Optimality of Nested Partitions and Its Application to Cluster Analysis

A partition of a set N of n distinct numbers is called nested if there do not exist four numbers a < b < c < d in N such that a and c are in one part while b and d in another. A partition is called a p-partition if the number of parts is specified at p, and a shape-partition if the sizes of the p parts are also specified. There are exponentially many p-partitions but only polynomially many nested p-partitions. In this paper we consider these notions in d-dimensional Euclidean spaces, and give a general condition on the cost structure for which an optimal shape-partition is always nested. We illustrate applications of our results to some clustering problems, generalize some known results in this way, and propose some open problems.