Clustering by Complete Enumeration

Abstract

A direct way of solving the cluster problem is to evaluate the objective function for each choice of clustering alternatives and then choose the partition yielding the optimal (minimum) value of the objective function. However, this procedure is not practical and is virtually impossible unless n (the number of objects) and m (the number of clusters) are small. This procedure is termed clustering by complete enumeration. If n = 8 and m = 4, for example, then the number of clustering alternatives is 1701; that is, there are 1701 ways of partitioning 8 objects into 4 subsets. The number of clustering alternatives, denoted by S(n,m), is a Stirling’s number of the second kind and can be computed by means of a formula to be derived in the next section. This chapter is only incidently related to the cluster problem and may be omitted by the reader.