Sum of Bitwise OR of all pairs in a given array

Given an array “arr[0..n-1]” of integers. The task is to calculate the sum of Bitwise OR of all pairs, i.e. calculate the sum of “arr[i] | arr[j]” for all the pairs in the given array where i < j. Here ‘|’ is bitwise OR operator. Expected time complexity is O(n).

Examples :

Input:  arr[] = {5, 10, 15}
Output: 15
Required Value = (5  |  10) + (5  |  15) + (10  |  15) 
               = 15 + 15 + 15 
               = 45

Input: arr[] = {1, 2, 3, 4}
Output: 3
Required Value = (1  |  2) + (1  |  3) + (1  |  4) + 
                 (2  |  3) + (2  |  4) + (3  |  4) 
               = 3 + 3 + 5 + 3 + 6 + 7
               = 27

A Brute Force approach is to run two loops and time complexity is O(n2).

C++

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// A Simple C++ program to compute sum of bitwise OR
// of all pairs
#include <bits/stdc++.h>
using namespace std;
  
// Returns value of "arr[0]  | arr[1] + arr[0] | arr[2] +
// ... arr[i] | arr[j] + ..... arr[n-2] |  arr[n-1]"
int pairORSum(int arr[], int n)
{
    int ans = 0; // Initialize result
  
    // Consider all pairs (arr[i], arr[j) such that
    // i < j
    for (int i = 0; i < n; i++)
        for (int j = i + 1; j < n; j++)
            ans += arr[i] | arr[j];
  
    return ans;
}
  
// Driver program to test above function
int main()
{
    int arr[] = { 1, 2, 3, 4 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << pairORSum(arr, n) << endl;
    return 0;
}

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Java

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// A Simple Java program to compute
// sum of bitwise OR of all pairs
import java.io.*;
  
class GFG {
  
    // Returns value of "arr[0] | arr[1] +
    // arr[0] | arr[2] + ... arr[i] | arr[j] +
    // ..... arr[n-2] | arr[n-1]"
    static int pairORSum(int arr[], int n)
    {
        int ans = 0; // Initialize result
  
        // Consider all pairs (arr[i], arr[j)
        // such that i < j
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++)
                ans += arr[i] | arr[j];
  
        return ans;
    }
  
    // Driver program to test above function
    public static void main(String args[])
    {
        int arr[] = { 1, 2, 3, 4 };
        int n = arr.length;
        System.out.println(pairORSum(arr, n));
    }
}

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Python3

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# A Simple Python 3 program to compute
# sum of bitwise OR of all pairs
  
# Returns value of "arr[0] | arr[1] +
# arr[0] | arr[2] + ... arr[i] | arr[j] +
# ..... arr[n-2] | arr[n-1]"
def pairORSum(arr, n) :
    ans = 0 # Initialize result
  
    # Consider all pairs (arr[i], arr[j) 
    # such that i < j
    for i in range(0, n) :
        for j in range((i + 1), n) :
            ans = ans + arr[i] | arr[j]
  
    return ans
  
# Driver program to test above function
arr = [1, 2, 3, 4]
n = len(arr) 
print(pairORSum(arr, n))

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C#

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// A Simple C# program to compute
// sum of bitwise OR of all pairs
using System;
  
class GFG {
  
    // Returns value of "arr[0] | arr[1] +
    // arr[0] | arr[2] + ... arr[i] | arr[j] +
    // ..... arr[n-2] | arr[n-1]"
    static int pairORSum(int[] arr, int n)
    {
  
        int ans = 0; // Initialize result
  
        // Consider all pairs (arr[i], arr[j)
        // such that i < j
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++)
                ans += arr[i] | arr[j];
  
        return ans;
    }
  
    // Driver program to test above function
    public static void Main()
    {
        int[] arr = { 1, 2, 3, 4 };
        int n = arr.Length;
        Console.Write(pairORSum(arr, n));
    }
}

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PHP

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<?php
// A Simple PHP program to 
// compute sum of bitwise 
// OR of all pairs
  
// Returns value of "arr[0] |
// arr[1] + arr[0] | arr[2] + 
// ... arr[i] | arr[j] + .....
// arr[n-2] | arr[n-1]"
  
function pairORSum($arr, $n)
{
    // Initialize result
    $ans = 0; 
  
    // Consider all pairs (arr[i], 
    // arr[j) such that i < j
    for ($i = 0; $i < $n; $i++)
        for ( $j = $i + 1; $j < $n; $j++)
        $ans += $arr[$i] | $arr[$j];
  
    return $ans;
}
  
// Driver Code
$arr = array(1, 2, 3, 4);
$n = sizeof($arr) ;
echo pairORSum($arr, $n), "\n";
  
?>

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Output :

27

An Efficient Solution can solve this problem in O(n) time. The assumption here is that integers are represented using 32 bits.

The idea is to count number of set bits at every i’th position (i>=0 && i<=31). Any i'th bit of the AND of two numbers is 1 iff the corresponding bit in both the numbers is equal to 1.

Let k1 be the count of set bits at i'th position. Total number of pairs with i'th set bit would be k1C2 = k1*(k1-1)/2 (Count k1 means there are k1 numbers which have i’th set bit). Every such pair adds 2i to total sum. Similarly, there are total k0 values which don’t have set bits at i’th position. Now each element (which have not set the bit at the i’th position can make pair with k1 elements (ie., those elements which have set bits at the i’th position), So there are total k1 * k0 pairs and every such pair also adds 2i to total sum.

sum = sum + (1<<i) * (k1*(k1-1)/2) + (1<<i) * (k1*k0)

This idea is similar to finding sum of bit differences among all pairs.

Below is the implementation of the above approach:

C++

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// An efficient C++ program to compute sum of bitwise OR
// of all pairs
#include <bits/stdc++.h>
using namespace std;
typedef long long int LLI;
  
// Returns value of "arr[0] | arr[1] + arr[0] | arr[2] +
// ... arr[i] | arr[j] + ..... arr[n-2] | arr[n-1]"
  
LLI pairORSum(LLI arr[], LLI n)
{
  
    LLI ans = 0; // Initialize result
    // Traverse over all bits
    for (LLI i = 0; i < 32; i++) {
        // Count number of elements with the i'th bit set(ie., 1)
        LLI k1 = 0; // Initialize the count
  
        // Count number of elements with i’th bit not-set(ie., 0) `
        LLI k0 = 0; // Initialize the count
  
        for (LLI j = 0; j < n; j++) {
  
            if ((arr[j] & (1 << i))) // if i'th bit is set
                k1++;
            else
                k0++;
        }
        // There are k1 set bits, means k1(k1-1)/2 pairs. k1C2
        // There are k0 not-set bits and k1 set bits so total pairs will be k1*k0.
  
        // Every pair adds 2^i to the answer. Therefore,
  
        ans = ans + (1 << i) * (k1 * (k1 - 1) / 2) + (1 << i) * (k1 * k0);
    }
  
    return ans;
}
  
// Driver program to test the above function
  
int main()
  
{
  
    LLI arr[] = { 1, 2, 3, 4 };
  
    LLI n = sizeof(arr) / sizeof(arr[0]);
  
    cout << pairORSum(arr, n) << endl;
  
    return 0;
}

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Java

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// An efficient Java program to compute
// sum of bitwise OR of all pairs
import java.io.*;
  
class GFG {
  
    // Returns value of "arr[0] | arr[1] + arr[0] | arr[2] +
    // ... arr[i] | arr[j] + ..... arr[n-2] | arr[n-1]"
    static int pairORSum(int arr[], int n)
    {
        int ans = 0; // Initialize result
        // Traverse over all bits
        for (int i = 0; i < 32; i++) {
            // Count number of elements with the ith bit set(ie., 1)
            int k1 = 0; // Initialize the count
  
            // Count number of elements with ith bit not-set(ie., 0) `
            int k0 = 0; // Initialize the count
  
            for (int j = 0; j < n; j++) {
  
                if ((arr[j] & (1 << i)) != 0) // if i'th bit is set
                    k1++;
                else
                    k0++;
            }
            // There are k1 set bits, means k1(k1-1)/2 pairs. k1C2
            // There are k0 not-set bits and k1 set bits so total pairs will be k1*k0.
            // Every pair adds 2^i to the answer. Therefore,
  
            ans = ans + (1 << i) * (k1 * (k1 - 1) / 2) + (1 << i) * (k1 * k0);
        }
  
        return ans;
    }
  
    // Driver program to test above function
    public static void main(String args[])
    {
        int arr[] = { 1, 2, 3, 4 };
        int n = arr.length;
        System.out.println(pairORSum(arr, n));
    }
}

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Python3

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# An efficient Python 3 program to
# compute the sum of bitwise OR of all pairs
  
# Returns value of "arr[0] | arr[1] + arr[0] | arr[2] +
# ... arr[i] | arr[j] + ..... arr[n-2] | arr[n-1]"
  
def pairORSum(arr, n) :
    # Initialize result
    ans = 0
    # Traverse over all bits
    for i in range(0, 32) :
        # Count number of elements with the i'th bit set(ie., 1)
        k1 = 0
          
        # Count number of elements with i’th bit not-set(ie., 0) `
        k0 = 0
          
        for j in range(0, n) :
              
            if( (arr[j] & (1<<i)) ):   # if i'th bit is set
                k1 = k1 + 1
                  
            else
                k0 = k0 + 1
        # There are k1 set bits, means k1(k1-1)/2 pairs. k1C2
        # There are k0 not-set bits and k1 set bits so total pairs will be k1 * k0.
  
        # Every pair adds 2 ^ i to the answer. Therefore,
          
        ans = ans + (1<<i) * (k1*(k1-1)//2) + (1<<i) * (k1 * k0)
      
    return ans
      
# Driver program to test above function
arr = [1, 2, 3, 4]
n = len(arr) 
print(pairORSum(arr, n))

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C#

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// An efficient C# program to compute
// sum of bitwise OR of all pairs
using System;
  
class GFG {
  
    // Returns value of "arr[0] | arr[1] + arr[0] | arr[2] +
    // ... arr[i] | arr[j] + ..... arr[n-2] | arr[n-1]"
    static int pairORSum(int[] arr, int n)
    {
        int ans = 0; // Initialize result
        // Traverse over all bits
        for (int i = 0; i < 32; i++) {
            // Count number of elements with the ith bit set(ie., 1)
            int k1 = 0; // Initialize the count
  
            // Count number of elements with ith bit not-set(ie., 0) `
            int k0 = 0; // Initialize the count
  
            for (int j = 0; j < n; j++) {
                // if i'th bit is set
                if ((arr[j] & (1 << i)) != 0)
                    k1++;
                else
                    k0++;
            }
            // There are k1 set bits, means k1(k1-1)/2 pairs. k1C2
            // There are k0 not-set bits and k1 set bits so total pairs will be k1*k0.
            // Every pair adds 2^i to the answer. Therefore,
  
            ans = ans + (1 << i) * (k1 * (k1 - 1) / 2) + (1 << i) * (k1 * k0);
        }
  
        return ans;
    }
  
    // Driver program to test above function
    public static void Main()
    {
        int[] arr = new int[] { 1, 2, 3, 4 };
        int n = arr.Length;
  
        Console.Write(pairORSum(arr, n));
    }
}

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PHP

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<?php
// An efficient PHP program to compute
// sum of bitwise OR of all pairs
  
  
// Returns value of "arr[0] | arr[1] + arr[0] | arr[2] +
// ... arr[i] | arr[j] + ..... arr[n-2] | arr[n-1]"
      
function pairORSum($arr, $n)
{
    $ans = 0; // Initialize result
    // Traverse over all bits
    for ( $i = 0; $i < 32; $i++){
        // Count number of elements with the ith bit set(ie., 1)
        $k1 = 0; // Initialize the count
  
        // Count number of elements with ith bit not-set(ie., 0) `
        $k0 = 0; // Initialize the count
  
        for ( $j = 0; $j < $n; $j++){
  
            if ( ($arr[$j] & (1 << $i)))  // if i'th bit is set
                $k1++;
            else
                $k0++;
    }
    // There are k1 set bits, means k1(k1-1)/2 pairs. k1C2
    // There are k0 not-set bits and k1 set bits so total pairs will be k1*k0.
    // Every pair adds 2^i to the answer. Therefore,
  
    $ans = $ans + (1<<$i) * ($k1*($k1-1)/2) + (1<<$i) * ($k1*$k0) ;
  
}
  
return $ans;
  
}
  
    // Driver Code
    $arr = array(1, 2, 3, 4);
    $n = sizeof($arr);
    echo pairORSum($arr, $n) ;
  
?>

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Output:

27


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