A High Quartet Distance Construction

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 2 3 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 2 3 + o(1) N 4 , and it always exceeds the 2 3 N 4 bound.

On the quartet distance given partial information

“…As the lower bound of 2 3 n 4 can be easily obtained by the random labeling argument (there are also highly explicit constructions achieving this lower bound [6]), Conjecture 1.1 implies, perhaps surprisingly, that the average distance between two random trees is asymptotically the same as the maximum distance. Alon, Snir, and Yuster [2] improved the upper bound on the maximum quartet distance proving it is asymptotically smaller than 9 10 n 4 .…”

On the quartet distance given partial information

“…As the lower bound of can be easily obtained by the random labeling argument (there are also highly explicit constructions achieving this lower bound [6]), Conjecture 1.1 implies, perhaps surprisingly, that the average distance between two random trees is asymptotically the same as the maximum distance. Alon, Snir, and Yuster [2] improved the upper bound on the maximum quartet distance proving it is asymptotically smaller than .…”