Extreme rays of the ℓ∞-nearest ultrametric tropical polytope

Yu, Luyan

doi:10.1016/j.laa.2019.10.026

Linear Algebra and its Applications

2020

DOI: 10.1016/j.laa.2019.10.026

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Extreme rays of the ℓ∞-nearest ultrametric tropical polytope

Luyan Yu

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Cited by 3 publications

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Tropical Geometric Variation of Tree Shapes

Lin

Monod

Yoshida

2022

Discrete Comput Geom

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We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree’s branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

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Tropical Geometric Variation of Tree Shapes

Lin

Monod

Yoshida

2022

Discrete Comput Geom

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L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

Bernstein¹

2020

SIAM J. Discrete Math.

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Given a dissimilarity map δ on finite set X, the set of ultrametrics (equidistant tree metrics) which are l ∞ -nearest to δ is a tropical polytope. We give an interior description of this tropical polytope. It was shown by Ardila and Klivans [4] that the set of all ultrametrics on a finite set of size n is the Bergman fan associated to the matroid underlying the complete graph on n vertices. Therefore, we derive our results in the more general context of Bergman fans of matroids. This added generality allows our results to be used on dissimilarity maps where only a subset of the entries are known.

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On matrix semiring over the extended tropical semiring

Roy,

Chakraborty,

Mandal

et al. 2024

Asian-European J. Math.

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In this paper, we first discuss about the basic semiring properties; subsemirings, ideals, [Formula: see text]-ideals, [Formula: see text]-ideals, units, structure preserving properties of both extended tropical semiring [Formula: see text] and semiring of tropical matrices [Formula: see text]. Using the known fact that the subsemimodules of [Formula: see text]-semimodule [Formula: see text] and the ideals of [Formula: see text] are in a canonical correspondence, we study different types of subsemimodules of [Formula: see text]-semimodule [Formula: see text], left (right) ideals, left (right) [Formula: see text]-ideals, left (right) [Formula: see text]-ideals of [Formula: see text]. We prove that both [Formula: see text] and [Formula: see text] are local semirings, Noetherian (Artinian), [Formula: see text]-Noetherian ([Formula: see text]-Artinian), [Formula: see text]-Noetherian ([Formula: see text]-Artinian). Finally, we deduce that [Formula: see text] is neither left nor right Noetherian (Artinian) but left as well as right [Formula: see text]-Noetherian ([Formula: see text]-Artinian).

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Extreme rays of the ℓ∞-nearest ultrametric tropical polytope

Cited by 3 publications

References 14 publications

Tropical Geometric Variation of Tree Shapes

Tropical Geometric Variation of Tree Shapes

L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

On matrix semiring over the extended tropical semiring

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