Count distinct Bitwise OR of all Subarrays

Given an array A of non-negative integers, where 0 \leq A[i] \leq 10^{9}                 . The task is to count number of distinct possible results obtained by taking the bitwise OR of all the elements in all possible Subarrays.

Examples: 

Input: A = [1, 2]
Output: 3
Explanation: The possible subarrays are [1], [2], [1, 2].
These Bitwise OR of subarrays are 1, 2, 3.
There are 3 distinct values, so the answer is 3.

Input: A = [1, 2, 4]
Output: 6
Explanation: The possible distinct values are 1, 2, 3, 4, 6, and 7.

Approach: The Naive approach is to generate all possible subarrays and take bitwise OR of all elements in the subarray. Store each result in set and return length of the set. 

Efficient Approach: We can make the above approach better. The Naive approach is to calculate all possible result where, res(i, j) = A[i] | A[i+1] | ... | A[j]. However we can speed this up by taking note of the fact that res(i, j+1) = res(i, j) | A[j+1]. At the kth step, say we have all of the res(i, k) in some set pre. Then we can find the next pre set (for k -> k+1) by using res(i, k+1) = res(i, k) | A[k+1].

However, the number of unique values in this set pre is atmost 32, since the list res(k, k), res(k-1, k), res(k-2, k), ... is monotone increasing, and any subsequent values that are different from previous must have more 1's in it's binary representation which can have maximum of 32 ones.

Below is the implementation of above approach. 

#include <iostream>
#include <vector>
#include <set>

using namespace std;

// function to return count of distinct bitwise OR
int subarrayBitwiseOR(vector<int>& A) {

  // res contains distinct values
  set<int> res;

  set<int> pre;
  pre.insert(0);

  for (int x : A) {
    set<int> temp;
    for (int y : pre) {
      temp.insert(x | y);
    }
    temp.insert(x);
    pre = temp;
    res.insert(pre.begin(), pre.end());
  }

  return res.size();
}

// Driver program
int main() {
  vector<int> A = {1, 2, 4};

  // print required answer
  cout << subarrayBitwiseOR(A) << endl;

  return 0;
}

import java.util.*;

class Main {

    // function to return count of distinct bitwise OR
    public static int subarrayBitwiseOR(int[] A) {

        // res contains distinct values
        Set<Integer> res = new HashSet<>();

        Set<Integer> pre = new HashSet<>();
        pre.add(0);

        for (int x : A) {
            Set<Integer> temp = new HashSet<>();
            for (int y : pre) {
                temp.add(x | y);
            }
            temp.add(x);
            pre = temp;
            res.addAll(pre);
        }

        return res.size();
    }

    // Driver program
    public static void main(String[] args) {
        int[] A = {1, 2, 4};

        // print required answer
        System.out.println(subarrayBitwiseOR(A));
    }
}

# Python implementation of the above approach

# function to return count of distinct bitwise OR
def subarrayBitwiseOR(A):

    # res contains distinct values
    res = set()

    pre = {0}

    for x in A:
        pre = {x | y for y in pre} | {x}
        res |= pre

    return len(res)


# Driver program
A = [1, 2, 4]

# print required answer
print(subarrayBitwiseOR(A))

# This code is written by
# Sanjit_Prasad

using System;
using System.Collections.Generic;

class MainClass {
    // function to return count of distinct bitwise OR
    public static int subarrayBitwiseOR(int[] A) {
        // res contains distinct values
        HashSet<int> res = new HashSet<int>();

        HashSet<int> pre = new HashSet<int>();
        pre.Add(0);

        foreach (int x in A) {
            HashSet<int> temp = new HashSet<int>();
            foreach (int y in pre) {
                temp.Add(x | y);
            }
            temp.Add(x);
            pre = temp;
            res.UnionWith(pre);
        }

        return res.Count;
    }

    // Driver program
    public static void Main(string[] args) {
        int[] A = {1, 2, 4};

        // print required answer
        Console.WriteLine(subarrayBitwiseOR(A));
    }
}

// JavaScript implementation of the above approach

// function to return count of distinct bitwise OR
function subarrayBitwiseOR(A) {

    // res contains distinct values
    let res = new Set();

    let pre = new Set([0]);

    for (let x of A) {
        let temp = new Set();
        for (let y of pre) {
            temp.add(x | y);
        }
        temp.add(x);
        res = new Set([...res, ...temp]);
        pre = new Set(temp);
    }

    return res.size;
}

// Driver program
let A = [1, 2, 4];

// print required answer
console.log(subarrayBitwiseOR(A));

6

Time Complexity: O(N*log(K)), where N is the length of A, and K is the maximum size of elements in A.

C++ implementation of the above approach.

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;

// function to calculate count of 
// distinct bitwise OR of all 
// subarrays.
int distintBitwiseOR(int arr[], int n)
{
    unordered_set<int> ans, prev;

    for (int i = 0; i < n; i++) {
        unordered_set<int> ne;

        for (auto x : prev)
            ne.insert(arr[i] | x);
        ne.insert(arr[i]);

        for (auto x : ne)
            ans.insert(x);

        prev = ne;
    }

    return ans.size();
}

// Driver Code
int main()
{
    int n = 3;
    int arr[] = { 1, 2, 4 };

    cout << distintBitwiseOR(arr, n);

    return 0;
}

// Java implementation of the above approach
import java.io.*;
import java.util.*;

class GFG
{
  
  // function to calculate count of
// distinct bitwise OR of all
// subarrays.
 static int distintBitwiseOR(int arr[], int n)
  {
   
     HashSet<Integer>ans = new HashSet<>();
        HashSet<Integer>prev = new HashSet<>();
        for(int i = 0; i < n; i++)
        {
            HashSet<Integer>ne = new HashSet<>();
            ne.add(arr[i]);
            for(int x :prev)
            {
                ne.add(arr[i]|x);
            }
            for(int x :ne)
            {
              ans.add(x);
            }
           
            prev = ne;
        }
        return ans.size();
    }
 
    // Driver code
    public static void main (String[] args) {
         int n = 3;
         int arr[] = { 1, 2, 4 };
         System.out.println(distintBitwiseOR(arr, n));
         
    }
}

// This code is contributed by iramkhalid24.

# Python implementation of the above approach

# function to calculate count of
# distinct bitwise OR of all
# subarrays.
def distintBitwiseOR(arr,n):

    ans,prev = set(), set()

    for i in range(n):
        ne = set()

        for x in prev:
            ne.add(arr[i] | x)
        ne.add(arr[i])

        for x in ne:
            ans.add(x)

        prev = ne

    return len(ans)

# Driver Code
n = 3
arr = [ 1, 2, 4 ]

print(distintBitwiseOR(arr, n))

# This code is written by Shinjanpatra

// C# implementation of the above approach
using System;
using System.Collections.Generic;

public class GFG
{

  // function to calculate count of
  // distinct bitwise OR of all
  // subarrays.
  static int distintBitwiseOR(int[] arr, int n)
  {

    HashSet<int> ans = new HashSet<int>();
    HashSet<int> prev = new HashSet<int>();

    for (int i = 0; i < n; i++) {
      HashSet<int> ne = new HashSet<int>();
      ne.Add(arr[i]);
      foreach(var x in prev) { ne.Add(arr[i] | x); }

      foreach(var x in ne) { ans.Add(x); }

      prev = ne;
    }
    return ans.Count;
  }

  // Driver code
  public static void Main(string[] args)
  {
    int n = 3;
    int[] arr = { 1, 2, 4 };

    // Function call
    Console.WriteLine(distintBitwiseOR(arr, n));
  }
}

// This code is contributed by phasing17

<script>

// JavaScript implementation of the above approach

// function to calculate count of
// distinct bitwise OR of all
// subarrays.
function distintBitwiseOR(arr,n)
{
    let ans = new Set(), prev = new Set();

    for (let i = 0; i < n; i++) {
        let ne = new Set();

        for (let x of prev)
            ne.add(arr[i] | x);
        ne.add(arr[i]);

        for (let x of ne)
            ans.add(x);

        prev = ne;
    }

    return ans.size;
}

// Driver Code

let n = 3;
let arr = [ 1, 2, 4 ];

document.write(distintBitwiseOR(arr, n));

// This code is written by Shinjanpatra

</script>

6

Time Complexity: O(N*K) where N is the length of A, and K is the maximum size of elements in A.

Auxiliary Space: O(K)