Implement Hierarchical Clustering solved by 57

Sept. 13, 2015, 10:41 p.m. by Rosalind Team

HierarchicalClustering, whose pseudocode is shown below, progressively generates n different partitions of the underlying data into clusters, all represented by a tree in which each node is labeled by a cluster of genes. The first partition has n single-element clusters represented by the leaves of the tree, with each element forming its own cluster. The second partition merges the two “closest” clusters into a single cluster consisting of two elements. In general, the i-th partition merges the two closest clusters from the (i - 1)-th partition and has n - i + 1 clusters. We hope this algorithm looks familiar — it is UPGMA (from “Implement UPGMA”) in disguise.

HierarchicalClustering(D, n)
 
Clustersn single-element clusters labeled 1, ... , n

 construct a graph T with n isolated nodes labeled by single elements 1, ... , n
while there is more than one cluster 
  find the two closest clusters Ci and Cj
 
  merge Ci and Cj into a new cluster Cnew with |Ci| + |Cj| elements
  add a new node labeled by cluster Cnew to T
  connect node Cnew to Ci and Cj by directed edges

  remove the rows and columns of D corresponding to Ci and Cj
  remove Ci and Cj from Clusters

  add a row/column to D for Cnew by computing D(Cnew, C) for each C in Clusters
  add Cnew to Clusters 
    assign root in T as a node with no incoming edges
    return T

Note that we have not yet defined how HierarchicalClustering computes the distance D(Cnew, C) between a newly formed cluster Cnew and each old cluster C. In practice, clustering algorithms vary in how they compute these distances, with results that can vary greatly. One commonly used approach defines the distance between clusters C1 and C2 as the smallest distance between any pair of elements from these clusters,

Dmin(C1,C2) = minall points i in cluster C1, all points j in cluster C2 Di, j  .

The distance function that we encountered with UPGMA uses the average distance between elements in two clusters,

$$D_\text{avg}(C_1, C_2) = \dfrac{\sum_{\text{all points }i\text{ in cluster }C_1} ~\sum_{\text{all points }j\text{ in cluster }C_2} D_{i,j}}{|C_1| \cdot |C_2|}$$

Implement Hierarchical Clustering

Given: An integer n, followed by an nxn distance matrix.

Return: The result of applying HierarchicalClustering to this distance matrix (using Davg), with each newly created cluster listed on each line.

Sample Dataset

7
0.00 0.74 0.85 0.54 0.83 0.92 0.89
0.74 0.00 1.59 1.35 1.20 1.48 1.55
0.85 1.59 0.00 0.63 1.13 0.69 0.73
0.54 1.35 0.63 0.00 0.66 0.43 0.88
0.83 1.20 1.13 0.66 0.00 0.72 0.55
0.92 1.48 0.69 0.43 0.72 0.00 0.80
0.89 1.55 0.73 0.88 0.55 0.80 0.00

Sample Output

4 6
5 7
3 4 6
1 2
5 7 3 4 6
1 2 5 7 3 4 6

Extra Dataset

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