Refined Buneman Trees

Algorithm 6 The algorithm that calculates the single linkage clustering tree
for x
Require: δ is a distance measure on X
Ensure: T is root in the single linkage clustering tree for x ∈ X
1: δ ′ = F ARRIST RANSF ORM(δ, x)
2: Q is a quad tree overlaid δ ′
3: C is a list of clusters corresponding to δ ′
4: while |C| > 1 do
5: (c 1 ,c 2 )=MINIMUM(Q)
6: c ′ = c 1 ∪ c 2
7: for c ′′ ∈ C, c ′′ ≠ c 1 ,c 2 do
8: δ c ′ ′ ,c = ′′ min{δ′ c 1,c ′′,δ′ c 2,c ′′}
9: end for
10: C = C ∪{c ′ }\{c 1 ,c 2 }
11: UPDATE(Q)
12: end while
13: T = C
Thus, the algorithm for computing the single linkage clustering tree for x
becomes the algorithm found in Algorithm 6.
Analysing Algorithm 6 is straightforward: initialising δ ′ and Q takes time
O(n 2 ). We iterate n times in the while loop, performing constant time search
for the minimum entry in the distance matrix, linear time update of the new
row/ column for C ′ with regards to the remaining clusters in the matrix, linear
time removal of C 1 and C 2 , and linear time updating the quad tree data
structure. All in all we have a O(n 2 ) implementation of the single linkage clustering
tree algorithm. Thus, we can safely replace anchored Buneman trees
with single linkage clustering trees in the main algorithm, since is has the same
running time complexity. Also, we have argued that the anchored Buneman
tree is contained in the single linkage clustering tree. And since we are building
overapproximations of the refined Buneman tree, the extra splits from the single
linkage clustering tree will not affect the algorithm adversely.
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