Abstract
In this work we propose the partial Wiener index as one possible measure of branching in phylogenetic evolutionary trees. We establish the connection between the generalized Robinson–Foulds (RF) metric for measuring the similarity of phylogenetic trees and partial Wiener indices by expressing the number of conflicting pairs of edges in the generalized RF metric in terms of partial Wiener indices. To do so we compute the minimum and maximum value of the partial Wiener index \(W\left(T,r, n\right)\), where \(T\) is a binary rooted tree with root \(r\) and \(n\) leaves. Moreover, under the Yule probabilistic model, we show how to compute the expected value of \(W\left(T,r, n\right)\). As a direct consequence, we give exact formulas for the upper bound and the expected number of conflicting pairs. By doing so we provide a better theoretical understanding of the computational complexity of the generalized RF metric.
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Notes
In the analysis of algorithms asymptotic notations, that ignores the constant factors and low order terms in expressions, are commonly used to evaluate the performance of an algorithm. We will use Θ(·) notation in order to simplify expressions, i.e. W(n) = Θ(nlog n) states that W(n) is asymptotically (i.e. for large n) 'behaving' as the nlogn function. For the precise definition of Θ notation we refer an interested reader to (Cormen et al. 2009. chapter 3).
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Authors gratefully acknowledge significant contribution of the anonymous referee whose advices immensely improved this paper.
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Vukičević, D., Matijević, D. The Connection of the Generalized Robinson–Foulds Metric with Partial Wiener Indices. Acta Biotheor 71, 5 (2023). https://doi.org/10.1007/s10441-023-09457-7
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DOI: https://doi.org/10.1007/s10441-023-09457-7