高四重奏距离结构
A High Quartets Distance Construction
摘要：
给定$N$标记叶子上的两个二叉树，树之间的四元组距离是不一致的四元组的数量。通过随机排列叶子，两棵树之间的预期四元组距离为$\frac{2}{3}\binom{N}{4}$。然而，没有强明确的建设达到这个界限渐近已知。我们考虑完整的，平衡的二叉树$N=2^n$叶，标记为$n$长位序列。在一棵树的前缀顺序，并在另一棵树的后缀顺序的叶子，我们表明，由此产生的四重距离是$\left(\frac{2}{3} + o(1)\right)\binom{N}{4}$，它总是超过了$\frac{2}{3}\binom{N}{4}$界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
