Data Structures and Algorithms with Object-Oriented Design Patterns in Python

Partitions

Consider the finite universal set . A partition of U is a finite set of sets with the following properties:

The sets , , ..., are pairwise disjoint. That is, for all values of i and j such that .
The sets , , ..., span the universe U. That is,

For example, consider the universe . There are exactly five partitions of U:

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In general, given a universe U of size n>0, i.e., |U|=n, there are partitions of U, where is the so-called Stirling number of the second kind which denotes the number of ways to partition a set of n elements into m nonempty disjoint subsets.

Applications which use partitions typically start with an initial partition and refine that partition either by joining or by splitting elements of the partition according to some application-specific criterion. The result of such a computation is the partition obtained when no more elements can be split or joined.

In this chapter we shall consider only applications that begin with the initial partition of U in which each item in U is in a separate element of the partition. Thus, the initial partition consists of |U| sets, each of size one (like above). Furthermore, we restrict the applications in that we only allow elements of a partition to be joined--we do not allow elements to split.

The two operations to be performed on partitions are:

find: Given an item in the universe, say , find the element of the partition that contains i. That is, find such that .
join: Given two distinct elements of a partition P, say and such that , create a new partition P' by removing the two elements and from P and replacing them with a single element .

For example, consider the partition . The result of the operation is the set because 3 is a member of . Furthermore, when we join sets and , we get the partition .