关于邻接算法的最优性
On the optimality of the neighbor-joining algorithm
摘要：
遗传学中常用的邻接（NJ）算法是一种贪婪算法，用于寻找与相异性映射相关的平衡最小进化（BME）树。从这个角度来看，NJ是“最优的”时，算法输出的树，最小化平衡的最小进化准则。我们利用NJ树拓扑和BME树拓扑是由相异映射空间${\R}_{+}^{n \choose 2}$的多面体剖分决定的事实来研究邻接算法的最优性。特别是，我们调查和比较的多面体细分$n \leq 8$。一个关键的要求是在高维的球形多面体的体积的测量，我们使用蒙特卡罗方法和多面体算法的组合获得。我们表明，高度不相关的树可以在BME重建的协同优化，NJ地区是不凸的。我们得到的$l_2$半径的邻居加入$n=5$，我们推测，邻居加入算法恢复BME树的能力取决于直径的BME树。
英文摘要：
The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map. From this point of view, NJ is ``optimal'' when the algorithm outputs the tree which minimizes the balanced minimum evolution criterion. We use the fact that the NJ tree topology and the BME tree topology are determined by polyhedral subdivisions of the spaces of dissimilarity maps ${\R}_{+}^{n \choose 2}$ to study the optimality of the neighbor-joining algorithm. In particular, we investigate and compare the polyhedral subdivisions for $n \leq 8$. A key requirement is the measurement of volumes of spherical polytopes in high dimension, which we obtain using a combination of Monte Carlo methods and polyhedral algorithms. We show that highly unrelated trees can be co-optimal in BME reconstruction, and that NJ regions are not convex. We obtain the $l_2$ radius for neighbor-joining for $n=5$ and we conjecture that the ability of the neighbor-joining algorithm to recover the BME tree depends on the diameter of the BME tree.
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