Refined Buneman Trees

Algorithm 11 A naive algorithm that computes the refined Buneman tree
Require: δ is a distance matrix of size n × n
Ensure: S = RB(δ)
1: S = ∅
2: for σ ∈ σ(X) do
3: Q = ∅
4: for q ∈ q(σ) do
5: Q = Q ∪ q
6: end for
7: SORT(Q)
8: w = 1
n−3
9: if w>0 then
10: S = S ∪ σ
11: end if
12: end for
∑ n−3
i=1 β Q[i](δ)
13.2 Implementation highlights
There are two important issues in this algorithm: we need to avoid considering
duplicate splits, and we need to avoid considering duplicate quartets. Both have
built-in symmetries.
Firstly, let us consider splits as bitvectors. We can start out with the bitvector
containing all zeros, and count through splits by bit-flipping. We flip from
low-order end to high-order end every bit that is ’1’. When we reach a bit that is
’0’, we flip it to ’1’ and stop. Then we have the next split. To avoid duplicates,
we can just restrict ourselves to counting through the first n − 1 bits, leaving
the nth bit as ’0’.
Regarding quartets, we need to recognize that quartets are symmetric on
either side of their central edge, i.e. the quartet ab|cd isthesamequartetas
ba|cd, ab|dc and ba|dc. For the split σ = U|V we say that a, b ∈ U and c, d ∈ V ,
and to avoid duplicates we just have to make sure that a ≤ b and c ≤ d.
13.3 Correctness of the reference implementation
We are going to use this reference implementation of the refined Buneman tree
algorithm to demonstrate the correctness of our implementation of the refined
Buneman tree algorithm described in [BFÖ+ 03]. But first we must convince
ourselves that the reference implementation is correct.
To this end, the author has written a test program for the reference implementation.
The test program generate a random distance matrix, and during
the computation of the reference implementation the program will output:
• the distance matrix.
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