How Can We Recursively Find All Partitions of a Set?

Finding All Partitions of a Set

In mathematics, a partition of a set is a collection of disjoint subsets (blocks or cells) whose union is the original set. Finding all partitions of a set is a classic combinatorial problem with applications in many areas.

Recursive Solution

A recursive solution can effectively solve this problem. The algorithm starts by generating all possible two-part partitions of the given set. For each two-part partition, the second part is further partitioned into two parts, yielding three-part partitions. This process continues recursively until all partitions are found.

Implementation

Here's a C# implementation of the recursive algorithm:

using System;
using System.Collections.Generic;
using System.Linq;

namespace Partitioning {
    public static class Partitioning {
        public static IEnumerable<T[][]> GetAllPartitions<T>(T[] elements) {
            return GetAllPartitions(new T[][]{}, elements);
        }

        private static IEnumerable<T[][]> GetAllPartitions<T>(
            T[][] fixedParts, T[] suffixElements)
        {
            // A trivial partition consists of the fixed parts
            // followed by all suffix elements as one block
            yield return fixedParts.Concat(new[] { suffixElements }).ToArray();

            // Get all two-group-partitions of the suffix elements
            // and sub-divide them recursively
            var suffixPartitions = GetTuplePartitions(suffixElements);
            foreach (Tuple<T[], T[]> suffixPartition in suffixPartitions) {
                var subPartitions = GetAllPartitions(
                    fixedParts.Concat(new[] { suffixPartition.Item1 }).ToArray(),
                    suffixPartition.Item2);
                foreach (var subPartition in subPartitions) {
                    yield return subPartition;
                }
            }
        }

        private static IEnumerable<Tuple<T[], T[]>> GetTuplePartitions<T>(
            T[] elements)
        {
            // No result if less than 2 elements
            if (elements.Length < 2) yield break;

            // Generate all 2-part partitions
            for (int pattern = 1; pattern < 1 << (elements.Length - 1); pattern++) {
                // Create the two result sets and
                // assign the first element to the first set
                List<T>[] resultSets = {
                    new List<T> { elements[0] }, new List<T>() };
                // Distribute the remaining elements
                for (int index = 1; index < elements.Length; index++) {
                    resultSets[(pattern >> (index - 1)) & 1].Add(elements[index]);
                }

                yield return Tuple.Create(
                    resultSets[0].ToArray(), resultSets[1].ToArray());
            }
        }
    }
}

Explanation

The GetAllPartitions method takes an input set elements and generates all possible partitions. It first calls GetTuplePartitions to generate all two-part partitions of the subset elements. For each two-part partition, it recursively calls GetAllPartitions. This recursive process continues until all partitions are found.

The GetTuplePartitions method generates all possible two-part partitions of a set. It does this by iterating through all possible bit patterns (i.e., binary numbers) that represent the assignment of elements to two partitions.

Example

For the set {1, 2, 3}, the GetAllPartitions method would generate the following partitions:

{ {1}, {2}, {3} }
{ {1, 2}, {3} }
{ {1, 3}, {2} }
{ {1}, {2, 3} }
{ {1, 2, 3} }

This algorithm efficiently generates all partitions of a set, making it a valuable tool in various applications, such as combinatorial optimization and data analysis.

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