Refined Buneman Trees

Chapter 13
The Reference
Implementation
For testing purposes the author has implemented a package reference, which
in a simple and transparent (and hopelessly inefficient) manner calculates the
refined Buneman tree.
The reference implementation is intended to provide a basis for comparison
against the implementation of the [BFÖ+ 03] algorithm. To justify that the
latter has been implemented correctly, we argue first that our reference implementation
is correct, and thereafter we demonstrate that on the same input,
the two implementations agree completely. These two arguments together form
a basis for concluding that the [BFÖ+ 03] algorithm has been implemented correctly.
13.1 A simple refined Buneman tree algorithm
Algorithm 11 is a very simple algorithm that calculates the refined Buneman
tree. It has been adapted directly from the definition of the refined Buneman
tree.
Line 1 initialises a set S to be the empty set. In line 2, we iterate over all
possible splits in σ(X), avoiding duplicates by interpreting splits as bitvectors,
and counting (using bit-flipping) through exactly half the possible splits. For
each split σ we in line 3 initialise a set of quartets Q to the empty set. In line 4
we iterate over all possible quartets in q(σ) and add them to Q in line 5. In line
7wesortQ in increasing order so that in line 8 we can sum over the n − 3least
scoring quartets to find the refined Buneman index. In lines 9–10 we report the
splits with positive refined Buneman index.
The complexity of this algorithm is quite horrible: first we iterate over O(2 n )
splits, and for each split we iterate over O(n 4 )quartets. Thealgorithm therefore
performs in time O(2 n n 4 ).
74