Data Modelling & AI Data Structure & Algorithm

Count pairs with Bitwise-AND as even number

30 July 2024

2

Given an array of $N$ integers. The task is to find the number of pairs (i, j) such that A[i] & A[j] is even.

Examples:

Input: N = 4, A[] = { 5, 1, 3, 2 }
Output: 3
Since pair of A[] are:
( 5, 1 ), ( 5, 3 ), ( 5, 2 ), ( 1, 3 ), ( 1, 2 ), ( 3, 2 )
5 AND 1 = 1, 5 AND 3 = 1, 5 AND 2 = 0,
1 AND 3 = 1, 1 AND 2 = 0,
3 AND 2 = 2
Total even pair A( i, j ) = 3

Input : N = 6, A[] = { 5, 9, 0, 6, 7, 3 }
Output : 9
Since pair of A[] =
( 5, 9 ) = 1, ( 5, 0 ) = 0, ( 5, 6 ) = 4, ( 5, 7 ) = 5, ( 5, 3 ) = 1,
( 9, 0 ) = 0, ( 9, 6 ) = 0, ( 9, 7 ) = 1, ( 9, 3 ) = 1,
( 0, 6 ) = 0, ( 0, 7 ) = 0, ( 0, 3 ) = 0,
( 6, 7 ) = 6, ( 6, 3 ) = 2,
( 7, 3 ) = 3

A Naive Approach is to check for every pair and print the count of pairs which are even.

Below is the implementation of the above approach:

C++

// C++ program to count pair with
// bitwise-AND as even number
 
#include <iostream>
using namespace std;
 
// Function to count number of pairs EVEN bitwise AND
int findevenPair(int A[], int N)
{
    int i, j;
 
    // variable for counting even pairs
    int evenPair = 0;
 
    // find all pairs
    for (i = 0; i < N; i++) {
        for (j = i + 1; j < N; j++) {
 
            // find AND operation
            // to check evenpair
            if ((A[i] & A[j]) % 2 == 0)
                evenPair++;
        }
    }
 
    // return number of even pair
    return evenPair;
}
 
// Driver Code
int main()
{
 
    int a[] = { 5, 1, 3, 2 };
    int n = sizeof(a) / sizeof(a[0]);
 
    cout << findevenPair(a, n) << endl;
 
    return 0;
}

C

// C program to count pair with
// bitwise-AND as even number
#include <stdio.h>
 
// Function to count number of pairs EVEN bitwise AND
int findevenPair(int A[], int N)
{
    int i, j;
 
    // variable for counting even pairs
    int evenPair = 0;
 
    // find all pairs
    for (i = 0; i < N; i++) {
        for (j = i + 1; j < N; j++) {
 
            // find AND operation
            // to check evenpair
            if ((A[i] & A[j]) % 2 == 0)
                evenPair++;
        }
    }
 
    // return number of even pair
    return evenPair;
}
 
// Driver Code
int main()
{
 
    int a[] = { 5, 1, 3, 2 };
    int n = sizeof(a) / sizeof(a[0]);
    
    printf("%d\n",findevenPair(a, n));
 
    return 0;
}
 
// This code is contributed by kothavvsaakash.

Java

// Java program to count pair with
// bitwise-AND as even number
import java.io.*;
 
class GFG
{
 
// Function to count number of
// pairs EVEN bitwise AND
static int findevenPair(int []A,
                        int N)
{
    int i, j;
 
    // variable for counting even pairs
    int evenPair = 0;
 
    // find all pairs
    for (i = 0; i < N; i++) 
    {
        for (j = i + 1; j < N; j++)
        {
 
            // find AND operation
            // to check evenpair
            if ((A[i] & A[j]) % 2 == 0)
                evenPair++;
        }
    }
 
    // return number of even pair
    return evenPair;
}
 
// Driver Code
public static void main (String[] args)
{
    int []a = { 5, 1, 3, 2 };
    int n = a.length;
     
    System.out.println(findevenPair(a, n));
}
}
 
// This code is contributed by anuj_67..

Python3

# Python 3 program to count pair with
# bitwise-AND as even number
 
# Function to count number of 
# pairs EVEN bitwise AND
def findevenPair(A, N):
 
    # variable for counting even pairs
    evenPair = 0
 
    # find all pairs
    for i in range(0, N):
        for j in range(i + 1, N):
             
            # find AND operation to 
            # check evenpair
            if ((A[i] & A[j]) % 2 == 0):
                evenPair += 1
 
    # return number of even pair
    return evenPair
 
# Driver Code
a = [ 5, 1, 3, 2 ]
n = len(a)
 
print(findevenPair(a, n))
 
# This code is contributed 
# by PrinciRaj1992

C#

// C# program to count pair with
// bitwise-AND as even number
using System;
 
class GFG
{
 
// Function to count number of
// pairs EVEN bitwise AND
static int findevenPair(int []A,
                        int N)
{
    int i, j;
 
    // variable for counting even pairs
    int evenPair = 0;
 
    // find all pairs
    for (i = 0; i < N; i++) 
    {
        for (j = i + 1; j < N; j++)
        {
 
            // find AND operation
            // to check evenpair
            if ((A[i] & A[j]) % 2 == 0)
                evenPair++;
        }
    }
 
    // return number of even pair
    return evenPair;
}
 
// Driver Code
public static void Main ()
{
    int []a = { 5, 1, 3, 2 };
    int n = a.Length;
     
    Console.WriteLine(findevenPair(a, n));
}
}
 
// This code is contributed by anuj_67..

PHP

<?php
// PHP program to count pair with
// bitwise-AND as even number
 
// Function to count number of 
// pairs EVEN bitwise AND
function findevenPair($A, $N)
{
 
    // variable for counting even pairs
    $evenPair = 0;
     
    // find all pairs
    for ($i = 0; $i < $N; $i++)
    {
        for ($j = $i + 1; $j < $N; $j++)
        {
            // find AND operation
            // to check evenpair
            if (($A[$i] & $A[$j]) % 2 == 0)
                $evenPair++;
        }
    }
 
    // return number of even pair
    return $evenPair;
}
 
// Driver Code
$a = array(5, 1, 3, 2 );
$n = sizeof($a);
echo findevenPair($a, $n);
 
// This code is contributed by akt_mit
?>

Javascript

<script>
// Javascript program to count pair with
// bitwise-AND as even number
 
// Function to count number of pairs EVEN bitwise AND
function findevenPair(A, N)
{
    let i, j;
 
    // variable for counting even pairs
    let evenPair = 0;
 
    // find all pairs
    for (i = 0; i < N; i++) {
        for (j = i + 1; j < N; j++) {
 
            // find AND operation
            // to check evenpair
            if ((A[i] & A[j]) % 2 == 0)
                evenPair++;
        }
    }
 
    // return number of even pair
    return evenPair;
}
 
// Driver Code
 
let a = [ 5, 1, 3, 2 ];
let n = a.length;
 
document.write(findevenPair(a, n));
 
// This code is contributed by souravmahato348.
</script>

Output

Time Complexity: O(N²)
Auxiliary Space: O(1)

An Efficient Approach will be to observe that the bitwise AND of two numbers will be even if atleast one of the two numbers is even. So, count all the odd numbers in the array say count. Now, number of pairs with both odd elements will be count*(count-1)/2.

Therefore, number of elements with atleast one even element will be:

Total Pairs - Count of pair with both odd elements

Below is the implementation of the above approach:

C++

// C++ program to count pair with
// bitwise-AND as even number
 
#include <iostream>
using namespace std;
 
// Function to count number of pairs
// with EVEN bitwise AND
int findevenPair(int A[], int N)
{
    int count = 0;
 
    // count odd numbers
    for (int i = 0; i < N; i++)
        if (A[i] % 2 != 0)
            count++;
 
    // count odd pairs
    int oddCount = count * (count - 1) / 2;
 
    // return number of even pair
    return (N * (N - 1) / 2) - oddCount;
}
 
// Driver Code
int main()
{
 
    int a[] = { 5, 1, 3, 2 };
    int n = sizeof(a) / sizeof(a[0]);
 
    cout << findevenPair(a, n) << endl;
 
    return 0;
}

C

// C program to count pair with
// bitwise-AND as even number
#include <stdio.h>
 
// Function to count number of pairs
// with EVEN bitwise AND
int findevenPair(int A[], int N)
{
    int count = 0;
 
    // count odd numbers
    for (int i = 0; i < N; i++)
        if (A[i] % 2 != 0)
            count++;
 
    // count odd pairs
    int oddCount = count * (count - 1) / 2;
 
    // return number of even pair
    return (N * (N - 1) / 2) - oddCount;
}
 
// Driver Code
int main()
{
 
    int a[] = { 5, 1, 3, 2 };
    int n = sizeof(a) / sizeof(a[0]);
     
    printf("%d\n",findevenPair(a, n));
 
    return 0;
}
 
// This code is contributed by kothavvsaakash.

Java

// Java program to count pair with
// bitwise-AND as even number
 
 
import java.io.*;
 
class GFG {
  
 
 
// Function to count number of pairs
// with EVEN bitwise AND
static int findevenPair(int A[], int N)
{
    int count = 0;
 
    // count odd numbers
    for (int i = 0; i < N; i++)
        if (A[i] % 2 != 0)
            count++;
 
    // count odd pairs
    int oddCount = count * (count - 1) / 2;
 
    // return number of even pair
    return (N * (N - 1) / 2) - oddCount;
}
 
// Driver Code
 
    public static void main (String[] args) {
            int a[] = { 5, 1, 3, 2 };
    int n =a.length;
 
    System.out.print( findevenPair(a, n));
    }
}
// This code is contributed by anuj_67..

Python3

# Python 3 program to count pair with
# bitwise-AND as even number
 
# Function to count number of pairs
# with EVEN bitwise AND
def findevenPair(A, N):
    count = 0
 
    # count odd numbers
    for i in range(0, N):
        if (A[i] % 2 != 0):
            count += 1
 
    # count odd pairs
    oddCount = count * (count - 1) / 2
 
    # return number of even pair
    return (int)((N * (N - 1) / 2) - oddCount)
 
# Driver Code
a = [5, 1, 3, 2 ]
n = len(a)
 
print(findevenPair(a, n))
 
# This code is contributed 
# by PrinciRaj1992 

C#

// C# program to count pair with
// bitwise-AND as even number
 
using System;
 
public class GFG{
     
// Function to count number of pairs
// with EVEN bitwise AND
static int findevenPair(int []A, int N)
{
    int count = 0;
 
    // count odd numbers
    for (int i = 0; i < N; i++)
        if (A[i] % 2 != 0)
            count++;
 
    // count odd pairs
    int oddCount = count * (count - 1) / 2;
 
    // return number of even pair
    return (N * (N - 1) / 2) - oddCount;
}
 
// Driver Code
 
    static public void Main (){
    int []a = { 5, 1, 3, 2 };
    int n =a.Length;
    Console.WriteLine( findevenPair(a, n));
    }
}
// This code is contributed by ajit

PHP

<?php
// PHP program to count pair with
// bitwise-AND as even number
 
// Function to count number of pairs
// with EVEN bitwise AND
function findevenPair($A, $N)
{
    $count = 0;
 
    // count odd numbers
    for ($i = 0; $i < $N; $i++)
        if ($A[$i] % 2 != 0)
            $count++;
 
    // count odd pairs
    $oddCount = $count * ($count - 1) / 2;
 
    // return number of even pair
    return ($N * ($N - 1) / 2) - $oddCount;
}
 
// Driver Code
$a = array(5, 1, 3, 2);
$n = sizeof($a);
 
echo findevenPair($a, $n) . "\n";
 
// This code is contributed 
// by Akanksha Rai
?>

Javascript

<script>
 
// Javascript program to count pair with
// bitwise-AND as even number
 
// Function to count number of pairs
// with EVEN bitwise AND
function findevenPair(A, N)
{
    let count = 0;
 
    // Count odd numbers
    for(let i = 0; i < N; i++)
        if (A[i] % 2 != 0)
            count++;
 
    // Count odd pairs
    let oddCount = parseInt((count * 
                            (count - 1)) / 2);
 
    // Return number of even pair
    return parseInt((N * (N - 1)) / 2) - oddCount;
}
 
// Driver Code
let a = [ 5, 1, 3, 2 ];
let n = a.length;
 
document.write(findevenPair(a, n));
 
// This code is contributed by subhammahato348
 
</script>

Output

Time Complexity: O(N)
Auxiliary Space: O(1)

Another Method: we observe that bitwise AND of any two numbers will be even if any one of them will be even, no matter what is another number, if one number is even bitwise ANd of that number with any number will be even.

so if our array has an even number count as evenNumCount then the total no. of pair whose bitwise AND is even will be

(n-1)*evenNumCount - (evenNumCount*(evenNumCount-1))/2

here second term (evenNumCount*(evenNumCount-1))/2 will be subtracted from actual ans as we counted (even, even) pair twice.

C++

// C++ program to count pair with bitwise-AND as even number
#include <iostream>
using namespace std;
 
// Function to count number of pairs with EVEN bitwise AND
int findevenPair(int A[], int N)
{
    int evenNumCount = 0;
 
    // count odd numbers
    for (int i = 0; i < N; i++)
        if (A[i] % 2 == 0)
            evenNumCount++;
 
    // return number of even pair
    return (N - 1) * evenNumCount - ((evenNumCount * (evenNumCount - 1)) / 2);
}
 
// Driver Code
int main()
{
 
    int a[] = { 5, 1, 3, 2 };
    int n = sizeof(a) / sizeof(a[0]);
 
    cout << findevenPair(a, n) << endl;
 
    return 0;
}

Java

// Java program to count pair with bitwise-AND as even
// number
import java.util.*;
 
public class Main {
    // Function to count number of pairs with EVEN bitwise
    // AND
    static int findevenPair(int A[], int N)
    {
        int evenNumCount = 0;
 
        // count odd numbers
        for (int i = 0; i < N; i++)
            if (A[i] % 2 == 0)
                evenNumCount++;
 
        // return number of even pair
        return (N - 1) * evenNumCount
            - ((evenNumCount * (evenNumCount - 1)) / 2);
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int a[] = { 5, 1, 3, 2 };
        int n = a.length;
 
        System.out.println(findevenPair(a, n));
    }
}

C#

using System;
 
public class MainClass {
    // Function to count number of pairs with EVEN bitwise
    // AND
    public static int FindevenPair(int[] A, int N)
    {
        int evenNumCount = 0;
        // count odd numbers
        for (int i = 0; i < N; i++)
            if (A[i] % 2 == 0)
                evenNumCount++;
 
        // return number of even pair
        return (N - 1) * evenNumCount
            - ((evenNumCount * (evenNumCount - 1)) / 2);
    }
 
    // Driver Code
    public static void Main()
    {
 
        int[] a = { 5, 1, 3, 2 };
        int n = a.Length;
 
        Console.WriteLine(FindevenPair(a, n));
    }
}

Javascript

// JavaScript program to count pair with bitwise-AND as even number
function findevenPair(A, N) {
    let evenNumCount = 0;
     
    // count odd numbers
    for (let i = 0; i < N; i++) {
        if (A[i] % 2 === 0) {
            evenNumCount++;
        }
    }
     
    // return number of even pair
    return (N - 1) * evenNumCount - ((evenNumCount * (evenNumCount - 1)) / 2);
}
 
// Driver Code
const a = [5, 1, 3, 2];
const n = a.length;
 
console.log(findevenPair(a, n));

Python3

# Function to count number of pairs with EVEN bitwise AND
def findevenPair(A, N):
    evenNumCount = 0
 
    # count odd numbers
    for i in range(N):
        if A[i] % 2 == 0:
            evenNumCount += 1
 
    # return number of even pair
    return (N - 1) * evenNumCount - ((evenNumCount * (evenNumCount - 1)) // 2)
 
# Driver Code
a = [5, 1, 3, 2]
n = len(a)
 
print(findevenPair(a, n))

Output

Time Complexity: O(N)
Auxiliary Space: O(1)

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Count pairs with Bitwise-AND as even number

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