Find all combinations of two equal sum subsequences

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Given an array arr[] of integers, the task is to find all possible ways the array could be split into two subsequences such that the sum of the elements in both the subsequences is equal. Each number in the array must belong to only one of the two subsequences. Print all possible combinations of two subsequences.

Examples:

Input: arr[] = {1, 2, 3, 9, 4, 5}
Output:
1 2 9 and 3 4 5
1 2 4 5 and 3 9
3 9 and 1 2 4 5
3 4 5 and 1 2 9

Input: arr[] = {4, -1, 2, -1}
Output:
4 -1 -1 and 2
2 and 4 -1 -1

Approach: A simple solution is to form all possible pairs of subsequences using recursion and compare the sum of elements in both the subsequences. For each array element, we have two choices, we can either take the current element in the first subsequence or the second.

Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
#define MAX 100000
 
// Function to print the subsequence elements
void print(int g1[], int a, int g2[], int b)
{
 
    // Prints elements of the 1st subarray
    for (int i = 0; i < a; i++) {
        cout << g1[i] << " ";
    }
    cout << "and ";
 
    // Prints elements of the 2nd subarray
    for (int i = 0; i < b; i++) {
        cout << g2[i] << " ";
    }
    cout << endl;
}
 
// Function that returns true if the sum of the
// elements of arrays g1[] and g2[] is equal
bool checksum(int g1[], int a, int g2[], int b)
{
    int i, x;
 
    // Adding elements of the 1st array
    for (i = 0, x = 0; i < a; i++) {
        x += g1[i];
    }
 
    // Subtracting elements of the 2nd array
    for (i = 0; i < b; i++) {
        x -= g2[i];
    }
 
    // If x is 0 then the sum of elements
    // in both the arrays is equal
    return (x == 0);
}
 
// Function to find all valid subsequences
void formgroups(int arr[], int x, int g1[], int a,
                int g2[], int b, int n)
{
    // Base Case
    if (x == n) {
 
        // If sum of the two subsequences
        // is equal then print the elements
        if (checksum(g1, a, g2, b)) {
 
            // Print the subsequences
            print(g1, a, g2, b);
        }
        return;
    }
 
    // Recursive Case
 
    // Choose current element to be a
    // part of the first subsequence
    g1[a] = arr[x];
    formgroups(arr, x + 1, g1, a + 1, g2, b, n);
 
    // Choose current element to be a
    // part of the second subsequence
    g2[b] = arr[x];
    formgroups(arr, x + 1, g1, a, g2, b + 1, n);
}
 
// Driver code
int main()
{
    int arr[] = { 1, 2, 3, 9, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    int g1[MAX], g2[MAX];
    formgroups(arr, 0, g1, 0, g2, 0, n);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
 
class GFG 
{
static int MAX = 100000;
 
// Function to print the 
// subsequence elements
static void print(int g1[], int a, 
                  int g2[], int b)
{
 
    // Prints elements of the 1st subarray
    for (int i = 0; i < a; i++)
    {
        System.out.print(g1[i] + " ");
    }
    System.out.print("and ");
 
    // Prints elements of the 2nd subarray
    for (int i = 0; i < b; i++) 
    {
        System.out.print(g2[i] + " ");
    }
    System.out.println();
}
 
// Function that returns true if 
// the sum of the elements of 
// arrays g1[] and g2[] is equal
static boolean checksum(int g1[], int a, 
                        int g2[], int b)
{
    int i, x;
 
    // Adding elements of the 1st array
    for (i = 0, x = 0; i < a; i++) 
    {
        x += g1[i];
    }
 
    // Subtracting elements of 
    // the 2nd array
    for (i = 0; i < b; i++)
    {
        x -= g2[i];
    }
 
    // If x is 0 then the sum of elements
    // in both the arrays is equal
    return (x == 0);
}
 
// Function to find all valid subsequences
static void formgroups(int arr[], int x, 
                       int g1[], int a, 
                       int g2[], int b, int n)
{
    // Base Case
    if (x == n) 
    {
 
        // If sum of the two subsequences
        // is equal then print the elements
        if (checksum(g1, a, g2, b)) 
        {
 
            // Print the subsequences
            print(g1, a, g2, b);
        }
        return;
    }
 
    // Recursive Case
 
    // Choose current element to be a
    // part of the first subsequence
    g1[a] = arr[x];
    formgroups(arr, x + 1, g1, 
                    a + 1, g2, b, n);
 
    // Choose current element to be a
    // part of the second subsequence
    g2[b] = arr[x];
    formgroups(arr, x + 1, g1, a, 
                g2, b + 1, n);
}
 
// Driver code
public static void main(String[] args) 
{
    int arr[] = { 1, 2, 3, 9, 4, 5 };
    int n = arr.length;
 
    int []g1 = new int[MAX];
    int []g2 = new int[MAX];
    formgroups(arr, 0, g1, 0, g2, 0, n);
}
}
 
// This code is contributed by PrinciRaj1992 

Python3

# Python 3 implementation of the approach
MAX = 100000
 
# Function to print the subsequence elements
def prints(g1, a, g2, b):
     
    # Prints elements of the 1st subarray
    for i in range(a):
        print(g1[i], end = " ")
     
    print("and ", end = "")
 
    # Prints elements of the 2nd subarray
    for i in range(b):
        print(g2[i], end = " ")
     
    print("\n", end = "")
 
# Function that returns true if the sum of the
# elements of arrays g1[] and g2[] is equal
def checksum(g1, a, g2, b):
     
    # Adding elements of the 1st array
    x = 0
    for i in range(0, a, 1):
        x += g1[i]
 
    # Subtracting elements of the 2nd array
    for i in range(b):
        x -= g2[i]
 
    # If x is 0 then the sum of elements
    # in both the arrays is equal
    return (x == 0)
 
# Function to find all valid subsequences
def formgroups(arr, x, g1, a, g2, b, n):
     
    # Base Case
    if (x == n):
         
        # If sum of the two subsequences
        # is equal then print the elements
        if (checksum(g1, a, g2, b)):
             
            # Print the subsequences
            prints(g1, a, g2, b)
 
        return
 
    # Recursive Case
 
    # Choose current element to be a
    # part of the first subsequence
    g1[a] = arr[x]
    formgroups(arr, x + 1, g1, a + 1, g2, b, n)
 
    # Choose current element to be a
    # part of the second subsequence
    g2[b] = arr[x]
    formgroups(arr, x + 1, g1, a, g2, b + 1, n)
 
# Driver code
if __name__ == '__main__':
    arr = [1, 2, 3, 9, 4, 5]
    n = len(arr)
    g1 = [0 for i in range(MAX)]
    g2 = [0 for i in range(MAX)]
    formgroups(arr, 0, g1, 0, g2, 0, n)
 
# This code is contributed by
# Surendra_Gangwar

C#

// C# implementation of the approach
using System;
 
class GFG 
{
static int MAX = 100000;
 
// Function to print the 
// subsequence elements
static void print(int []g1, int a, 
                  int []g2, int b)
{
 
    // Prints elements of the 1st subarray
    for (int i = 0; i < a; i++)
    {
        Console.Write(g1[i] + " ");
    }
    Console.Write("and ");
 
    // Prints elements of the 2nd subarray
    for (int i = 0; i < b; i++) 
    {
        Console.Write(g2[i] + " ");
    }
    Console.WriteLine();
}
 
// Function that returns true if 
// the sum of the elements of 
// arrays g1[] and g2[] is equal
static bool checksum(int []g1, int a, 
                     int []g2, int b)
{
    int i, x;
 
    // Adding elements of the 1st array
    for (i = 0, x = 0; i < a; i++) 
    {
        x += g1[i];
    }
 
    // Subtracting elements of 
    // the 2nd array
    for (i = 0; i < b; i++)
    {
        x -= g2[i];
    }
 
    // If x is 0 then the sum of elements
    // in both the arrays is equal
    return (x == 0);
}
 
// Function to find all valid subsequences
static void formgroups(int []arr, int x, 
                       int []g1, int a, 
                       int []g2, int b, int n)
{
    // Base Case
    if (x == n) 
    {
 
        // If sum of the two subsequences
        // is equal then print the elements
        if (checksum(g1, a, g2, b)) 
        {
 
            // Print the subsequences
            print(g1, a, g2, b);
        }
        return;
    }
 
    // Recursive Case
 
    // Choose current element to be a
    // part of the first subsequence
    g1[a] = arr[x];
    formgroups(arr, x + 1, g1, 
                    a + 1, g2, b, n);
 
    // Choose current element to be a
    // part of the second subsequence
    g2[b] = arr[x];
    formgroups(arr, x + 1, g1, a, 
                g2, b + 1, n);
}
 
// Driver code
public static void Main() 
{
    int []arr = { 1, 2, 3, 9, 4, 5 };
    int n = arr.Length;
 
    int []g1 = new int[MAX];
    int []g2 = new int[MAX];
    formgroups(arr, 0, g1, 0, g2, 0, n);
}
}
 
// This code is contributed by anuj_67...

PHP

<?php
// PHP implementation of the approach
const MAX = 100000; 
 
// Function to print the subsequence elements 
function printi($g1, $a, $g2, $b) 
{ 
 
    // Prints elements of the 1st subarray 
    for ($i = 0; $i < $a; $i++)
    { 
        echo ($g1[$i]);
        echo " ";
    } 
    echo ("and "); 
 
    // Prints elements of the 2nd subarray 
    for ($i = 0; $i < $b; $i++)
    { 
        echo ($g2[$i]);
        echo " ";
    } 
    echo "\n"; 
} 
 
// Function that returns true if the sum of the 
// elements of arrays g1[] and g2[] is equal 
function checksum($g1, $a, $g2, $b) 
{ 
 
    // Adding elements of the 1st array 
    for ($i = 0, $x = 0; $i < $a; $i++) 
    { 
        $x += $g1[$i]; 
    } 
 
    // Subtracting elements of the 2nd array 
    for ($i = 0; $i < $b; $i++) 
    { 
        $x -= $g2[$i]; 
    } 
 
    // If x is 0 then the sum of elements 
    // in both the arrays is equal 
    return ($x == 0); 
} 
 
// Function to find all valid subsequences 
function formgroups($arr, $x, $g1, $a, 
                          $g2, $b, $n) 
{ 
    // Base Case 
    if ($x == $n) 
    { 
 
        // If sum of the two subsequences 
        // is equal then print the elements 
        if (checksum($g1, $a, $g2, $b)) 
        { 
 
            // Print the subsequences 
            printi($g1, $a, $g2, $b); 
        } 
        return; 
    } 
 
    // Recursive Case 
 
    // Choose current element to be a 
    // part of the first subsequence 
    $g1[$a] = $arr[$x]; 
    formgroups($arr, $x + 1, $g1, 
               $a + 1, $g2, $b, $n); 
 
    // Choose current element to be a 
    // part of the second subsequence 
    $g2[$b] = $arr[$x]; 
    formgroups($arr, $x + 1, $g1, $a, 
                $g2, $b + 1, $n); 
} 
 
// Driver code 
$arr = array(1, 2, 3, 9, 4, 5); 
$n = count($arr);
 
$g1 = array();
$g2 = array();
 
formgroups($arr, 0, $g1, 0, $g2, 0, $n); 
 
// This code is contributed by Naman_Garg.
?>

Javascript

<script>
 
// Javascript implementation of the approach
var MAX = 100000;
 
// Function to print the subsequence elements
function print(g1, a, g2, b)
{
     
    // Prints elements of the 1st subarray
    for(var i = 0; i < a; i++)
    {
        document.write( g1[i] + " ");
    }
    document.write("and ");
 
    // Prints elements of the 2nd subarray
    for(var i = 0; i < b; i++)
    {
        document.write( g2[i] + " ");
    }
    document.write("<br>");
}
 
// Function that returns true if the sum of the
// elements of arrays g1[] and g2[] is equal
function checksum(g1, a, g2, b)
{
    var i, x;
 
    // Adding elements of the 1st array
    for(i = 0, x = 0; i < a; i++) 
    {
        x += g1[i];
    }
 
    // Subtracting elements of the 2nd array
    for(i = 0; i < b; i++)
    {
        x -= g2[i];
    }
 
    // If x is 0 then the sum of elements
    // in both the arrays is equal
    return (x == 0);
}
 
// Function to find all valid subsequences
function formgroups(arr, x, g1, a, g2, b, n)
{
     
    // Base Case
    if (x == n)
    {
         
        // If sum of the two subsequences
        // is equal then print the elements
        if (checksum(g1, a, g2, b)) 
        {
             
            // Print the subsequences
            print(g1, a, g2, b);
        }
        return;
    }
 
    // Recursive Case
 
    // Choose current element to be a
    // part of the first subsequence
    g1[a] = arr[x];
    formgroups(arr, x + 1, g1, a + 1, 
               g2, b, n);
 
    // Choose current element to be a
    // part of the second subsequence
    g2[b] = arr[x];
    formgroups(arr, x + 1, g1, a, g2, 
                    b + 1, n);
}
 
// Driver code
var arr = [ 1, 2, 3, 9, 4, 5 ];
var n = arr.length;
 
var g1 = Array(MAX).fill(0), 
    g2 = Array(MAX).fill(0);
formgroups(arr, 0, g1, 0, g2, 0, n);
 
// This code is contributed by noob2000
 
</script>

Output:

1 2 9 and 3 4 5 
1 2 4 5 and 3 9 
3 9 and 1 2 4 5 
3 4 5 and 1 2 9

Time Complexity: O(2ⁿ)

Auxiliary Space: O(n), where n is the length of the input array

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