Refined Buneman Trees

Chapter 7
TheSingleLinkage
Clustering Tree
Single linkage clustering trees play an important part in this work as a replacement
for anchored Buneman trees. The refined Buneman tree algorithm
described in this thesis is based on Lemma 5, where anchored Buneman trees
need to be merged with refined Buneman trees to create new refined Buneman
trees. However, since in the first part of the algorithm we are only required
to build overapproximations of the refined Buneman tree, we can use single
linkage clustering trees instead of anchored Buneman trees since single linkage
clustering trees contain anchored Buneman trees.
7.1 Replacing the anchored Buneman tree
The anchored Buneman tree can be computed in time O(n 2 ), according to
Lemma 3, by first constructing a single linkage clustering tree that is a superset
of B x (δ), and then pruning that tree ([BB99], section 3). But, since we are
creating an overapproximation of the refined Buneman tree by incrementally
considering splits from anchored Buneman trees, we can replace the anchored
Buneman trees altogether with the unpruned single linkage clustering trees. The
extra splits might or might not become part of our overapproximation, but they
will be weeded out later on in the algorithm, and they do not asymptotically
change the size of the overapproximation. The tree cannot grow beyond O(n)
splits. And luckily, the single linkage clustering tree is extremely simple to
compute.
7.2 Calculating the single linkage clustering tree
As mentioned, the single linkage clustering tree is very simple to calculate.
Pseudo code for the algorithm, adapted from [BG91], section 3.2.7, is listed in
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