Given two numbers n and k where n represents a number of elements in a set, find a 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}}
Recursive Solution
-
Approach: Firstly, let’s define a recursive solution to find the solution for nth element. There are two cases.
- The previous n – 1 elements are divided into k partitions, i.e S(n-1, k) ways. Put this nth element into one of the previous k partitions. So, count = k * S(n-1, k)
- The previous n – 1 elements are divided into k – 1 partitions, i.e S(n-1, k-1) ways. Put the nth element into a new partition ( single element partition).So, count = S(n-1, k-1)
- Total count = k * S(n-1, k) + S(n-1, k-1).
-
Algorithm:
- Create a recursive function which accepts two parameters, n and k. The function returns total number of partitions of n elements into k sets.
- Handle the base cases. If n = 0 or k = 0 or k > n return 0 as there cannot be any subset. If n is equal to k or k is equal to 1 return 1.
- Else calculate the value as follows: S(n, k) = k*S(n-1, k) + S(n-1, k-1), i.e call recursive function with the required parameter and calculate the value of S(n, k).
- Return the sum.
- Implementation:
C++
// 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;
} |
Java
// Java program to count number // of partitions of a set with // n elements into k subsets import java.io.*;
class GFG
{ // Returns count of different
// partitions of n elements in
// k subsets
public static 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
public static void main(String args[])
{
System.out.println(countP( 3 , 2 ));
}
} //This code is contributed by Anshika Goyal. |
Python 3
# A Python3 program to count number # of partitions of a set with n # elements into k subsets # Returns count of different partitions # of n elements in k subsets def countP(n, k):
# Base cases
if (n = = 0 or k = = 0 or k > n):
return 0
if (k = = 1 or 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 Code if __name__ = = "__main__" :
print (countP( 3 , 2 ))
# This code is contributed # by Akanksha Rai(Abby_akku) |
C#
// C# program to count number // of partitions of a set with // n elements into k subsets using System;
class GFG {
// Returns count of different
// partitions of n elements in
// k subsets
public static 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
public static void Main()
{
Console.WriteLine(countP(3, 2));
}
} // This code is contributed by anuj_67. |
Javascript
<script> // Javascript program to count number // of partitions of a set with // n elements into k subsets // Returns count of different
// partitions of n elements in
// k subsets
function countP(n, 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
document.write(countP(3, 2));
// This code is contributed by avanitrachhadiya2155
</script> |
PHP
<?php // A PHP program to count // number of partitions of // a set with n elements // into k subsets // Returns count of different // partitions of n elements // in k subsets function countP( $n , $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 Code
echo countP(3, 2);
// This code is contributed by aj_36 ?> |
- Output:
3
-
Complexity Analysis:
-
Time complexity: O(2^n).
For every value of n, two recursive function is called. More specifically the Time complexity is exponential. -
Space complexity: O(n)(Due to call stack), since n extra space has been taken.
-
Time complexity: O(2^n).
Efficient Solution
- Approach: The time complexity of above recursive solution is exponential. The solution can be optimized by reducing the overlapping subproblems. Below is recursion tree of countP(10,7). The subproblem countP(8,6) or CP(8,6) is called multiple times.
-
So this problem has both properties (see Type 1 and Type 2) 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 shown recursive formula.
Next comes the reduction of the sub-problems to optimize the complexity of the problem. This can be done in two ways:- bottom-up manner: This keeps the recursive structure intact and stores the values in a hashmap or a 2D array. Then compute the value only once and when the function is called next return the value.
- top-down manner: This keeps a 2D array of size n*k, where dp[i][j] represents a total number of partitions of i elements into j sets. Fill in the base cases for dp[i][0] and dp[0][i]. For a value (i,j), the values of dp[i-1][j] and dp[i-1][j-1] is needed. So fill the DP from row 0 to n and column 0 to k.
-
Algorithm:
- Create a Dp array dp[n+1][k+1] of size ( n + 1 )* ( k + 1 ) .
- Fill the values of basic cases. For all values of i from 0 to n fill dp[i][0] = 0 and for all values of i from 0 to k fill dp[0][k] = 0
- Run a nested loop, the outer loop from 1 to n, and inner loop from 1 to k.
- For index i and j (outer loop and inner loop respectively), calculate dp[i][j] = j * dp[i – 1][j] + dp[i – 1][j – 1] and if j == 1 or i == j, calculate dp[i][j] = 1.
- Print values dp[n][k]
- Implementation:
C++
// 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;
} |
Java
// A Dynamic Programming based Java program to count // number of partitions of a set with n elements // into k subsets import java.util.*;
import java.io.*;
class GFG{
// Returns count of different partitions of n // elements in k subsets static int countP( int n, int k)
{ // Table to store results of subproblems
int [][] dp = new int [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 <= k; 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 public static void main(String[] args )
{ System.out.println(countP( 5 , 2 ));
} } // This code is contributed by Rajput-Ji |
Python3
# A Dynamic Programming based Python3 program # to count number of partitions of a set with # n elements into k subsets # Returns count of different partitions # of n elements in k subsets def countP(n, k):
# Table to store results of subproblems
dp = [[ 0 for i in range (k + 1 )]
for j in range (n + 1 )]
# Base cases
for i in range (n + 1 ):
dp[i][ 0 ] = 0
for i in range (k + 1 ):
dp[ 0 ][k] = 0
# Fill rest of the entries in
# dp[][] in bottom up manner
for i in range ( 1 , n + 1 ):
for j in range ( 1 , k + 1 ):
if (j = = 1 or 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 Code if __name__ = = '__main__' :
print (countP( 5 , 2 ))
# This code is contributed by # Surendra_Gangwar |
C#
// A Dynamic Programming based C# program // to count number of partitions of a // set with n elements into k subsets using System;
class GFG
{ // Returns count of different partitions of n // elements in k subsets static int countP( int n, int k)
{ // Table to store results of subproblems
int [,] dp = new int [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 <= k; 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 code public static void Main( )
{ Console.Write(countP(5, 2));
} } // This code is contributed by Ita_c. |
Javascript
<script> // A Dynamic Programming based Javascript program to count // number of partitions of a set with n elements // into k subsets // Returns count of different partitions of n
// elements in k subsets
function countP(n,k)
{
// Table to store results of subproblems
let dp = new Array(n+1);
for (let i = 0; i < n + 1; i++)
{
dp[i] = new Array(k+1);
for (let j = 0; j < k + 1; j++)
{
dp[i][j] = -1;
}
}
// Base cases
for (let i = 0; i <= n; i++)
dp[i][0] = 0;
for (let i = 0; i <= k; i++)
dp[0][k] = 0;
// Fill rest of the entries in dp[][]
// in bottom up manner
for (let i = 1; i <= n; i++)
for (let j = 1; j <= k; 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
document.write(countP(5, 2))
// This code is contributed by rag2127
</script> |
PHP
<?php // A Dynamic Programming based PHP // program to count number of // partitions of a set with n // elements into k subsets // Returns count of different // partitions of n elements in // k subsets function countP( $n , $k )
{ // Table to store results // of subproblems $dp [ $n + 1][ $k + 1] = array ( array ());
// Base cases for ( $i = 0; $i <= $n ; $i ++)
$dp [ $i ][0] = 0;
for ( $i = 0; $i <= $k ; $i ++)
$dp [0][ $k ] = 0;
// Fill rest of the entries in // dp[][] in bottom up manner for ( $i = 1; $i <= $n ; $i ++)
for ( $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 Code echo countP(5, 2);
// This code is contributed by jit_t ?> |
- Output:
15
-
Complexity Analysis:
-
Time complexity:O(n*k).
The 2D dp array of size n*k is filled, so the time Complexity is O(n*k). -
Space complexity:O(n*k).
An extra 2D DP array is required.
-
Time complexity:O(n*k).
Efficient approach : Space optimization
In previous approach the current value dp[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.
Implementation steps:
- Create a 1D vector dp of size n+1.
- Set a base case dp[0] = 1 .
- Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
- Initialize variable prev and temp used to store the previous values from current computations.
- After every iteration assign the value of temp to prev for further iteration.
- At last return and print the final answer stored in dp[n].
Implementation:
C++
//c++ code for above approach #include<bits/stdc++.h> using namespace std;
// Returns count of different partitions of n // elements in k subsets int countP( int n, int k) {
// Vector to store results of subproblems
vector< int > dp(n+1, 0);
// Base cases
dp[0] = 1;
for ( int j = 1; j <= k; j++) {
int prev = dp[0];
for ( int i = 1; i <= n; i++) {
int temp = dp[i];
if (j == 1 || i == j) {
dp[i] = 1;
} else {
dp[i] = j * dp[i - 1] + prev;
}
prev = temp;
}
}
// return finala answer
return dp[n];
} // Driver program int main() {
cout << countP(5, 2);
return 0;
} |
Java
public class Main {
// Function to count different partitions of n elements
// in k subsets
static int countPartitions( int n, int k)
{
// Create an array to store results of subproblems
int [] dp = new int [n + 1 ];
// Base cases
dp[ 0 ] = 1 ;
for ( int j = 1 ; j <= k; j++) {
int prev = dp[ 0 ];
for ( int i = 1 ; i <= n; i++) {
int temp = dp[i];
if (j == 1 || i == j) {
dp[i] = 1 ;
}
else {
dp[i] = j * dp[i - 1 ] + prev;
}
prev = temp;
}
}
// Return the final answer
return dp[n];
}
// Driver program
public static void main(String[] args)
{
int n = 5 ;
int k = 2 ;
System.out.println( "Number of partitions: "
+ countPartitions(n, k));
}
} |
Python3
def countP(n, k):
# List to store results of subproblems
dp = [ 0 ] * (n + 1 )
# Base cases
dp[ 0 ] = 1
for j in range ( 1 , k + 1 ):
prev = dp[ 0 ]
for i in range ( 1 , n + 1 ):
temp = dp[i]
if j = = 1 or i = = j:
dp[i] = 1
else :
dp[i] = j * dp[i - 1 ] + prev
prev = temp
# Return the final answer
return dp[n]
# Driver program if __name__ = = "__main__" :
print (countP( 5 , 2 ))
|
C#
// c# code for above approach using System;
class GFG
{ // Returns count of different partitions of n
// elements in k subsets
static int CountP( int n, int k)
{
// store results of subproblems
int [] dp = new int [n + 1];
// Base cases
dp[0] = 1;
for ( int j = 1; j <= k; j++)
{
int prev = dp[0];
for ( int i = 1; i <= n; i++)
{
int temp = dp[i];
if (j == 1 || i == j)
{
dp[i] = 1;
}
else
{
// Otherwise, calculate the number of partitions using the recurrence relation:
// dp[i] = j * dp[i - 1] + prev
dp[i] = j * dp[i - 1] + prev;
}
prev = temp;
}
}
// The final answer is stored in dp[n]
return dp[n];
}
static void Main()
{
// Test the function with n=5 and k=2
int n = 5;
int k = 2;
int result = CountP(n, k);
Console.WriteLine(result);
}
} |
Javascript
function GFG(n, k) {
// Array to store results of
// subproblems
const dp = new Array(n + 1).fill(0);
// Base cases
dp[0] = 1;
for (let j = 1; j <= k; j++) {
let prev = dp[0];
for (let i = 1; i <= n; i++) {
const temp = dp[i];
if (j === 1 || i === j) {
dp[i] = 1;
} else {
dp[i] = j * dp[i - 1] + prev;
}
prev = temp;
}
}
// Return the final answer
return dp[n];
} // Driver program console.log(GFG(5, 2)); |
Output:
15
Time complexity: O(n*k).
Space complexity: O(n).
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