Refined Buneman Trees

and David Bryant ([BB99], section 3) states the complexity of computing the
anchored Buneman tree.
Lemma 2. Let δ be a distance measure on X. ThenB(δ) = ⋂ B x (δ)
Lemma 3. B x (δ) can be computed in time and space O(n 2 ).
2.9 Refined Buneman tree
Given a split σ for a set of size n, letm = |q(σ)| and let q 1 , ··· ,q m be an ordering
of the elements in q(σ) in non-decreasing order of their Buneman scores. Then
the refined Buneman index of a split σ is defined as:
μ σ (δ) = 1
n−3
∑
β qi (2.6)
n − 3
i=1
In other words, the refined Buneman index of a split is the average over
the n − 3 least scoring quartets. The choice of n − 3 is accredited to divine
intervention — apparently, that was the choice that would make the proof in
[MS99] work. The set of splits RB(δ) ={σ : μ σ (δ) > 0} is a compatible set of
splits ([MS99]). And thus the Refined Buneman Tree corresponding to a given
dissimilarity measure δ is defined to be the weighted unrooted tree whose edges
represent the splits σ ∈ RB(δ) and are weighted according to μ σ (δ).
When constructing the refined Buneman tree as described later in this work,
we shall rely heavily on Lemma 4. The lemma is due to [MS99], and it is used
to maintain compatibility in a set of splits overapproximating the set of refined
Buneman splits.
Lemma 4. Given two incompatible splits σ and σ ′ ,
μ σ (δ) ≤ 0 ∨ μ σ ′(δ) ≤ 0
and this can be computed in time O(n).
In the algorithm that computes the refined Buneman tree, we are going to
construct a compatible set of splits which is an overapproximation of the set
of refined Buneman splits. We shall do this for subsets of X of increasing size.
The way to do this is, after bootstrapping some compatible set of splits, we
shall introduce a set of candidates to go into the overapproximation. However,
to maintain compatibility in the set we shall test the candidate splits against
the existing splits, to find pairs that are incompatible.
Once we find a pair of incompatible splits, we shall use Lemma 4 to determine
which one does not belong in the set. We are not concerned with testing if
one of the splits in the pair actually belongs in the set of refined Buneman
splits. We are only interested in throwing away candidates that are clearly
incompatible. We can allow ourselves this luxury since we are only creating
an overapproximation of RB(δ| Xk ), not the set itself, and this saves valueable
x∈X
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