Suppose we are given a set of points P = {P1,P2,… ,Pn}, together with a distance function d on the set p; d is simply a function on pairs of points in p with the properties that d(Pi,Pj) = d(P,Pi)> 0 if i ≠ j, and that d(pi,p) = 0 for each i.
We define a hierarchical metric on p to be any distance function tthat can be constructed as follows. We build a rooted tree T with n leaves, and we associate with each node v of T (both leaves and internal nodes) a height hv. These heights must satisfy the properties that h(v) = 0 for each leaf v, and if u is the parent of v in T, then h(u) > h(v). We place each point in p at a distinct leaf in T. Now, for any pair of points pi and pj, their distance t(pi, pj) is defined as follows. We determine the least common ancestor v in T of the leaves containing pt and pj, and define t(pi, pj) = hv.
We say that a hierarchical metric tis consistent with our distance function d if, for all pairs i, j, we have t(pt,pj)
(i) τis consistent with d, and
(ii) if τʹis any other hierarchical metric consistent with d, then t’(pt,<t(pt, pj) for each pair of points pi and pj.
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