Print all the combinations of N elements by changing sign such that their sum is divisible by M

Given an array of N integers and an integer M. You can change the sign(positive or negative) of any element in the array. The task is to print all possible combinations of the array elements that can be obtained by changing the sign of the elements such that their sum is divisible by M.

Note: You have to take all of the array elements in each combination and in the same order as the elements present in the array. However, you can change the sign of elements.

Examples:



Input: a[] = {5, 6, 7}, M = 3
Output:
-5-6-7
+5-6+7
-5+6-7
+5+6+7

Input: a[] = {3, 5, 6, 8}, M = 5
Output:
-3-5+6-8
-3+5+6-8
+3-5-6+8
+3+5-6+8

Approach: The concept of power set is used here to solve this problem. Using power-set generate all possible combinations of signs that can be applied to the array of elements. If the sum obtained is divisible by M, then print the combination. Below are the steps:

  • Iterate for all possible combinations of ‘+’ and ‘-‘ using power set.
  • Iterate on the array elements and if the j-th bit from left is set, then assume the array element to be positive and if the bit is not set, then assume the array element to be negative. Refer here to check if bit any index is set or not.
  • If the sum is divisible by M, then the again traverse the array elements and print them along with sign( ‘+’ or ‘-‘ ).

Below is the implementation of the above approach:

C++

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#include <bits/stdc++.h>
using namespace std;
  
// Function to print all the combinations
void printCombinations(int a[], int n, int m)
{
  
    // Iterate for all combinations
    for (int i = 0; i < (1 << n); i++) {
        int sum = 0;
  
        // Initially 100 in binary if n is 3
        // as 1<<(3-1) = 100 in binary
        int num = 1 << (n - 1);
  
        // Iterate in the array and assign signs
        // to the array elements
        for (int j = 0; j < n; j++) {
  
            // If the j-th bit from left is set
            // take '+' sign
            if (i & num)
                sum += a[j];
            else
                sum += (-1 * a[j]);
  
            // Right shift to check if
            // jth bit is set or not
            num = num >> 1;
        }
  
        if (sum % m == 0) {
  
            // re-initialize
            num = 1 << (n - 1);
  
            // Iterate in the array elements
            for (int j = 0; j < n; j++) {
  
                // If the jth from left is set
                if ((i & num))
                    cout << "+" << a[j] << " ";
                else
                    cout << "-" << a[j] << " ";
  
                // right shift
                num = num >> 1;
            }
            cout << endl;
        }
    }
}
  
// Driver Code
int main()
{
    int a[] = { 3, 5, 6, 8 };
    int n = sizeof(a) / sizeof(a[0]);
    int m = 5;
  
    printCombinations(a, n, m);
    return 0;
}

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Java

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import java.io.*;
  
class GFG
{
      
// Function to print
// all the combinations
static void printCombinations(int a[], 
                              int n, int m)
{
  
    // Iterate for all
    // combinations
    for (int i = 0
             i < (1 << n); i++) 
    {
        int sum = 0;
  
        // Initially 100 in binary 
        // if n is 3 as 
        // 1<<(3-1) = 100 in binary
        int num = 1 << (n - 1);
  
        // Iterate in the array 
        // and assign signs to
        // the array elements
        for (int j = 0; j < n; j++)
        {
  
            // If the j-th bit 
            // from left is set
            // take '+' sign
            if ((i & num) > 0)
                sum += a[j];
            else
                sum += (-1 * a[j]);
  
            // Right shift to check if
            // jth bit is set or not
            num = num >> 1;
        }
  
        if (sum % m == 0
        {
  
            // re-initialize
            num = 1 << (n - 1);
  
            // Iterate in the 
            // array elements
            for (int j = 0; j < n; j++) 
            {
  
                // If the jth from
                // left is set
                if ((i & num) > 0)
                    System.out.print("+"
                                     a[j] + " ");
                else
                    System.out.print("-"
                                     a[j] + " ");
  
                // right shift
                num = num >> 1;
            }
              
        System.out.println();
        }
    }
}
  
// Driver code
public static void main(String args[])
{
    int a[] = { 3, 5, 6, 8 };
    int n = a.length;
    int m = 5;
  
    printCombinations(a, n, m);
}
}
  
// This code is contributed
// by inder_verma.

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Python3

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# Function to print
# all the combinations
def printCombinations(a, n, m):
  
    # Iterate for all
    # combinations
    for i in range(0, (1 << n)): 
      
        sum = 0
  
        # Initially 100 in binary 
        # if n is 3 as 
        # 1<<(3-1) = 100 in binary
        num = 1 << (n - 1)
  
        # Iterate in the array 
        # and assign signs to
        # the array elements
        for j in range(0, n): 
          
            # If the j-th bit 
            # from left is set
            # take '+' sign
            if ((i & num) > 0):
                sum += a[j]
            else:
                sum += (-1 * a[j])
  
            # Right shift to check if
            # jth bit is set or not
            num = num >> 1
  
        if (sum % m == 0): 
          
            # re-initialize
            num = 1 << (n - 1)
  
            # Iterate in the 
            # array elements
            for j in range(0, n): 
  
                # If the jth from
                # left is set
                if ((i & num) > 0):
                    print("+", a[j], end = " "
                                     sep = "")
                else:
                    print("-", a[j], end = " ",
                                     sep = "")
  
                # right shift
                num = num >> 1
            print("")
      
# Driver code
a = [ 3, 5, 6, 8 ]
n = len(a)
m = 5
printCombinations(a, n, m)
  
# This code is contributed
# by smita.

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C#

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// Print all the combinations 
// of N elements by changing 
// sign such that their sum 
// is divisible by M
using System;
  
class GFG
{
      
// Function to print
// all the combinations
static void printCombinations(int []a, 
                              int n, int m)
{
  
    // Iterate for all
    // combinations
    for (int i = 0; 
             i < (1 << n); i++) 
    {
        int sum = 0;
  
        // Initially 100 in binary 
        // if n is 3 as 
        // 1<<(3-1) = 100 in binary
        int num = 1 << (n - 1);
  
        // Iterate in the array 
        // and assign signs to
        // the array elements
        for (int j = 0; j < n; j++)
        {
  
            // If the j-th bit 
            // from left is set
            // take '+' sign
            if ((i & num) > 0)
                sum += a[j];
            else
                sum += (-1 * a[j]);
  
            // Right shift to check if
            // jth bit is set or not
            num = num >> 1;
        }
  
        if (sum % m == 0) 
        {
  
            // re-initialize
            num = 1 << (n - 1);
  
            // Iterate in the 
            // array elements
            for (int j = 0; j < n; j++) 
            {
  
                // If the jth from
                // left is set
                if ((i & num) > 0)
                    Console.Write("+"
                              a[j] + " ");
                else
                    Console.Write("-"
                              a[j] + " ");
  
                // right shift
                num = num >> 1;
            }
              
        Console.Write("\n");
        }
    }
}
  
// Driver code
public static void Main()
{
    int []a = { 3, 5, 6, 8 };
    int n = a.Length;
    int m = 5;
  
    printCombinations(a, n, m);
}
}
  
// This code is contributed
// by Smitha.

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PHP

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<?php
// Function to print all the combinations 
function printCombinations($a, $n, $m
  
    // Iterate for all combinations 
    for ($i = 0; $i < (1 << $n); $i++) 
    
        $sum = 0; 
  
        // Initially 100 in binary if n  
        // is 3 as 1<<(3-1) = 100 in binary 
        $num = 1 << ($n - 1); 
  
        // Iterate in the array and assign
        // signs to the array elements 
        for ($j = 0; $j < $n; $j++) 
        
  
            // If the j-th bit from left 
            // is set take '+' sign 
            if ($i & $num
                $sum += $a[$j]; 
            else
                $sum += (-1 * $a[$j]); 
  
            // Right shift to check if 
            // jth bit is set or not 
            $num = $num >> 1; 
        
  
        if ($sum % $m == 0) 
        
  
            // re-initialize 
            $num = 1 << ($n - 1); 
  
            // Iterate in the array elements 
            for ($j = 0; $j < $n; $j++) 
            
  
                // If the jth from left is set 
                if (($i & $num)) 
                    echo "+" , $a[$j] , " "
                else
                    echo "-" , $a[$j] , " "
  
                // right shift 
                $num = $num >> 1; 
            
        echo "\n";
        
    
  
// Driver Code 
$a = array( 3, 5, 6, 8 ); 
$n = sizeof($a); 
$m = 5; 
  
printCombinations($a, $n, $m); 
  
// This code is contributed by ajit
?>

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Output:

-3 -5 +6 -8 
-3 +5 +6 -8 
+3 -5 -6 +8 
+3 +5 -6 +8

Time Complexity: O(2N * N), where N is the number of elements.



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