Betweenness and Nonbetweenness

TL;DR

This paper investigates the minimum number of total orderings needed to satisfy certain betweenness and nonbetweenness conditions among objects, providing exact and asymptotic formulas and connecting these to phylogenetic tree structures.

Contribution

It derives exact and asymptotic formulas for betweenness and nonbetweenness functions, and introduces the extreme ternary constraint function as a generalization.

Findings

nbet(n) = ⌈log₂log₂n⌉ + 1

bet(n) = Θ(log n)

Minimum size of rooted phylogenetic trees set is Θ(log log n)

Abstract

The betweenness function $bet(n)$ is the minimum number of total orderings of $n$ objects such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is between $a$ and $c$. The nonbetweenness function $nbet(n)$ is the minimum number of total orderings such that for any three distinct objects $a$, $b$ and $c$, there is an ordering in which $b$ is not between $a$ and $c$. We show that $nbet(n) = \left\lceil \log_2\log_2n \right\rceil+1$ and $bet(n) = \Theta(\log n)$. Betweenness and Nonbetweenness are specific cases of a more general extreme value function called the `extreme ternary constraint function'. The asymptotic value of this generalisation is computed using the values of $nbet(n)$ and $bet(n)$. This result demonstrates that the minimum size of a set of rooted phylogenetic trees is consistent with all phylogenetic triplets is $\Theta(\log\log n)$.