Maximum bitwise OR value of subsequence of length K

Given an array arr[] of N positive integers and a number K, the task is to find the maximum value of bitwise OR of the subsequence of size K.

Examples:

Input: arr[] = {2, 5, 3, 6, 11, 13}, k = 3 
Output: 15 
Explanation:
The sub-sequence will maximum OR value is 2, 11, 13.

Input: arr[] = {5, 9, 7, 19}, k = 3 
Output: 31 
Explanation:
The maximum value of bitwise OR of the subsequence of size K = 3 is 31. 

Naive Approach: The naive approach is to generate all the subsequence of length K and find the Bitwise OR value of all subsequences. The maximum among all of them will be the answer.

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

Efficient Approach: To optimize the above method try to implement the Greedy Approach. Below are the steps: 

1. Initialize an integer array bit[] of size 32 with all value equal to 0.
2. Now iterate for each index of bit[] array from 31 to 0, and check if the i^th value of bit array is 0 then iterate in the given array and find an element which contributes maximum 1 to our bit array after taking it.
3. Take that element and change the bit array correspondingly, also decrease k each time by 1 if k > 0. Otherwise break out from the loop.
4. Now convert the bit[] array into a decimal number to get final answer.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <iostream>
using namespace std;

// Function to convert bit array to
// decimal number
int build_num(int bit[])
{
    int ans = 0;

    for (int i = 0; i < 32; i++)
        if (bit[i])
            ans += (1 << i);

    // Return the final result
    return ans;
}

// Function to find the maximum Bitwise
// OR value of subsequence of length K
int maximumOR(int arr[], int n, int k)
{
    // Initialize bit array of
    // size 32 with all value as 0
    int bit[32] = { 0 };

    // Iterate for each index of bit[]
    // array from 31 to 0, and check if
    // the ith value of bit array is 0
    for (int i = 31; i >= 0; i--) {

        if (bit[i] == 0 && k > 0) {
            int temp = build_num(bit);
            int temp1 = temp;
            int val = -1;

            for (int j = 0; j < n; j++) {

                // Check for maximum
                // contributing element
                if (temp1 < (temp | arr[j])) {
                    temp1 = temp | arr[j];
                    val = arr[j];
                }
            }

            // Update the bit array
            // if max_contributing
            // element is found
            if (val != -1) {

                // Decrement the value of K
                k--;
                for (int j = 0; j < 32; j++) {
                    if (val & (1 << j))
                        bit[j]++;
                }
            }
        }
    }

    // Return the result
    return build_num(bit);
}

// Driver Code
int main()
{
    // Given array arr[]
    int arr[] = { 5, 9, 7, 19 };

    // Length of subsequence
    int k = 3;
    int n = sizeof arr / sizeof arr[0];

    // Function Call
    cout << maximumOR(arr, n, k);
    return 0;
}

Java

// Java program for the above approach
class GFG{

// Function to convert bit array to
// decimal number
static int build_num(int []bit)
{
    int ans = 0;

    for(int i = 0; i < 32; i++)
       if (bit[i] == 1)
           ans += (1 << i);
           ans += 32;

    // Return the final result
    return ans;
}

// Function to find the maximum Bitwise
// OR value of subsequence of length K
static int maximumOR(int []arr, int n, int k)
{

    // Initialize bit array of
    // size 32 with all value as 0
    int bit[] = new int[32];

    // Iterate for each index of bit[]
    // array from 31 to 0, and check if
    // the ith value of bit array is 0
    for(int i = 31; i >= 0; i--)
    {
       if (bit[i] == 0 && k > 0)
       {
           int temp = build_num(bit);
           int temp1 = temp;
           int val = -1;

           for(int j = 0; j < n; j++)
           {

              // Check for maximum
              // contributing element
              if (temp1 < (temp | arr[j]))
              {
                  temp1 = temp | arr[j];
                  val = arr[j];
              }
           }

           // Update the bit array
           // if max_contributing
           // element is found
           if (val != -1)
           {

               // Decrement the value of K
               k--;
               for(int j = 0; j < 32; j++)
               {
                  bit[j]++;
               }
           }
       }
    }

    // Return the result
    return build_num(bit);
}

// Driver Code
public static void main(String[] args)
{

    // Given array arr[]
    int arr[] = { 5, 9, 7, 19 };

    // Length of subsequence
    int k = 3;
    int n = arr.length;

    // Function call
    System.out.println(maximumOR(arr, n, k));
}
}

// This code is contributed by rock_cool

Python3

# Python3 program to implement
# above approach

# Function to convert bit array to
# decimal number
def build_num(bit):

    ans = 0
    for i in range(0, 32):
        if (bit[i]):
            ans += (1 << i)

    # Return the final result
    return ans;

# Function to find the maximum Bitwise
# OR value of subsequence of length K
def maximumOR(arr, n, k):

    # Initialize bit array of
    # size 32 with all value as 0
    bit = [0] * 32

    # Iterate for each index of bit[]
    # array from 31 to 0, and check if
    # the ith value of bit array is 0
    for i in range(31, -1, -1):
        if (bit[i] == 0 and k > 0):
            temp = build_num(bit)
            temp1 = temp
            val = -1

            for j in range(0, n):

                # Check for maximum
                # contributing element
                if (temp1 < (temp | arr[j])):
                    temp1 = temp | arr[j]
                    val = arr[j]

            # Update the bit array
            # if max_contributing
            # element is found
            if (val != -1):

                # Decrement the value of K
                k -= 1
                for j in range(0, 32):
                    if (val & (1 << j)):
                        bit[j] += 1

    # Return the result
    return build_num(bit)

# Driver Code

# Given array arr[]
arr = [ 5, 9, 7, 19 ]

# Length of subsequence
k = 3;
n = len(arr)

# Function call
print(maximumOR(arr, n, k))

# This code is contributed by sanjoy_62

C#

// C# program for the above approach
using System;
class GFG{

// Function to convert bit array to
// decimal number
static int build_num(int []bit)
{
    int ans = 0;

    for(int i = 0; i < 32; i++)
       if (bit[i] == 1)
           ans += (1 << i);
           ans += 32;

    // Return the final result
    return ans;
}

// Function to find the maximum Bitwise
// OR value of subsequence of length K
static int maximumOR(int []arr, int n, int k)
{

    // Initialize bit array of
    // size 32 with all value as 0
    int []bit = new int[32];

    // Iterate for each index of bit[]
    // array from 31 to 0, and check if
    // the ith value of bit array is 0
    for(int i = 31; i >= 0; i--)
    {
       if (bit[i] == 0 && k > 0)
       {
           int temp = build_num(bit);
           int temp1 = temp;
           int val = -1;

           for(int j = 0; j < n; j++)
           {

              // Check for maximum
              // contributing element
              if (temp1 < (temp | arr[j]))
              {
                  temp1 = temp | arr[j];
                  val = arr[j];
              }
           }

           // Update the bit array
           // if max_contributing
           // element is found
           if (val != -1)
           {

               // Decrement the value of K
               k--;
               for(int j = 0; j < 32; j++)
               {
                  bit[j]++;
               }
           }
       }
    }

    // Return the result
    return build_num(bit);
}

// Driver Code
public static void Main()
{

    // Given array arr[]
    int []arr = { 5, 9, 7, 19 };

    // Length of subsequence
    int k = 3;
    int n = arr.Length;

    // Function call
    Console.Write(maximumOR(arr, n, k));
}
}

// This code is contributed by Code_Mech

Javascript

<script>

// Javascript program for the above approach

// Function to convert bit array to
// decimal number
function build_num(bit)
{
    let ans = 0;

    for(let i = 0; i < 32; i++)
        if (bit[i] > 0)
            ans += (1 << i);

    // Return the final result
    return ans;
}

// Function to find the maximum Bitwise
// OR value of subsequence of length K
function maximumOR(arr, n, k)
{

    // Initialize bit array of
    // size 32 with all value as 0
    let bit = new Array(32);
    bit.fill(0);

    // Iterate for each index of bit[]
    // array from 31 to 0, and check if
    // the ith value of bit array is 0
    for(let i = 31; i >= 0; i--)
    {
        if (bit[i] == 0 && k > 0)
        {
            let temp = build_num(bit);
            let temp1 = temp;
            let val = -1;

            for(let j = 0; j < n; j++)
            {

                // Check for maximum
                // contributing element
                if (temp1 < (temp | arr[j]))
                {
                    temp1 = temp | arr[j];
                    val = arr[j];
                }
            }

            // Update the bit array
            // if max_contributing
            // element is found
            if (val != -1)
            {

                // Decrement the value of K
                k--;
                for(let j = 0; j < 32; j++)
                {
                    if ((val & (1 << j)) > 0)
                        bit[j]++;
                }
            }
        }
    }

    // Return the result
    return build_num(bit);
}

// Driver code

// Given array arr[]
let arr = [ 5, 9, 7, 19 ];

// Length of subsequence
let k = 3;
let n = arr.length;

// Function Call
document.write(maximumOR(arr, n, k));

// This code is contributed by divyeshrabadiya07

</script>

Output:

31

Time Complexity: O(N*log N)
Auxiliary Space: O(1)
Similar article: Maximum Bitwise AND value of subsequence of length K