Number of ordered pairs such that (Ai & Aj) = 0

06 Jan, 2022
Given an array A[] of n integers, find out the number of ordered pairs such that Ai&Aj is zero, where 0<=(i,j)<n. Consider (i, j) and (j, i) to be different. 
Constraints: 1<=n<=104 1<=Ai<=104
Examples: 
Input : A[] = {3, 4, 2}
Output : 4
Explanation : The pairs are (3, 4) and (4, 2) which are 
counted as 2 as (4, 3) and (2, 4) are considered different. 

Input : A[]={5, 4, 1, 6}
Output : 4 
Explanation : (4, 1), (1, 4), (6, 1) and (1, 6) are the pairs
Simple approach: A simple approach is to check for all possible pairs and count the number of ordered pairs whose bitwise & returns 0. 
Below is the implementation of the above idea:  
C++
Java
Python3
C#
PHP
JavaScript
// CPP program to calculate the number// of ordered pairs such that their bitwise// and is zero#include <bits/stdc++.h>using namespace std; // Naive function to count the number// of ordered pairs such that their// bitwise and is 0int countPairs(int a[], int n){    int count = 0;     // check for all possible pairs    for (int i = 0; i < n; i++) {        for (int j = i + 1; j < n; j++)            if ((a[i] & a[j]) == 0)                 // add 2 as (i, j) and (j, i) are                // considered different                count += 2;    }     return count;} // Driver Codeint main(){    int a[] = { 3, 4, 2 };    int n = sizeof(a) / sizeof(a[0]);       cout << countPairs(a, n);       return 0;}
// Java program to calculate the number// of ordered pairs such that their bitwise// and is zero class GFG {         // Naive function to count the number    // of ordered pairs such that their    // bitwise and is 0    static int countPairs(int a[], int n)    {        int count = 0;         // check for all possible pairs        for (int i = 0; i < n; i++) {            for (int j = i + 1; j < n; j++)                if ((a[i] & a[j]) == 0)                     // add 2 as (i, j) and (j, i) are                    // considered different                    count += 2;        }         return count;    }     // Driver Code    public static void main(String arg[])    {        int a[] = { 3, 4, 2 };        int n = a.length;        System.out.print(countPairs(a, n));    }} // This code is contributed by Anant Agarwal.
# Python3 program to calculate the number# of ordered pairs such that their# bitwise and is zero # Naive function to count the number# of ordered pairs such that their# bitwise and is 0def countPairs(a, n):    count = 0     # check for all possible pairs    for i in range(0, n):        for j in range(i + 1, n):            if (a[i] & a[j]) == 0:                 # add 2 as (i, j) and (j, i) are                # considered different                count += 2    return count # Driver Codea = [ 3, 4, 2 ]n = len(a)print (countPairs(a, n)) # This code is contributed# by Shreyanshi Arun.
// C# program to calculate the number// of ordered pairs such that their// bitwise and is zerousing System; class GFG {         // Naive function to count the number    // of ordered pairs such that their    // bitwise and is 0    static int countPairs(int []a, int n)    {        int count = 0;         // check for all possible pairs        for (int i = 0; i < n; i++)        {            for (int j = i + 1; j < n; j++)                if ((a[i] & a[j]) == 0)                     // add 2 as (i, j) and (j, i)                    // arev considered different                    count += 2;        }         return count;    }     // Driver Code    public static void Main()    {        int []a = { 3, 4, 2 };        int n = a.Length;        Console.Write(countPairs(a, n));    }} // This code is contributed by nitin mittal.
<?php// PHP program to calculate the number// of ordered pairs such that their// bitwise and is zero // Naive function to count the number// of ordered pairs such that their// bitwise and is 0function countPairs($a, $n){    $count = 0;     // check for all possible pairs    for ($i = 0; $i < $n; $i++)    {        for ($j = $i + 1; $j < $n; $j++)            if (($a[$i] & $a[$j]) == 0)                 // add 2 as (i, j) and (j, i) are                // considered different                $count += 2;    }     return $count;} // Driver Code{    $a = array(3, 4, 2);    $n = sizeof($a) / sizeof($a[0]);    echo countPairs($a, $n);    return 0;} // This code is contributed by nitin mittal
<script>    // JavaScript program to calculate the number    // of ordered pairs such that their bitwise    // and is zero     // Naive function to count the number    // of ordered pairs such that their    // bitwise and is 0    const countPairs = (a, n) => {        let count = 0;         // check for all possible pairs        for (let i = 0; i < n; i++) {            for (let j = i + 1; j < n; j++)                if ((a[i] & a[j]) == 0)                     // add 2 as (i, j) and (j, i) are                    // considered different                    count += 2;        }         return count;    }     // Driver Code     let a = [3, 4, 2];    let n = a.length;    document.write(countPairs(a, n));     // This code is contributed by rakeshsahni </script>  ?>
Output: 
4
Time Complexity: O(n2)
Efficient approach: An efficient approach is to use Sum over Subsets Dynamic Programming method and count the number of ordered pairs. In the SOS DP we find out the pairs whose bitwise & returned 0. Here we need to count the number of pairs. Some key observations are the constraints, the maximum that an array element can be is 104. Calculating the mask up to (1<<15) will give us our answer. Use hashing to count the occurrence of every element. If the last bit is OFF, then relating to SOS dp, we will have a base case since there is only one possibility of OFF bit. 
dp[mask][0] = freq(mask)
If the last bit is set ON, then we will have the base case as:  
dp[mask][0] = freq(mask) + freq(mask^1)
We add freq(mask^1) to add the other possibility of OFF bit. Iterate over N=15 bits, which is the maximum possible.Let’s consider the i-th bit to be 0, then no subset can differ from the mask in the i-th bit as it would mean that the numbers will have a 1 at i-th bit where the mask has a 0 which would mean that it is not a subset of the mask. Thus we conclude that the numbers now differ in the first (i-1) bits only. Hence, 
DP(mask, i) = DP(mask, i-1)
Now the second case, if the i-th bit is 1, it can be divided into two non-intersecting sets. One containing numbers with i-th bit as 1 and differing from mask in the next (i-1) bits. Second containing numbers with ith bit as 0 and differing from mask
(2i) in next (i-1) bits. Hence,
DP(mask, i) = DP(mask, i-1) + DP(mask
2i, i-1).
DP[mask][i] stores the number of subsets of mask which differ from mask only in first i bits. Iterate for all array elements, and for every array element add the number of subsets (dp[ ( ( 1<<N ) – 1 ) ^ a[i] ][ N ]) to the number of pairs. N = maximum number of bits. 
Explanation of addition of dp[ ( ( 1<<N ) – 1 ) ^ a[i] ][N] to the number of pairs: Take an example of A[i] being 5, which is 101 in binary. For better understanding, assume N=3 in this case, therefore, the reverse of 101 will be 010 which on applying bitwise & gives 0. So (1<<3) gives 1000 which on subtraction from 1 gives 111. 111 101 gives 010 which is the reversed bit.So dp[((1<<N)-1)^a[i]][N] will have the number of subsets that returns 0 on applying bitwise & operator.
Below is the implementation of the above idea:  
C++
Java
Python3
C#
// CPP program to calculate the number// of ordered pairs such that their bitwise// and is zero #include <bits/stdc++.h>using namespace std; const int N = 15; // efficient function to count pairslong long countPairs(int a[], int n){    // stores the frequency of each number    unordered_map<int, int> hash;     long long dp[1 << N][N + 1];         memset(dp, 0, sizeof(dp)); // initialize 0 to all     // count the frequency of every element    for (int i = 0; i < n; ++i)        hash[a[i]] += 1;     // iterate for all possible values that a[i] can be    for (long long mask = 0; mask < (1 << N); ++mask) {         // if the last bit is ON        if (mask & 1)            dp[mask][0] = hash[mask] + hash[mask ^ 1];        else // is the last bit is OFF            dp[mask][0] = hash[mask];         // iterate till n        for (int i = 1; i <= N; ++i) {             // if mask's ith bit is set            if (mask & (1 << i))            {                dp[mask][i] = dp[mask][i - 1] +                        dp[mask ^ (1 << i)][i - 1];            }               else // if mask's ith bit is not set                dp[mask][i] = dp[mask][i - 1];        }    }     long long ans = 0;     // iterate for all the array element    // and count the number of pairs    for (int i = 0; i < n; i++)        ans += dp[((1 << N) - 1) ^ a[i]][N];     // return answer    return ans;} // Driver Codeint main(){    int a[] = { 5, 4, 1, 6 };    int n = sizeof(a) / sizeof(a[0]);    cout << countPairs(a, n);    return 0;}
// Java program to calculate// the number of ordered pairs// such that their bitwise// and is zeroimport java.util.*;class GFG{     static int N = 15;   // Efficient function to count pairspublic static int countPairs(int a[],                             int n){  // Stores the frequency of  // each number  HashMap<Integer,          Integer> hash = new HashMap<>();   int dp[][] = new int[1 << N][N + 1];   // Initialize 0 to all     // Count the frequency  // of every element  for (int i = 0; i < n; ++i)  {    if(hash.containsKey(a[i]))    {      hash.replace(a[i],      hash.get(a[i]) + 1);    }    else    {      hash.put(a[i], 1);    }  }   // Iterate for all possible  // values that a[i] can be  for (int mask = 0;           mask < (1 << N); ++mask)  {    // If the last bit is ON    if ((mask & 1) != 0)    {      if(hash.containsKey(mask))      {        dp[mask][0] = hash.get(mask);      }      if(hash.containsKey(mask ^ 1))      {        dp[mask][0] += hash.get(mask ^ 1);      }    }    else    {      // is the last bit is OFF      if(hash.containsKey(mask))      {        dp[mask][0] = hash.get(mask);       }    }     // Iterate till n    for (int i = 1; i <= N; ++i)    {      // If mask's ith bit is set      if ((mask & (1 << i)) != 0)      {        dp[mask][i] = dp[mask][i - 1] +                      dp[mask ^ (1 << i)][i - 1];      }          else      {        // If mask's ith bit is not set        dp[mask][i] = dp[mask][i - 1];      }    }  }   int ans = 0;   // Iterate for all the  // array element and  // count the number of pairs  for (int i = 0; i < n; i++)  {    ans += dp[((1 << N) - 1) ^ a[i]][N];  }   // return answer  return ans;} // Driver code   public static void main(String[] args){  int a[] = {5, 4, 1, 6};  int n = a.length;  System.out.print(countPairs(a, n));}} // This code is contributed by divyeshrabadiya07
# Python program to calculate the number# of ordered pairs such that their bitwise# and is zeroN = 15 # efficient function to count pairsdef countPairs(a, n):     # stores the frequency of each number    Hash = {}         # initialize 0 to all    dp = [[0 for i in range(N + 1)] for j in range(1 << N)]     # count the frequency of every element    for i in range(n):        if a[i] not in Hash:            Hash[a[i]] = 1        else:            Hash[a[i]] += 1     # iterate for all possible values that a[i] can be    mask = 0    while(mask < (1 << N)):        if mask not in Hash:            Hash[mask] = 0         # if the last bit is ON        if(mask & 1):            dp[mask][0] = Hash[mask] + Hash[mask ^ 1]                     else:    # is the last bit is OFF            dp[mask][0] = Hash[mask]         # iterate till n        for i in range(1, N + 1):             # if mask's ith bit is set            if(mask & (1 << i)):                dp[mask][i] = dp[mask][i - 1] + dp[mask ^ (1 << i)][i - 1]                             else:    # if mask's ith bit is not set                dp[mask][i] = dp[mask][i - 1]         mask += 1     ans = 0         # iterate for all the array element    # and count the number of pairs    for i in range(n):        ans += dp[((1 << N) - 1) ^ a[i]][N]             # return answer    return ans # Driver Codea = [5, 4, 1, 6]n = len(a)print(countPairs(a, n)) # This code is contributed by avanitrachhadiya2155
// C# program to calculate// the number of ordered pairs// such that their bitwise// and is zerousing System;using System.Collections.Generic; class GFG {         static int N = 15;        // Efficient function to count pairs    static int countPairs(int[] a, int n)    {      // Stores the frequency of      // each number      Dictionary<int, int> hash = new Dictionary<int, int>();            int[, ] dp = new int[1 << N, N + 1];            // Initialize 0 to all              // Count the frequency      // of every element      for (int i = 0; i < n; ++i)      {        if(hash.ContainsKey(a[i]))        {          hash[a[i]] += 1;        }        else        {          hash.Add(a[i], 1);        }      }            // Iterate for all possible      // values that a[i] can be      for (int mask = 0;               mask < (1 << N); ++mask)      {        // If the last bit is ON        if ((mask & 1) != 0)        {          if(hash.ContainsKey(mask))          {            dp[mask, 0] = hash[mask];          }          if(hash.ContainsKey(mask ^ 1))          {            dp[mask, 0] += hash[mask ^ 1];          }        }        else        {          // is the last bit is OFF          if(hash.ContainsKey(mask))          {            dp[mask, 0] = hash[mask];           }        }              // Iterate till n        for (int i = 1; i <= N; ++i)        {                     // If mask's ith bit is set          if ((mask & (1 << i)) != 0)          {            dp[mask, i] = dp[mask, i - 1] +                          dp[mask ^ (1 << i), i - 1];          }              else          {                         // If mask's ith bit is not set            dp[mask, i] = dp[mask, i - 1];          }        }      }            int ans = 0;            // Iterate for all the      // array element and      // count the number of pairs      for (int i = 0; i < n; i++)      {        ans += dp[((1 << N) - 1) ^ a[i], N];      }            // return answer      return ans;    }      // Driver code  static void Main()  {      int[] a = {5, 4, 1, 6};      int n = a.Length;      Console.WriteLine(countPairs(a, n));  }} // This code is contributed by divyesh072019
Output: 
4
Time Complexity: O(N*2N) where N=15 which is a maximum number of bits possible, since Amax=104.
nitin mittal
ManasChhabra2
shubham_singh
divyeshrabadiya07
divyesh072019
avanitrachhadiya2155
rakeshsahni
surinderdawra388
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