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Maximize sum of LSBs of Bitwise OR of all possible N/2 pairs from given Array

  • Last Updated : 04 Oct, 2021

Given an array arr[] consisting of N positive integers, where N is even, the task is to form N/2 pairs such that the sum of the Least Significant Bit of Bitwise OR of all these pairs is maximum.

Examples:

Input: arr[] = {1, 2, 3, 4, 5, 6, 7, 8}
Output: 8
Explanation:
On forming the pairs as (8, 4),(6, 2),(1, 3),(5, 7), the Bitwise OR of the pair is given by:
8 OR 4 = 12 and LSB = 4
6 OR 2 = 6 and LSB = 2
1 OR 3 = 3 and LSB = 1
5 OR 7 = 7 and LSB = 1
The sum of all the LSB is 4 + 2 + 1 + 1 = 8, which is maximum possible sum.

Input: arr[] = {1, 2, 3, 4, 5}
Output: 3

Approach: The given problem can be solved by finding the LSB of each array element arr[i] and store them in another array, say lsb_arr[] and sort this array in descending order.  Now, storing just the LSB of each array element is sufficient because in the answer, it is only requires to consider the LSB. So, only the LSB’s can be used for the Bitwise OR operation. Now, consider each pair (i, i + 1) and add the minimum of these two to the result. Follow the steps below to solve the given problem:

Below is the implementation of the above approach:

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function top get LSB value of v
int chk(int n)
{
 
    // Binary conversion
    vector<int> v;
 
    while (n != 0) {
        v.push_back(n % 2);
        n = n / 2;
    }
 
    for (int i = 0; i < v.size(); i++) {
        if (v[i] == 1) {
            return pow(2, i);
        }
    }
 
    return 0;
}
 
// Function to find the sum of LSBs of
// all possible pairs of the given array
void sumOfLSB(int arr[], int N)
{
 
    // Stores the LSB of array elements
    vector<int> lsb_arr;
    for (int i = 0; i < N; i++) {
 
        // Storing the LSB values
        lsb_arr.push_back(chk(arr[i]));
    }
    // Sort the array lab_arr[]
    sort(lsb_arr.begin(), lsb_arr.end(), greater<int>());
 
    int ans = 0;
 
    for (int i = 0; i < N - 1; i += 2) {
 
        // Taking pairwise sum to get
        // the maximum sum of LSB
        ans += (lsb_arr[i + 1]);
    }
 
    // Print the result
    cout << (ans);
}
 
// Driver Code
int main()
{
    int N = 5;
    int arr[] = { 1, 2, 3, 4, 5 };
 
    // Function Call
    sumOfLSB(arr, N);
}
 
// This code is contributed by Potta Lokesh

Java




// Java program for the above approach
import java.util.*;
class GFG
{
 
// Function top get LSB value of v
static int chk(int n)
{
 
    // Binary conversion
    Vector<Integer> v = new Vector<Integer>();
 
    while (n != 0) {
        v.add(n % 2);
        n = n / 2;
    }
 
    for (int i = 0; i < v.size(); i++) {
        if (v.get(i) == 1) {
            return (int) Math.pow(2, i);
        }
    }
 
    return 0;
}
 
// Function to find the sum of LSBs of
// all possible pairs of the given array
static void sumOfLSB(int arr[], int N)
{
 
    // Stores the LSB of array elements
    Vector<Integer> lsb_arr = new Vector<Integer>() ;
    for (int i = 0; i < N; i++) {
 
        // Storing the LSB values
        lsb_arr.add(chk(arr[i]));
    }
   
    // Sort the array lab_arr[]
    Collections.sort(lsb_arr);
 
    int ans = 0;
 
    for (int i = 0; i < N - 1; i += 2) {
 
        // Taking pairwise sum to get
        // the maximum sum of LSB
        ans += (lsb_arr.get(i + 1));
    }
 
    // Print the result
    System.out.print(ans);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 5;
    int arr[] = { 1, 2, 3, 4, 5 };
 
    // Function Call
    sumOfLSB(arr, N);
}
}
 
// This code contributed by shikhasingrajput

Python3




# Python program for the above approach
 
# Function top get LSB value of v
def chk(v):
 
    # Binary conversion
    v = list(bin(v)[2:])
    v.reverse()
     
    if('1' in v):
        v = v.index('1')
        return (2**v)
    else:
        return 0
 
# Function to find the sum of LSBs of
# all possible pairs of the given array
def sumOfLSB(arr, N):
 
    # Stores the LSB of array elements
    lsb_arr = []
    for i in range(N):
 
        # Storing the LSB values
        lsb_arr.append(chk(arr[i]))
 
    # Sort the array lab_arr[]
    lsb_arr.sort(reverse=True)
 
    ans = 0
 
    for i in range(0, N-1, 2):
 
        # Taking pairwise sum to get
        # the maximum sum of LSB
        ans += (lsb_arr[i+1])
 
    # Print the result
    print(ans)
 
# Driver Code
N = 5
arr = [1, 2, 3, 4, 5]
 
# Function Call
sumOfLSB(arr, N)

Javascript




<script>
    // JavaScript program for the above approach
 
    // Function top get LSB value of v
    const chk = (n) => {
 
        // Binary conversion
        let v = [];
 
        while (n != 0) {
            v.push(n % 2);
            n = parseInt(n / 2);
        }
 
        for (let i = 0; i < v.length; i++) {
            if (v[i] == 1) {
                return Math.pow(2, i);
            }
        }
 
        return 0;
    }
 
    // Function to find the sum of LSBs of
    // all possible pairs of the given array
    const sumOfLSB = (arr, N) => {
 
        // Stores the LSB of array elements
        let lsb_arr = [];
        for (let i = 0; i < N; i++) {
 
            // Storing the LSB values
            lsb_arr.push(chk(arr[i]));
        }
        // Sort the array lab_arr[]
         
        lsb_arr.sort((a, b) => a - b)
        let ans = 0;
 
        for (let i = 0; i < N - 1; i += 2) {
 
            // Taking pairwise sum to get
            // the maximum sum of LSB
            ans += (lsb_arr[i + 1]);
        }
 
        // Print the result
        document.write(ans);
    }
 
    // Driver Code
    let N = 5;
    let arr = [1, 2, 3, 4, 5];
 
    // Function Call
    sumOfLSB(arr, N);
     
    // This code is contributed by rakeshsahni
</script>

C#




// C# program for the above approach
using System;
using System.Collections.Generic;
 
public class GFG
{
   
// Function top get LSB value of v
static int chk(int n)
{
 
    // Binary conversion
    List<int> v = new List<int>();
 
    while (n != 0) {
        v.Add(n % 2);
        n = n / 2;
    }
     
      int j = 0;
    foreach(int i in v) {
        if (i == 1) {
            return (int) Math.Pow(2.0, (double)j);
        }
          j++;
    }
 
    return 0;
}
 
// Function to find the sum of LSBs of
// all possible pairs of the given array
static void sumOfLSB(int[] arr, int N)
{
 
    // Stores the LSB of array elements
      int[] lsb_arr = new int[N];
     
    for (int i = 0; i < N; i++) {
 
        // Storing the LSB values
        lsb_arr[i] = chk(arr[i]);
    }
   
    // Sort the array lab_arr[]
    Array.Sort(lsb_arr);
 
    int ans = 0;
 
    for (int i = 0; i < N - 1; i += 2) {
 
        // Taking pairwise sum to get
        // the maximum sum of LSB
        ans += (lsb_arr[i + 1]);
    }
 
    // Print the result
    Console.WriteLine(ans);
}
 
// Driver Code
static public void Main (){
 
    int N = 5;
    int[] arr = { 1, 2, 3, 4, 5 };
 
    // Function Call
    sumOfLSB(arr, N);
}
}
 
// This code is contributed by Dharanendra L V.
Output: 
3

 

Time Complexity: O(N*log N)
Auxiliary Space: O(N)


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