Refined Buneman Trees

Algorithm 5. Offhand, the algorithm looks like it would perform in time O(n 3 ).
We iterate through a loop, reducing the size of the distance matrix by one
(adding one entry and removing two) on each loop, until the distance matrix is
spent. Inside the loop, we in line 2 need to find the minimum entry in an n × n
matrix, normally a O(n 2 ) operation. However, if we overlay our distance matrix
with a quad tree search structure, we can get away with spending constant
time finding the minimum entry in the matrix, while spending only linear time
updating the search structure after we update the distance matrix in line 6.
More about the quad tree data structure in chapter 8.
Algorithm 5 The single linkage clustering tree algorithm
Require: δ is a distance measure on X
Ensure: C is the single linkage clustering tree for δ
1: C a set of clusters, one for each element in X
2: while |C| > 1 do
3: Choose the clusters c 1 ,c 2 ∈ C that minimize the quantity δ(c 1 ,c 2 ).
4: Create c ′ = c 1 ∪ c 2 .
5: Calculate distances from c ′ to all other clusters c ′′ ∈ C by setting
δ(c ′ ,c ′′ )=min{δ(c 1 ,c ′′ ),δ(c 2 ,c ′′ )}.
6: Erase c 1 and c 2 from C, addc ′ .
7: end while
Also, instead of actually inserting and removing entire rows and columns in
the distance matrix, we might reuse a row or column instead, and we might not
delete rows but rather keep track of clusters that are alive. Figure 7.1 illustrates
this point.
7.3 Converting the distance matrix
One important note about the use of the single linkage clustering algorithm
instead of the anchored Buneman tree algorithm is the distinction between distance
matrix and similarity matrix. The anchored Buneman tree for a species
x ∈ X is computed based on a distance measure δ on a X, while the single
linkage clustering tree for x ∈ X is computed based on an inverted similarity
measure on X with respect to x. In other words we must distinguish between
SLCT(δ, X) andSLCT(F x (δ),X).
Basically, this means that we have to transform δ before using it to calculate
the single linkage clustering tree. The transformation is described in [BB99],
section 3, where is it termed a Farris transformation:
F x (a, b) = 1 2 (δ ax + δ bx − δ ab )
where a, b ∈ X \{x}. The Farris transformation creates a similarity measure,
but this can be converted into a distance measure by e.g. changing signs.
52