Dynamic Programming - Subset Sum Problem

Subset Sum Problem | DP-25

Difficulty Level : Medium
Last Updated : 09 Nov, 2020

Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.

Example:

Input: set[] = {3, 34, 4, 12, 5, 2}, sum = 9
Output: True  
There is a subset (4, 5) with sum 9.

Input: set[] = {3, 34, 4, 12, 5, 2}, sum = 30
Output: False
There is no subset that add up to 30.

Method 1: Recursion.
Approach: For the recursive approach we will consider two cases.

Consider the last element and now the required sum = target sum – value of ‘last’ element and number of elements = total elements – 1
Leave the ‘last’ element and now the required sum = target sum and number of elements = total elements – 1

Following is the recursive formula for isSubsetSum() problem.

isSubsetSum(set, n, sum) 
= isSubsetSum(set, n-1, sum) || 
  isSubsetSum(set, n-1, sum-set[n-1])
Base Cases:
isSubsetSum(set, n, sum) = false, if sum > 0 and n == 0
isSubsetSum(set, n, sum) = true, if sum == 0

Let’s take a look at the simulation of above approach-:

set[]={3, 4, 5, 2}
sum=9
(x, y)= 'x' is the left number of elements,
'y' is the required sum
  
              (4, 9)
             {True}
           /        \  
        (3, 6)       (3, 9)
               
        /    \        /   \ 
     (2, 2)  (2, 6)   (2, 5)  (2, 9)
     {True}  
     /   \ 
  (1, -3) (1, 2)  
{False}  {True} 
         /    \
       (0, 0)  (0, 2)
       {True} {False}

C++

// A recursive solution for subset sum problem
#include <stdio.h>
 
// Returns true if there is a subset
// of set[] with sum equal to given sum
bool isSubsetSum(int set[], int n, int sum)
{
    // Base Cases
    if (sum == 0)
        return true;
    if (n == 0)
        return false;
 
    // If last element is greater than sum,
    // then ignore it
    if (set[n - 1] > sum)
        return isSubsetSum(set, n - 1, sum);
 
    /* else, check if sum can be obtained by any 
of the following:
      (a) including the last element
      (b) excluding the last element   */
    return isSubsetSum(set, n - 1, sum)
           || isSubsetSum(set, n - 1, sum - set[n - 1]);
}
 
// Driver code
int main()
{
    int set[] = { 3, 34, 4, 12, 5, 2 };
    int sum = 9;
    int n = sizeof(set) / sizeof(set[0]);
    if (isSubsetSum(set, n, sum) == true)
        printf("Found a subset with given sum");
    else
        printf("No subset with given sum");
    return 0;
}

Java

// A recursive solution for subset sum
// problem
class GFG {
 
    // Returns true if there is a subset
    // of set[] with sum equal to given sum
    static boolean isSubsetSum(int set[],
                               int n, int sum)
    {
        // Base Cases
        if (sum == 0)
            return true;
        if (n == 0)
            return false;
 
        // If last element is greater than
        // sum, then ignore it
        if (set[n - 1] > sum)
            return isSubsetSum(set, n - 1, sum);
 
        /* else, check if sum can be obtained 
        by any of the following
            (a) including the last element
            (b) excluding the last element */
        return isSubsetSum(set, n - 1, sum)
            || isSubsetSum(set, n - 1, sum - set[n - 1]);
    }
 
    /* Driver code */
    public static void main(String args[])
    {
        int set[] = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.length;
        if (isSubsetSum(set, n, sum) == true)
            System.out.println("Found a subset"
                               + " with given sum");
        else
            System.out.println("No subset with"
                               + " given sum");
    }
}
 
/* This code is contributed by Rajat Mishra */

Python3

# A recursive solution for subset sum
# problem
 
# Returns true if there is a subset
# of set[] with sun equal to given sum
 
 
def isSubsetSum(set, n, sum):
 
    # Base Cases
    if (sum == 0):
        return True
    if (n == 0):
        return False
 
    # If last element is greater than
    # sum, then ignore it
    if (set[n - 1] > sum):
        return isSubsetSum(set, n - 1, sum)
 
    # else, check if sum can be obtained
    # by any of the following
    # (a) including the last element
    # (b) excluding the last element
    return isSubsetSum(
        set, n-1, sum) or isSubsetSum(
        set, n-1, sum-set[n-1])
 
 
# Driver code
set = [3, 34, 4, 12, 5, 2]
sum = 9
n = len(set)
if (isSubsetSum(set, n, sum) == True):
    print("Found a subset with given sum")
else:
    print("No subset with given sum")
 
# This code is contributed by Nikita Tiwari.

C#

// A recursive solution for subset sum problem
using System;
 
class GFG {
    // Returns true if there is a subset of set[] with sum
    // equal to given sum
    static bool isSubsetSum(int[] set, int n, int sum)
    {
        // Base Cases
        if (sum == 0)
            return true;
        if (n == 0)
            return false;
 
        // If last element is greater than sum,
        // then ignore it
        if (set[n - 1] > sum)
            return isSubsetSum(set, n - 1, sum);
 
        /* else, check if sum can be obtained 
        by any of the following
        (a) including the last element
        (b) excluding the last element */
        return isSubsetSum(set, n - 1, sum) 
          || isSubsetSum(set, n - 1, sum - set[n - 1]);
    }
 
    // Driver code
    public static void Main()
    {
        int[] set = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.Length;
        if (isSubsetSum(set, n, sum) == true)
            Console.WriteLine("Found a subset with given sum");
        else
            Console.WriteLine("No subset with given sum");
    }
}
 
// This code is contributed by Sam007

PHP

<?php
// A recursive solution for subset sum problem
 
// Returns true if there is a subset of set
// with sun equal to given sum
function isSubsetSum($set, $n, $sum)
{
    // Base Cases
    if ($sum == 0)
        return true;
    if ($n == 0)
        return false;
     
    // If last element is greater
    // than sum, then ignore it
    if ($set[$n - 1] > $sum)
        return isSubsetSum($set, $n - 1, $sum);
     
    /* else, check if sum can be 
       obtained by any of the following
        (a) including the last element
        (b) excluding the last element */
    return isSubsetSum($set, $n - 1, $sum) || 
        isSubsetSum($set, $n - 1, 
                    $sum - $set[$n - 1]);
}
 
// Driver Code
$set = array(3, 34, 4, 12, 5, 2);
$sum = 9;
$n = 6;
 
if (isSubsetSum($set, $n, $sum) == true)
    echo"Found a subset with given sum";
else
    echo "No subset with given sum";
     
// This code is contributed by Anuj_67 
?>

Output

Found a subset with given sum

Complexity Analysis: The above solution may try all subsets of given set in worst case. Therefore time complexity of the above solution is exponential. The problem is in-fact NP-Complete (There is no known polynomial time solution for this problem).

Method 2: To solve the problem in Pseudo-polynomial time use the Dynamic programming.
So we will create a 2D array of size (arr.size() + 1) * (target + 1) of type boolean. The state DP[i][j] will be true if there exists a subset of elements from A[0….i] with sum value = ‘j’. The approach for the problem is:

if (A[i] > j)
DP[i][j] = DP[i-1][j]
else 
DP[i][j] = DP[i-1][j] OR DP[i-1][j-A[i]]

This means that if current element has value greater than ‘current sum value’ we will copy the answer for previous cases
And if the current sum value is greater than the ‘ith’ element we will see if any of previous states have already experienced the sum=’j’ OR any previous states experienced a value ‘j – A[i]’ which will solve our purpose.

The below simulation will clarify the above approach:

set[]={3, 4, 5, 2}
target=6
 
    0    1    2    3    4    5    6

0   T    F    F    F    F    F    F

3   T    F    F    T    F    F    F
     
4   T    F    F    T    T    F    F   
      
5   T    F    F    T    T    T    F

2   T    F    T    T    T    T    T

Below is the implementation of the above approach:

C++

// A Dynamic Programming solution
// for subset sum problem
#include <stdio.h>
 
// Returns true if there is a subset of set[]
// with sun equal to given sum
bool isSubsetSum(int set[], int n, int sum)
{
    // The value of subset[i][j] will be true if
    // there is a subset of set[0..j-1] with sum
    // equal to i
    bool subset[n + 1][sum + 1];
 
    // If sum is 0, then answer is true
    for (int i = 0; i <= n; i++)
        subset[i][0] = true;
 
    // If sum is not 0 and set is empty,
    // then answer is false
    for (int i = 1; i <= sum; i++)
        subset[0][i] = false;
 
    // Fill the subset table in botton up manner
    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= sum; j++) {
            if (j < set[i - 1])
                subset[i][j] = subset[i - 1][j];
            if (j >= set[i - 1])
                subset[i][j] = subset[i - 1][j]
                               || subset[i - 1][j - set[i - 1]];
        }
    }
 
    /*   // uncomment this code to print table
     for (int i = 0; i <= n; i++)
     {
       for (int j = 0; j <= sum; j++)
          printf ("%4d", subset[i][j]);
       printf("\n");
     }*/
 
    return subset[n][sum];
}
 
// Driver code
int main()
{
    int set[] = { 3, 34, 4, 12, 5, 2 };
    int sum = 9;
    int n = sizeof(set) / sizeof(set[0]);
    if (isSubsetSum(set, n, sum) == true)
        printf("Found a subset with given sum");
    else
        printf("No subset with given sum");
    return 0;
}
// This code is contributed by Arjun Tyagi.

Java

// A Dynamic Programming solution for subset
// sum problem
class GFG {
 
    // Returns true if there is a subset of
    // set[] with sun equal to given sum
    static boolean isSubsetSum(int set[],
                               int n, int sum)
    {
        // The value of subset[i][j] will be
        // true if there is a subset of
        // set[0..j-1] with sum equal to i
        boolean subset[][] = new boolean[sum + 1][n + 1];
 
        // If sum is 0, then answer is true
        for (int i = 0; i <= n; i++)
            subset[0][i] = true;
 
        // If sum is not 0 and set is empty,
        // then answer is false
        for (int i = 1; i <= sum; i++)
            subset[i][0] = false;
 
        // Fill the subset table in botton
        // up manner
        for (int i = 1; i <= sum; i++) {
            for (int j = 1; j <= n; j++) {
                subset[i][j] = subset[i][j - 1];
                if (i >= set[j - 1])
                    subset[i][j] = subset[i][j]
                                   || subset[i - set[j - 1]][j - 1];
            }
        }
 
        /* // uncomment this code to print table
        for (int i = 0; i <= sum; i++)
        {
        for (int j = 0; j <= n; j++)
            System.out.println (subset[i][j]);
        } */
 
        return subset[sum][n];
    }
 
    /* Driver code*/
    public static void main(String args[])
    {
        int set[] = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.length;
        if (isSubsetSum(set, n, sum) == true)
            System.out.println("Found a subset"
                               + " with given sum");
        else
            System.out.println("No subset with"
                               + " given sum");
    }
}
 
/* This code is contributed by Rajat Mishra */

Python3

# A Dynamic Programming solution for subset 
# sum problem Returns true if there is a subset of 
# set[] with sun equal to given sum 
 
# Returns true if there is a subset of set[] 
# with sun equal to given sum
def isSubsetSum(set, n, sum):
     
    # The value of subset[i][j] will be 
    # true if there is a
    # subset of set[0..j-1] with sum equal to i
    subset =([[False for i in range(sum + 1)] 
            for i in range(n + 1)])
     
    # If sum is 0, then answer is true 
    for i in range(n + 1):
        subset[i][0] = True
         
    # If sum is not 0 and set is empty, 
    # then answer is false 
    for i in range(1, sum + 1):
         subset[0][i]= False
             
    # Fill the subset table in botton up manner
    for i in range(1, n + 1):
        for j in range(1, sum + 1):
            if j<set[i-1]:
                subset[i][j] = subset[i-1][j]
            if j>= set[i-1]:
                subset[i][j] = (subset[i-1][j] or
                                subset[i - 1][j-set[i-1]])
     
    # uncomment this code to print table 
    # for i in range(n + 1):
    # for j in range(sum + 1):
    # print (subset[i][j], end =" ")
    # print()
    return subset[n][sum]
         
# Driver code
if __name__=='__main__':
    set = [3, 34, 4, 12, 5, 2]
    sum = 9
    n = len(set)
    if (isSubsetSum(set, n, sum) == True):
        print("Found a subset with given sum")
    else:
        print("No subset with given sum")
         
# This code is contributed by 
# sahil shelangia. 

C#

// A Dynamic Programming solution for
// subset sum problem
using System;
 
class GFG {
    // Returns true if there is a subset
    // of set[] with sun equal to given sum
    static bool isSubsetSum(int[] set, int n, int sum)
    {
        // The value of subset[i][j] will be true if there
        // is a subset of set[0..j-1] with sum equal to i
        bool[, ] subset = new bool[sum + 1, n + 1];
 
        // If sum is 0, then answer is true
        for (int i = 0; i <= n; i++)
            subset[0, i] = true;
 
        // If sum is not 0 and set is empty,
        // then answer is false
        for (int i = 1; i <= sum; i++)
            subset[i, 0] = false;
 
        // Fill the subset table in bottom up manner
        for (int i = 1; i <= sum; i++) {
            for (int j = 1; j <= n; j++) {
                subset[i, j] = subset[i, j - 1];
                if (i >= set[j - 1])
                    subset[i, j] = subset[i, j]
                                   || subset[i - set[j - 1], j - 1];
            }
        }
 
        return subset[sum, n];
    }
 
    // Driver code
    public static void Main()
    {
        int[] set = { 3, 34, 4, 12, 5, 2 };
        int sum = 9;
        int n = set.Length;
        if (isSubsetSum(set, n, sum) == true)
            Console.WriteLine("Found a subset with given sum");
        else
            Console.WriteLine("No subset with given sum");
    }
}
// This code is contributed by Sam007

PHP

<?php
// A Dynamic Programming solution for 
// subset sum problem
 
// Returns true if there is a subset of
// set[] with sun equal to given sum
function isSubsetSum( $set, $n, $sum)
{
    // The value of subset[i][j] will
    // be true if there is a subset of
    // set[0..j-1] with sum equal to i
    $subset = array(array());
 
    // If sum is 0, then answer is true
    for ( $i = 0; $i <= $n; $i++)
        $subset[$i][0] = true;
 
    // If sum is not 0 and set is empty,
    // then answer is false
    for ( $i = 1; $i <= $sum; $i++)
        $subset[0][$i] = false;
 
    // Fill the subset table in botton
    // up manner
    for ($i = 1; $i <= $n; $i++)
    {
        for ($j = 1; $j <= $sum; $j++)
        {
            if($j < $set[$i-1])
                $subset[$i][$j] = 
                      $subset[$i-1][$j];
            if ($j >= $set[$i-1])
                $subset[$i][$j] = 
                       $subset[$i-1][$j] || 
                       $subset[$i - 1][$j - 
                               $set[$i-1]];
        }
    }
 
    /* // uncomment this code to print table
    for (int i = 0; i <= n; i++)
    {
    for (int j = 0; j <= sum; j++)
        printf ("%4d", subset[i][j]);
    printf("n");
    }*/
 
    return $subset[$n][$sum];
}
 
// Driver code
$set = array(3, 34, 4, 12, 5, 2);
$sum = 9;
$n = count($set);
 
if (isSubsetSum($set, $n, $sum) == true)
    echo "Found a subset with given sum";
else
    echo "No subset with given sum";
 
// This code is contributed by anuj_67.
?>

Output

Found a subset with given sum

Complexity Analysis:

Time Complexity: O(sum*n), where sum is the ‘target sum’ and ‘n’ is the size of array.
Auxiliary Space: O(sum*n), as the size of 2-D array is sum*n.

Subset Sum Problem in O(sum) space
Perfect Sum Problem (Print all subsets with given sum)
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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