Single-Linkage Clustering: The Algorithm

The algorithm is an agglomerative scheme that erases rows and columns in the proximity matrix as old clusters are merged into new ones.

The N*N proximity matrix is D = [d(i,j)]. The clusterings are assigned sequence numbers 0,1,......, (n-1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is denoted d [(r),(s)].

The algorithm is composed of the following steps:
 

1. Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
   
2. Find the least dissimilar pair of clusters in the current clustering, say pair (r), (s), according to
   
   d[(r),(s)] = min d[(i),(j)]
   
   where the minimum is over all pairs of clusters in the current clustering.
   
3. Increment the sequence number : m = m +1. Merge clusters (r) and (s) into a single cluster to form the next clustering m. Set the level of this clustering to
   
   L(m) = d[(r),(s)]
   
4. Update the proximity matrix, D, by deleting the rows and columns corresponding to clusters (r) and (s) and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted (r,s) and old cluster (k) is defined in this way:
   
   d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)]
   
5. If all objects are in one cluster, stop. Else, go to step 2.