A High Quartets Distance Construction

TL;DR

This paper presents a novel explicit construction of two binary trees with leaves labeled by bit sequences, achieving a quartet distance close to the theoretical maximum of two-thirds of all quartets, surpassing previous bounds.

Contribution

The authors introduce a new explicit method for constructing binary trees with leaves labeled by bit sequences that asymptotically reach the maximum quartet distance.

Findings

Quartet distance approaches (2/3) * binomial(N, 4)

Construction always exceeds the 2/3 bound

Explicit construction matches asymptotic optimality

Abstract

Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is $\frac{2}{3}\binom{N}{4}$. However, no strongly explicit construction reaching this bound asymptotically was known. We consider complete, balanced binary trees on $N=2^n$ leaves, labeled by $n$ long bit sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is $\left(\frac{2}{3} + o(1)\right)\binom{N}{4}$, and it always exceeds the $\frac{2}{3}\binom{N}{4}$ bound.