Recognizing Treelike k-Dissimilarities

摘要

A k-dissimilarity D on a finite set X, {pipe}X{pipe} ≥ k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edgeweighted 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 (2004) 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 2 k-element subset of X arises from some tree, and that 2 k 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.