由三重覆盖距离重建全分辨树
Reconstructing fully-resolved trees from triplet cover distances
摘要：
一个经典的结果是：任何具有正权边且没有2度顶点的有限树，唯一地由每对叶子之间的加权路径距离决定。此外，叶对的（小的）严格子集$\cl$可能足以重构树及其边权重，只要给出$\cl$中的叶对之间的距离。已知对于所有内部顶点的度都为3的树，任何具有此性质的集合$\cl$必须形成$T$的{ \em覆盖}--即，对于$T$的每个内部顶点$v$，$\cl$必须包含来自$T-v$的三个分量中的每一对的一对叶。在这里，我们提供了一个部分的逆这个结果表明，如果一组$\cl$的叶对形成了一定类型的覆盖这样一个树$T$然后$T$和它的边缘权重可以唯一地确定从$\cl$的叶对之间的距离。此外，存在用于实现该重建的多项式时间算法。结果建立了一个特殊的情况下，最近的一个问题，关于“三重覆盖”，并有关进化基因组学中出现的问题。
英文摘要：
It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset $\cl$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in $\cl$. It is known that any set $\cl$ with this property for a tree in which all interior vertices have degree 3 must form a {\em cover} for $T$ -- that is, for each interior vertex $v$ of $T$, $\cl$ must contain a pair of leaves from each pair of the three components of $T-v$. Here we provide a partial converse of this result by showing that if a set $\cl$ of leaf pairs forms a cover of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in $\cl$. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning `triplet covers', and is relevant to a problem arising in evolutionary genomics.
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