# Number of subsets with same AND, OR and XOR values in an Array

• Difficulty Level : Hard
• Last Updated : 18 Nov, 2021

Given an array arr[] of size N consisting of non-negative integers, the task is to find the number of non-empty subsets of the array such that the bitwise AND, bitwise OR and bitwise XOR values of the subsequence are equal to each other.

Note: Since the answer can be large, mod it with 1000000007.

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Examples:

Input: arr[] = [1, 3, 2, 1, 2, 1]
Output:
Explanation:
One of the subsequences with equal bitwise Xor, bitwise or and bitwise AND is {1, 1, 1}.

Input: arr = [2, 3, 4, 5]
Output:

Naive Approach The naive approach for this problem is to traverse through all the subsets of the array in an iterative manner, and for each subset find the bitwise AND, OR and XOR value and check whether they are equal or not. Finally, return the count of such equal subsets.

Below is the implementation of the above approach:

## C++

 `// C++ implementation to find the number``// of subsets with equal bitwise AND,``// OR and XOR values` `#include ``using` `namespace` `std;``const` `int` `mod = 1000000007;` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``int` `countSubsets(``int` `a[], ``int` `n)``{``    ``int` `answer = 0;` `    ``// Traverse through all the subsets``    ``for` `(``int` `i = 0; i < (1 << n); i++) {` `        ``int` `bitwiseAND = -1;``        ``int` `bitwiseOR = 0;``        ``int` `bitwiseXOR = 0;` `        ``// Finding the subsets with the bits``        ``// of 'i' which are set``        ``for` `(``int` `j = 0; j < n; j++) {` `            ``// Computing the bitwise AND``            ``if` `(i & (1 << j)) {``                ``if` `(bitwiseAND == -1)``                    ``bitwiseAND = a[j];``                ``else``                    ``bitwiseAND &= a[j];` `                ``// Computing the bitwise OR``                ``bitwiseOR |= a[j];` `                ``// Computing the bitwise XOR``                ``bitwiseXOR ^= a[j];``            ``}``        ``}` `        ``// Comparing all the three values``        ``if` `(bitwiseAND == bitwiseOR``            ``&& bitwiseOR == bitwiseXOR)``            ``answer = (answer + 1) % mod;``    ``}``    ``return` `answer;``}` `// Driver code``int` `main()``{``    ``int` `N = 6;``    ``int` `A[N] = { 1, 3, 2, 1, 2, 1 };` `    ``cout << countSubsets(A, N);` `    ``return` `0;``}`

## Java

 `// Java implementation to find the number``// of subsets with equal bitwise AND,``// OR and XOR values``import` `java.io.*;` `class` `GFG {``static` `int` `mod = ``1000000007``;` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``static` `int` `countSubsets(``int` `a[], ``int` `n)``{``    ``int` `answer = ``0``;` `    ``// Traverse through all the subsets``    ``for` `(``int` `i = ``0``; i < (``1` `<< n); i++) {` `        ``int` `bitwiseAND = -``1``;``        ``int` `bitwiseOR = ``0``;``        ``int` `bitwiseXOR = ``0``;` `        ``// Finding the subsets with the bits``        ``// of 'i' which are set``        ``for` `(``int` `j = ``0``; j < n; j++) {` `            ``// Computing the bitwise AND``            ``if` `((i & (``1` `<< j)) == ``0``) {``                ``if` `(bitwiseAND == -``1``)``                    ``bitwiseAND = a[j];``                ``else``                    ``bitwiseAND &= a[j];` `                ``// Computing the bitwise OR``                ``bitwiseOR |= a[j];` `                ``// Computing the bitwise XOR``                ``bitwiseXOR ^= a[j];``            ``}``        ``}` `        ``// Comparing all the three values``        ``if` `(bitwiseAND == bitwiseOR``            ``&& bitwiseOR == bitwiseXOR)``            ``answer = (answer + ``1``) % mod;``    ``}``    ``return` `answer;``}` `// Driver Code``public` `static` `void` `main (String[] args)``{``    ``int` `N = ``6``;``    ``int` `A[] = { ``1``, ``3``, ``2``, ``1``, ``2``, ``1` `};` `    ``System.out.print(countSubsets(A, N));``}``}` `// This code is contributed by shivanisinghss2110`

## Python3

 `# Python3 implementation to find the number``# of subsets with equal bitwise AND,``# OR and XOR values` `mod ``=` `1000000007``;` `# Function to find the number of``# subsets with equal bitwise AND,``# OR and XOR values``def` `countSubsets(a, n) :` `    ``answer ``=` `0``;` `    ``# Traverse through all the subsets``    ``for` `i ``in` `range``(``1` `<< n) :` `        ``bitwiseAND ``=` `-``1``;``        ``bitwiseOR ``=` `0``;``        ``bitwiseXOR ``=` `0``;` `        ``# Finding the subsets with the bits``        ``# of 'i' which are set``        ``for` `j ``in` `range``(n) :` `            ``# Computing the bitwise AND``            ``if` `(i & (``1` `<< j)) :``                ``if` `(bitwiseAND ``=``=` `-``1``) :``                    ``bitwiseAND ``=` `a[j];``                ``else` `:``                    ``bitwiseAND &``=` `a[j];` `                ``# Computing the bitwise OR``                ``bitwiseOR |``=` `a[j];` `                ``# Computing the bitwise XOR``                ``bitwiseXOR ^``=` `a[j];` `        ``# Comparing all the three values``        ``if` `(bitwiseAND ``=``=` `bitwiseOR ``and` `bitwiseOR ``=``=` `bitwiseXOR) :``            ``answer ``=` `(answer ``+` `1``) ``%` `mod;``    ` `    ``return` `answer;` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:``    ` `    ``N ``=` `6``;``    ``A ``=` `[ ``1``, ``3``, ``2``, ``1``, ``2``, ``1` `];` `    ``print``(countSubsets(A, N));` `# This code is contributed by AnkitRai01`

## C#

 `// C# implementation to find the number``// of subsets with equal bitwise AND,``// OR and XOR values``using` `System;` `class` `GFG {``static` `int` `mod = 1000000007;`` ` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``static` `int` `countSubsets(``int` `[]a, ``int` `n)``{``    ``int` `answer = 0;`` ` `    ``// Traverse through all the subsets``    ``for` `(``int` `i = 0; i < (1 << n); i++) {`` ` `        ``int` `bitwiseAND = -1;``        ``int` `bitwiseOR = 0;``        ``int` `bitwiseXOR = 0;`` ` `        ``// Finding the subsets with the bits``        ``// of 'i' which are set``        ``for` `(``int` `j = 0; j < n; j++) {`` ` `            ``// Computing the bitwise AND``            ``if` `((i & (1 << j)) == 0) {``                ``if` `(bitwiseAND == -1)``                    ``bitwiseAND = a[j];``                ``else``                    ``bitwiseAND &= a[j];`` ` `                ``// Computing the bitwise OR``                ``bitwiseOR |= a[j];`` ` `                ``// Computing the bitwise XOR``                ``bitwiseXOR ^= a[j];``            ``}``        ``}`` ` `        ``// Comparing all the three values``        ``if` `(bitwiseAND == bitwiseOR``            ``&& bitwiseOR == bitwiseXOR)``            ``answer = (answer + 1) % mod;``    ``}``    ``return` `answer;``}`` ` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `N = 6;``    ``int` `[]A = { 1, 3, 2, 1, 2, 1 };`` ` `    ``Console.Write(countSubsets(A, N));``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output:
`7`

Time Complexity: O(N * 2N) where N is the size of the array.

Auxiliary Space: O(1)

Efficient Approach: The efficient approach lies behind the property of bitwise operations.

• Using the property of bitwise AND, and bitwise OR we can say that if a & b == a | b, then a is equal to b. So if the AND and OR values of the subset are equal then all the elements of the subset are identical (say x). So the AND and OR values are equal to x.
• Since all the values of subsequence are equal to each other, two case arise for XOR value:
1. Subset size is odd: The XOR value equals to x.
2. Subset size is even: The XOR values equals to 0.
• Therefore, from the above observation, we can come to the conclusion that all odd-sized subsequences/subsets with equal elements follow the property.
• In addition to this if all the elements of the subset are 0, then then the subset will follow the property (irrespective of subset size). So all the subsets which have only 0 as their element will be added to the answer.
• If frequency of some element is K, the then number of odd sized subsets it can form is 2K – 1, and the total non-empty subsets it can form is 2K – 1.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the number``// of subsets with equal bitwise``// AND, OR and XOR values` `#include ``using` `namespace` `std;``const` `int` `mod = 1000000007;` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``int` `countSubsets(``int` `a[], ``int` `n)``{``    ``int` `answer = 0;` `    ``// Precompute the modded powers``    ``// of two for subset counting``    ``int` `powerOfTwo;``    ``powerOfTwo = 1;` `    ``// Loop to iterate and find the modded``    ``// powers of two for subset counting``    ``for` `(``int` `i = 1; i < 100005; i++)``        ``powerOfTwo[i]``            ``= (powerOfTwo[i - 1] * 2)``              ``% mod;` `    ``// Map to store the frequency of``    ``// each element``    ``unordered_map<``int``, ``int``> frequency;` `    ``// Loop to compute the frequency``    ``for` `(``int` `i = 0; i < n; i++)``        ``frequency[a[i]]++;` `    ``// For every element > 0, the number of``    ``// subsets formed using this element only``    ``// is equal to 2 ^ (frequency[element]-1).``    ``// And for 0, we have to find all``    ``// the subsets, so 2^(frequency[element]) -1``    ``for` `(``auto` `el : frequency) {` `        ``// If element is greater than 0``        ``if` `(el.first != 0)``            ``answer``                ``= (answer % mod``                   ``+ powerOfTwo[el.second - 1])``                  ``% mod;` `        ``else``            ``answer``                ``= (answer % mod``                   ``+ powerOfTwo[el.second]``                   ``- 1 + mod)``                  ``% mod;``    ``}``    ``return` `answer;``}` `// Driver code``int` `main()``{``    ``int` `N = 6;``    ``int` `A[N] = { 1, 3, 2, 1, 2, 1 };` `    ``cout << countSubsets(A, N);` `    ``return` `0;``}`

## Java

 `// Java program to find the number``// of subsets with equal bitwise``// AND, OR and XOR values`` `  `import` `java.util.*;` `class` `GFG{``static` `int` `mod = ``1000000007``;`` ` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``static` `int` `countSubsets(``int` `a[], ``int` `n)``{``    ``int` `answer = ``0``;`` ` `    ``// Precompute the modded powers``    ``// of two for subset counting``    ``int` `[]powerOfTwo = ``new` `int``[``100005``];``    ``powerOfTwo[``0``] = ``1``;`` ` `    ``// Loop to iterate and find the modded``    ``// powers of two for subset counting``    ``for` `(``int` `i = ``1``; i < ``100005``; i++)``        ``powerOfTwo[i]``            ``= (powerOfTwo[i - ``1``] * ``2``)``              ``% mod;`` ` `    ``// Map to store the frequency of``    ``// each element``    ``HashMap frequency = ``new` `HashMap();`` ` `    ``// Loop to compute the frequency``    ``for` `(``int` `i = ``0``; i < n; i++)``        ``if``(frequency.containsKey(a[i])){``            ``frequency.put(a[i], frequency.get(a[i])+``1``);``        ``}``else``{``            ``frequency.put(a[i], ``1``);``    ``}`` ` `    ``// For every element > 0, the number of``    ``// subsets formed using this element only``    ``// is equal to 2 ^ (frequency[element]-1).``    ``// And for 0, we have to find all``    ``// the subsets, so 2^(frequency[element]) -1``    ``for` `(Map.Entry el : frequency.entrySet()) {`` ` `        ``// If element is greater than 0``        ``if` `(el.getKey() != ``0``)``            ``answer``                ``= (answer % mod``                   ``+ powerOfTwo[el.getValue() - ``1``])``                  ``% mod;`` ` `        ``else``            ``answer``                ``= (answer % mod``                   ``+ powerOfTwo[el.getValue()]``                   ``- ``1` `+ mod)``                  ``% mod;``    ``}``    ``return` `answer;``}`` ` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `N = ``6``;``    ``int` `A[] = { ``1``, ``3``, ``2``, ``1``, ``2``, ``1` `};`` ` `    ``System.out.print(countSubsets(A, N));`` ` `}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 program to find the number``# of subsets with equal bitwise``# AND, OR and XOR values``mod ``=` `1000000007` `# Function to find the number of``# subsets with equal bitwise AND,``# OR and XOR values``def` `countSubsets(a, n):``    ` `    ``answer ``=` `0``    ` `    ``# Precompute the modded powers``    ``# of two for subset counting``    ``powerOfTwo ``=` `[``0` `for` `x ``in` `range``(``100005``)]``    ``powerOfTwo[``0``] ``=` `1` `    ``# Loop to iterate and find the modded``    ``# powers of two for subset counting``    ``for` `i ``in` `range``(``1``, ``100005``):``        ``powerOfTwo[i] ``=` `(powerOfTwo[i ``-` `1``] ``*` `2``) ``%` `mod``        ` `    ``# Map to store the frequency of``    ``# each element``    ``frequency ``=` `{}``    ` `    ``# Loop to compute the frequency``    ``for` `i ``in` `range``(``0``, n):``        ``if` `a[i] ``in` `frequency:``            ``frequency[a[i]] ``+``=` `1``        ``else``:``            ``frequency[a[i]] ``=` `1``            ` `    ``# For every element > 0, the number of``    ``# subsets formed using this element only``    ``# is equal to 2 ^ (frequency[element]-1).``    ``# And for 0, we have to find all``    ``# the subsets, so 2^(frequency[element]) -1``    ``for` `key, value ``in` `frequency.items():``        ` `        ``# If element is greater than 0``        ``if` `(key !``=` `0``):``            ``answer ``=` `(answer ``%` `mod ``+``                  ``powerOfTwo[value ``-` `1``]) ``%` `mod``        ``else``:``            ``answer ``=` `(answer ``%` `mod ``+``                 ``powerOfTwo[value] ``-` `1` `+` `mod)``%` `mod``                 ` `    ``return` `answer` `# Driver code``N ``=` `6``A ``=` `[ ``1``, ``3``, ``2``, ``1``, ``2``, ``1` `]` `print``(countSubsets(A, N))` `# This code is contributed by amreshkumar3`

## C#

 `// C# program to find the number``// of subsets with equal bitwise``// AND, OR and XOR values``using` `System;``using` `System.Collections.Generic;` `class` `GFG{``    ` `static` `int` `mod = 1000000007;` `// Function to find the number of``// subsets with equal bitwise AND,``// OR and XOR values``static` `int` `countSubsets(``int` `[]a, ``int` `n)``{``    ``int` `answer = 0;` `    ``// Precompute the modded powers``    ``// of two for subset counting``    ``int` `[]powerOfTwo = ``new` `int``;``    ``powerOfTwo = 1;` `    ``// Loop to iterate and find the modded``    ``// powers of two for subset counting``    ``for``(``int` `i = 1; i < 100005; i++)``       ``powerOfTwo[i] = (powerOfTwo[i - 1] * 2) % mod;` `    ``// Map to store the frequency``    ``// of each element``    ``Dictionary<``int``, ``int``> frequency = ``new` `Dictionary<``int``, ``int``>();` `    ``// Loop to compute the frequency``    ``for``(``int` `i = 0; i < n; i++)``       ``if``(frequency.ContainsKey(a[i]))``       ``{``           ``frequency[a[i]] = frequency[a[i]] + 1;``       ``}``       ``else``       ``{``           ``frequency.Add(a[i], 1);``       ``}` `    ``// For every element > 0, the number of``    ``// subsets formed using this element only``    ``// is equal to 2 ^ (frequency[element]-1).``    ``// And for 0, we have to find all``    ``// the subsets, so 2^(frequency[element]) -1``    ``foreach` `(KeyValuePair<``int``, ``int``> el ``in` `frequency)``    ``{``        ` `        ``// If element is greater than 0``        ``if` `(el.Key != 0)``            ``answer = (answer % mod +``                      ``powerOfTwo[el.Value - 1]) % mod;``        ``else``            ``answer = (answer % mod +``                      ``powerOfTwo[el.Value] - 1 +``                      ``mod) % mod;``    ``}``    ``return` `answer;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `N = 6;``    ``int` `[]A = { 1, 3, 2, 1, 2, 1 };` `    ``Console.Write(countSubsets(A, N));``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``
Output:
`7`

Time Complexity: O(N), where N is the size of the array.

Auxiliary Space: O(N + 105)

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