted graph in such a way that the total weight of the edges is minimal. We discuss metric spaces with = 3,4,5 points in detail and show that the already known classification for these cases can be obtained by simple operations on the associated graph of the given metric space. Keywords: Finite metric spaces, Embeddings, Weighted graphs SONLU METRİK UZAYLARIN ÇİZGELERE OPTİMAL GÖMÜLMESİ ÖZBu çalışmada, sonlu bir metrik uzaydan, ağırlıklı bir çizgeye, çizge kenarlarının toplam ağırlığı en az olacak biçimde gerçekleşen gömme dönüşümlerini gözönüne alıyoruz. Nokta sayıları üç, dört ve beş olan metrik uzaylar üzerinde bu tipten gömme dönüşümlerini detaylı olarak inceliyor ve bu durumlar için bilinen sınıflandırmaların, metrik uzaylara karşılık gelen çizgeler üzerinde tanımlanan bazı basit operasyonlar yardımıyla elde edilebileceğini gösteriyoruz.

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A constructive algorithm for realizing a distance matrix

Cited by 13 publications

References 31 publications

Realization of distance matrices by graphs of genus 1

Realization of distance matrices by graphs of genus 1

Compact mixed integer linear programming models to the minimum weighted tree reconstruction problem

Optimal Embeddings of Finite Metric Spaces Into Graphs

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