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Minimize Array Sum by replacing pairs with (X, Y) keeping their bitwise OR same

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  • Difficulty Level : Easy
  • Last Updated : 31 May, 2022
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Given an array arr[] of size N. Find the minimum sum of the array after performing given operations any number of times:

  • Select two different indices i, j (1 ≤ i < j ≤ N),
  • Replace arr[i] and arr[j] with X and Y respectively (X, Y>0), such that arr[i] | arr[j] = X | Y, where | denotes the bitwise OR operation

Examples:

Input: arr[] = {1, 3, 2}
Output: 3
Explanation: arr = {1, 3, 2}  {bitwise OR of array elements is 3.}, sum=6
Replace 1 and 3 with 3 and 0 as {1|3 = 3 = 3|0}.
arr = {3, 0, 2}, {bitwise OR of array elements is still 3}, sum = 5
Replace 3 and 2 with 3 and 0 as {3|2 = 3 = 3|0}
arr = {3, 0, 0} {or of all array elements is still 3), sum= 3.
This is the minimum sum possible.

Input: arr[] = {3, 5, 6}
Output: 7

 

Approach: The solution to the problem is based on the following observation: 

If (arr[i], arr[j]) is replaced with ((arr[i] | arr[j]), 0), the bitwise OR value will remain the same and the sum will be minimized.

Illustration:

Consider: arr[] = {1, 3, 2}

Initial Array sum = 6

Operation 1: Replace (1, 3) with (1|3 , 0):
    -> Bitwise OR of (1, 3) = 3
    -> Replacing 1 with bitwise OR value 3 and 3 with 0 respectively
    -> New pair will be (3, 0), whose bitwise OR value will be 3|0 = 3 (same as original pair (1, 3)).
    -> Updated Array: {3, 0, 2}
    -> Updated Array sum = 5

Operation 2: Similarly replace (3, 2) with (3|2 , 0):
    -> Bitwise OR of (3, 2) = 3
    -> Replacing 3 with bitwise OR value 3 and 2 with 0 respectively
    -> New pair will be (3, 0), whose bitwise OR value will be 3|0 = 3 (same as original pair (3, 2)).
    -> Updated Array: {3, 0, 0}
    -> Updated Array sum = 3

Now no more operations can be done and the Array sum cannot be reduced further from 3. 

Therefore final minimized Array sum will be the bitwise OR of all Array elements.

Follow the steps mentioned below:

  • Traverse the array from the starting.
  • Calculate the bitwise OR of all the array elements.
  • Return this as the answer.

Below is the implementation of the above approach,

C++




// C++ code to implement the approach
 
#include <bits/stdc++.h>
using namespace std;
typedef long long int ll;
 
// Function minsum() which will calculate
// the minimum sum of the given array
ll minsum(ll array[], ll n, ll sum)
{
    for (int i = 0; i < n; i++) {
 
        // Calculating the or of
        // all the elements in the array
        sum |= array[i];
    }
    return sum;
}
 
// Driver code
int main()
{
    ll array[] = { 1, 3, 2 };
 
    // Calculating the size of array
    ll n = sizeof(array) / sizeof(array[0]);
 
    // Initialising a variable sum with zero
    ll sum = 0;
   
    // Function call
    cout << minsum(array, n, sum) << "\n";
 
    return 0;
}


Java




// JAVA code to implement the approach
import java.util.*;
class GFG
{
 
  // Function minsum() which will calculate
  // the minimum sum of the given array
  public static long minsum(long array[], long n,
                            long sum)
  {
    for (int i = 0; i < n; i++) {
 
      // Calculating the or of
      // all the elements in the array
      sum |= array[i];
    }
    return sum;
  }
 
  // Driver code
  public static void main(String[] args)
  {
    long array[] = { 1, 3, 2 };
 
    // Calculating the size of array
    long n = array.length;
 
    // Initialising a variable sum with zero
    long sum = 0;
 
    // Function call
    System.out.println(minsum(array, n, sum));
  }
}
 
// This code is contributed by Taranpreet


Python3




# python3 code to implement the approach
 
# Function minsum() which will calculate
# the minimum sum of the given array
def minsum(array, n, sum):
 
    for i in range(0, n):
 
        # Calculating the or of
        # all the elements in the array
        sum |= array[i]
 
    return sum
 
# Driver code
if __name__ == "__main__":
 
    array = [1, 3, 2]
 
    # Calculating the size of array
    n = len(array)
 
    # Initialising a variable sum with zero
    sum = 0
 
    # Function call
    print(minsum(array, n, sum))
 
# This code is contributed by rakeshsahni


C#




// C# code to implement the approach
using System;
public class GFG{
 
  // Function minsum() which will calculate
  // the minimum sum of the given array
  static long minsum(long[] array, long n,
                     long sum)
  {
    for (int i = 0; i < n; i++) {
 
      // Calculating the or of
      // all the elements in the array
      sum |= array[i];
    }
    return sum;
  }
 
  // Driver code
  static public void Main ()
  {
 
    long[] array = { 1, 3, 2 };
 
    // Calculating the size of array
    long n = array.Length;
 
    // Initialising a variable sum with zero
    long sum = 0;
 
    // Function call
    Console.Write(minsum(array, n, sum));
  }
}
 
// This code is contributed by hrithikgarg03188.


Javascript




<script>
      // JavaScript code for the above approach
 
      // Function minsum() which will calculate
      // the minimum sum of the given array
      function minsum(array, n, sum) {
          for (let i = 0; i < n; i++) {
 
              // Calculating the or of
              // all the elements in the array
              sum |= array[i];
          }
          return sum;
      }
 
      // Driver code
      let array = [1, 3, 2];
 
      // Calculating the size of array
      let n = array.length;
 
      // Initialising a variable sum with zero
      let sum = 0;
 
      // Function call
      document.write(minsum(array, n, sum) + "<br>");
 
   // This code is contributed by Potta Lokesh
  </script>


Output

3

 
 Time Complexity: O(N), as we are using a loop to traverse N times.

Auxiliary Space: O(1), as we are not using any extra space.

 


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