We consider the problem of fitting an n x n distance matrix D by a tree metric T. Let E be the distance to the closest tree metric under the L, norm, that is, E = minT{II T7D \loo). F' t irs we P resent an O(n2) algorithm for finding an additive tree T such that ]I T, D ljoo 5 3~. Second we show that it is NP-hard to find a tree T such that I] T, D IId0 < ZE. This paper presents the first algorithm for this problem with a performance guarantee. One of the most common methods for clustering numeric data involves fitting the data to a tree mettic, which is defined by a weighted tree spanning the points of the metric, the distance between two points being the sum of the weights of the edges of the path between them. Not surprisingly, this problem, the so-called Numedical Taxonomy problem, has received a great deal of attention (see [2] and [7] for extensive surveys) with work dating as far back as the beginning of the century [l].
The only non-trivial approximation result is the 3-approximation of Agarwala etal REF for the closest tree metric under the ∞ norm.
On the approximability of numerical taxonomy (fitting distances by tree metrics)
SODA '96
Mathematics Computer Science