On NP-Hardness in Hierarchical Clustering

Křivánek, Mirko; Morávek, Jaroslav

doi:10.1007/978-3-642-51883-6_26

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…, n} find an equidistant tree T on n leaves such that 1≤i<j≤n (d(i, j) − x(i, j)) 2 is minimized. It is known that this problem (as well as the unrooted nonequidistant version) is NP complete [13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Least Squares Methods for Equidistant Tree Reconstruction

Fahey,

Hosten,

Krieger

et al. 2008

Preprint

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UPGMA is a heuristic method identifying the least squares equidistant phylogenetic tree given empirical distance data among n taxa. We study this classic algorithm using the geometry of the space of all equidistant trees with n leaves, also known as the Bergman complex of the graphical matroid for the complete graph K n . We show that UPGMA performs an orthogonal projection of the data onto a maximal cell of the Bergman complex. We also show that the equidistant tree with the least (Euclidean) distance from the data is obtained from such an orthogonal projection, but not necessarily given by UPGMA. Using this geometric information we give an extension of the UPGMA algorithm. We also present a branch and bound method for finding the best equidistant tree. Finally, we prove that there are distance data among n taxa which project to at least (n − 1)! equidistant trees.

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Section: Introductionmentioning

confidence: 99%

Least Squares Methods for Equidistant Tree Reconstruction

Fahey,

Hosten,

Krieger

et al. 2008

Preprint

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“…If the node is partially captured by the columnar front nЈ ϭ n ϩ I ⌬t [5] (i.e., there is an intersection between the front and the cell nRЈ ϭ nR ϩ nЈ⌬R [6] associated with the node), the increment of the equiaxed volume given by the equiaxed model is multiplied by the nR 2 Ј ϭ nR 2 ϩ 2 nRЈ⌬R [7] fraction captured:…”

Section: Introductionmentioning

confidence: 99%

“…The iron carbon phase diagram illusModels describing the solidification of white and gray iron trated in Figure 1 and the kinetics diagram of Figure 2 have been proposed by several authors. [2][3][4][5][6][7][8] The approach illustrate the difference in thermodynamic equilibrium and generally taken [2,[5][6][7][8] consists of coupling a finite-difference growth kinetics of the two eutectic structures.…”

Section: Introductionmentioning

confidence: 99%

A two-dimensional model for the description of the columnar-to-equiaxed transition in competing gray and white iron eutectics and its application to calender rolls

Jacot¹,

Maijer

Cockcroft

2000

Metall Mater Trans A

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A two-dimensional (2-D) microstructural model has been developed for the description of the solidification of gray and white iron eutectics at the scale of a casting process. The model deals with the competition that can occur between the white and gray iron microstructures and between the columnar and equiaxed morphologies. The evolution of the volume fraction of gray and white iron and the respective proportions of columnar and equiaxed morphologies are calculated at each node of a finiteelement (FE) mesh of the casting. The model is coupled with the commercial FE software ABAQUS, which provides the solution of the heat-flow problem. Columnar solidification is described with a new front tracking algorithm that allows the undercooling at the interface and the non-steady-state stage of growth to be calculated accurately. Equiaxed solidification is described with a deterministic model based on nucleation and growth laws. The model is first applied to a casting of a simple geometry in order to investigate possible numerical problems associated with the front tracking algorithm. The model is then applied to the solidification of a reduced-scale cast-iron calender roll. The results of the simulation are compared with measurements performed in an instrumented casting of 0.4 m in diameter and 1 m in length. A comparison between calculated and measured volume fractions of white and gray iron is presented as a function of the radial position in the casting. It shows that the model is able to predict properly the transition from white to gray iron, which occurs at approximately 20 mm from the external surface of the roll. Comparisons between calculated and measured temperature evolutions and equiaxed grain densities of equiaxed grains are also presented and show satisfactory agreement.

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Computational complexity of inferring phylogenies from dissimilarity matrices

Day

1987

Bltn Mathcal Biology

129

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On NP-Hardness in Hierarchical Clustering

Cited by 4 publications

References 2 publications

Least Squares Methods for Equidistant Tree Reconstruction

Least Squares Methods for Equidistant Tree Reconstruction

A two-dimensional model for the description of the columnar-to-equiaxed transition in competing gray and white iron eutectics and its application to calender rolls

Computational complexity of inferring phylogenies from dissimilarity matrices

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