Length-minimizing LED Trees
Mariana Sarkociov\'a Reme\v{s}\'ikov\'a, Peter Sarkoci, M\'aria, Trnovsk\'a

TL;DR
This paper introduces length-minimizing LED trees, a special Euclidean tree structure relevant to phylogeny, and proves their uniqueness along with geometric properties, with an application in historical linguistics.
Contribution
It provides a uniqueness proof for length-minimizing LED trees and explores their geometric and topological properties, extending Steiner tree concepts.
Findings
Proved the uniqueness of length-minimizing LED trees.
Explored geometric and topological properties of the feasible set.
Demonstrated an application in historical linguistics.
Abstract
In this paper, we introduce a specific type of Euclidean tree called LED (Leaves of Equal Depth) tree. LED trees can be used in computational phylogeny, since they are a natural representative of the time evolution of a set of species in a feature space. This work is focused on LED trees that are length minimizers for a given set of leaves (species) and a given isomorphism type (the hierarchical structure of ancestors). The underlying minimization problem can be seen as a variant of the classical Euclidean Steiner tree problem. Even though it has a convex objective function, it is rather non-trivial, since it has a non-convex feasible set. The main contribution of this paper is that we provide a uniqueness result for this problem. Moreover, we explore some geometrical and topological properties of the feasible set and we prove several geometrical characteristics of the length minimizers…
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Taxonomy
TopicsAdvanced Graph Theory Research
