识别树状k-相异点
Recognizing Treelike k-Dissimilarities
摘要：
有限集X上的k-相异度D，|X| >= k，是从X的大小为k的子集的集合到实数的映射。这样的映射自然地产生于具有叶集X的边加权树T：给定X的大小为k的子集Y，D（Y）被定义为具有叶集Y的T的最小子树的总长度。在k = 2的情况下，众所周知，以这种方式产生的2-相异度可以由所谓的“4点条件”表征。然而，在k > 2的情况下，Pachter和Speyer最近提出了以下问题：给定一个任意的k-相异度，我们如何测试这个映射是否来自树？在本文中，我们给出了这个问题的答案，证明了对于k >= 3，集合X上的k-相异度来自树当且仅当它对X的每个2k-元素子集的限制来自某个树，并且2k是确保这种情况的最小可能子集大小。作为一个推论，我们表明，存在一个多项式时间算法，以确定当一个k-相异度从一棵树。我们还给出了一个6点条件，用于确定何时从树中产生3-相异度，这类似于前面提到的4点条件。
英文摘要：
A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.
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