Refined Buneman Trees

the pseudo-code for the algorithm is given in Algorithm 4.
The algorithm is very long and complicated, but it can be broken up into
three parts: lines 1–3 deal with initializing linked lists representing “global
matrix fronts” on matrices which contain diagonal quartets. Lines 4–17 populate
the matrix fronts. In lines 18–28 we search the matrices, finding the minimum
quartets that we need to calculate the refined Buneman indices for the edges
represented by the matrices. And in lines 29–34 we report refined Buneman
splits. Before we explain the algorithm, we need to study the nature of diagonal
quartets.
5.2.1 Searching for minimum diagonals
We know from previous sections that a quartet induces two diagonal quartets.
And clearly, given a diagonal quartet we can identify its “parent” quartet. In
the pruning part of our algorithm, we are interesting in finding the n−3quartets
with minimum score induced by each edge in T . We will do this by searching
for diagonal quartets instead, but we need to ensure that we never identify the
same quartet twice (for example by identifying it from seeing both of its diagonal
quartets).
We will use a convention that says we shall only identify a quartet if we
see its minimum diagonal. In case we see a diagonal quartet which is not a
minimum diagonal, we shall disregard it.
Another important property of diagonal quartets is this: if we fix a and c,
such that a and c lie on different sides of a fixed edge e, we can search for b and
d independently to minimize the score of the diagonal quartet ab||cd induced by
e. Byfixinga and c we can rewrite the score of a diagonal quartet into a sum
of two functions f a,c and g a,c , such that f a,c only depends on b and g a,c only
depends on d. Clearly, such a function takes its minimum only when f a,c and
g a,c are minimal.
η ab||cd = 1 2 (δ bc − δ ab + δ ad − δ dc )=f a,c (b)+g a,c (d)
where f a,c (b) =(δ bc − δ ab )/2 andg a,c (d) =(δ ad − δ dc )/2.
Not only can we find the diagonal quartet with minimum score in this way.
But also, we can search for the “next minimum”, i.e. the diagonal quartet with
minimum score when discounting the actual minimum. We can do this in a
general way: say we have some diagonal quartet ab i ||cd j with score η abi||cd j
.
Imagine we have considered all diagonal quartets with scores less than η abi||cd j
,
and now we wish to consider ab i ||cd j and then find the next diagonal quartet
with minimum score.
The way to do this is to search for b i+1 such that η abi+1||cd j
≥ η abi||cd j
is the
minimum among all choices of b i+1 , and similarly for d j+1 , η abi||cd j+1
≥ η abi||cd j
must be the smallest among choices of d j+1 . One of those will be the next
minimum. Note that the indices refer to an ordering of increasing f a,c and g a,c
respectively, not ordering as members of X.
36