邻域连接算法的最优性与平衡最小进化多面体的面
Optimality of the Neighbor Joining Algorithm and Faces of the Balanced Minimum Evolution Polytope
摘要：
平衡最小进化（Balanced Minimum Evolution，BME）是一种基于统计一致距离的方法，用于从分子数据的比对中重建系统发育树。在2000年，Pauplin证明了BME方法等价于在BME多面体上优化线性泛函，从Pauplin公式得到的BME向量的凸包适用于所有二叉树。BME方法与邻居连接（NJ）算法有关，现在已知该算法是BME原理的贪婪优化。此外，先前已经研究了NJ和BME算法，以了解NJ算法何时返回少量分类群的BME树。在本文中，我们的目的是阐明结构的BME多面体和加强知识之间的联系BME方法和NJ算法。我们首先证明，任何子树修剪嫁接移动从一个二叉树到另一个二叉树对应的BME多面体的边缘。此外，我们描述了一个完整的家庭的脸参数化不相交的分支。我们表明，这些{ \em进化枝面}是较小的维BME多面体本身。最后，我们证明了，对于任何顺序的连接节点形成一棵树，存在一个相关的距离矩阵（即，相异度图），NJ算法针对该相异度图返回BME树。更强地，我们表明，BME锥和每一个NJ锥关联到一棵树$T$有一个交集的积极措施。
英文摘要：
Balanced minimum evolution (BME) is a statistically consistent distance-based method to reconstruct a phylogenetic tree from an alignment of molecular data. In 2000, Pauplin showed that the BME method is equivalent to optimizing a linear functional over the BME polytope, the convex hull of the BME vectors obtained from Pauplin's formula applied to all binary trees. The BME method is related to the Neighbor Joining (NJ) algorithm, now known to be a greedy optimization of the BME principle. Further, the NJ and BME algorithms have been studied previously to understand when the NJ Algorithm returns a BME tree for small numbers of taxa. In this paper we aim to elucidate the structure of the BME polytope and strengthen knowledge of the connection between the BME method and NJ Algorithm. We first prove that any subtree-prune-regraft move from a binary tree to another binary tree corresponds to an edge of the BME polytope. Moreover, we describe an entire family of faces parametrized by disjoint clades. We show that these {\em clade-faces} are smaller dimensional BME polytopes themselves. Finally, we show that for any order of joining nodes to form a tree, there exists an associated distance matrix (i.e., dissimilarity map) for which the NJ Algorithm returns the BME tree. More strongly, we show that the BME cone and every NJ cone associated to a tree $T$ have an intersection of positive measure.
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