Count number of ways to partition a set into k subsets

3.3

Given two numbers n and k where n represents number of elements in a set, find number of ways to partition the set into k subsets.

Example:

Input: n = 3, k = 2
Output: 3
Explanation: Let the set be {1, 2, 3}, we can partition
             it into 2 subsets in following ways
             {{1,2}, {3}},  {{1}, {2,3}},  {{1,3}, {2}}  

Input: n = 3, k = 1
Output: 1
Explanation: There is only one way {{1, 2, 3}}

We strongly recommend you to minimize your browser and try this yourself first.

Let S(n, k) be total number of partitions of n elements into k sets. Value of S(n, k) can be defined recursively as,

S(n, k) = k*S(n-1, k) + S(n-1, k-1) 

S(n, k) is called Stirling numbers of the second kind


How does above recursive formula work?

When we add a (n+1)’th element to k partitions, there are two possibilities.
1) It is added as a single element set to existing partitions, i.e, S(n, k-1)
2) It is added to all sets of every partition, i.e., k*S(n, k)

Therefore S(n+1, k) = k*S(n, k) + S(n, k-1) which means S(n, k) = k*S(n-1, k) + S(n-1, k-1)

Below is recursive solution based on above formula.

// A C++ program to count number of partitions
// of a set with n elements into k subsets
#include<iostream>
using namespace std;

// Returns count of different partitions of n
// elements in k subsets
int countP(int n, int k)
{
  // Base cases
  if (n == 0 || k == 0 || k > n)
     return 0;
  if (k == 1 || k == n)
      return 1;

  // S(n+1, k) = k*S(n, k) + S(n, k-1)
  return  k*countP(n-1, k) + countP(n-1, k-1);
}

// Driver program
int main()
{
   cout <<  countP(3, 2);
   return 0;
}

Output:

3

The time complexity of above recursive solution is exponential. The solution can be optimized as there are overlapping subproblems. For example, below is recursion tree of countP(10,7). The subproblem countP(8,6) or CP(8,6) is called multiple times.

 CP() represents countP()

             CP(10,7)
           /        \
       CP(9,7)       CP(9,6) 
       /    \       /    \
  CP(8,7) CP(8,6) CP(8,6)  CP(8,5)
    / \    / \     / \       / \

  Partial Recursion Tree for countP(10, 7) 
  to highlight overlapping subproblems.

So this problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array dp[][] in bottom up manner using the above recursive formula.

Below is C++ implementation of Dynamic Programming Solution.

// A Dynamic Programming based C++ program to count
// number of partitions of a set with n elements
// into k subsets
#include<iostream>
using namespace std;

// Returns count of different partitions of n
// elements in k subsets
int countP(int n, int k)
{
  // Table to store results of subproblems
  int dp[n+1][k+1];

  // Base cases
  for (int i=0; i<=n; i++)
     dp[i][0] = 0;
  for (int i=0; i<=k; i++)
     dp[0][k] = 0;

  // Fill rest of the entries in dp[][]
  // in bottom up manner
  for (int i=1; i<=n; i++)
     for (int j=1; j<=i; j++)
       if (j == 1 || i == j)
          dp[i][j] = 1;
       else
          dp[i][j] = j*dp[i-1][j] + dp[i-1][j-1];

  return dp[n][k];
}

// Driver program
int main()
{
   cout <<  countP(5, 2);
   return 0;
}

Output:

15

Time Complexity: O(n x k)
Auxiliary Space: O(n x k)

Similar Article: Bell Numbers

This article is contributed by Rajeev Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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