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Maximum Bitwise AND value of subsequence of length K
  • Difficulty Level : Basic
  • Last Updated : 19 Aug, 2019

Given an array a of size N and an integer K. The task is to find the maximum bitwise and value of elements of any subsequence of length K.

Note: a[i] <= 109

Examples:

Input: a[] = {10, 20, 15, 4, 14}, K = 4
Output: 4
{20, 15, 4, 14} is the subsequence with highest ‘&’ value.

Input: a[] = {255, 127, 31, 5, 24, 37, 15}, K = 5
Output: 8



Naive Approach: A naive approach is to recursively find the bitwise and value of all subsequences of length K and the maximum among all of them will be the answer.

Efficient Approach: An efficient approach is to solve it using bit properties. Below are the steps to solve the problem:

  • Iterate from the left(initially left = 31 as 232 > 109 ) till we find > K numbers in the vector temp (initially temp = arr) whose i-th bit is set. Update the new set of numbers to temp array
  • If we do not get > K numbers, the & value of any K elements in the temp array will be the maximum & value possible.
  • Repeat Step-1 with left re-initialized as first-bit + 1.
  • till K elements are possible

Below is the implementation of the above approach:

C++

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// C++ program to find the sum of
// the addition of all possible subsets.
#include <bits/stdc++.h>
using namespace std;
  
// Function to perform step-1
vector<int> findSubset(vector<int>& temp, int& last, int k)
{
    vector<int> ans;
  
    // Iterate from left till 0
    // till we get a bit set of K numbers
    for (int i = last; i >= 0; i--) {
        int cnt = 0;
  
        // Count the numbers whose
        // i-th bit is set
        for (auto it : temp) {
            int bit = it & (1 << i);
            if (bit > 0)
                cnt++;
        }
  
        // If the array has numbers>=k
        // whose i-th bit is set
        if (cnt >= k) {
            for (auto it : temp) {
                int bit = it & (1 << i);
                if (bit > 0)
                    ans.push_back(it);
            }
  
            // Update last
            last = i - 1;
  
            // Return the new set of numbers
            return ans;
        }
    }
  
    return ans;
}
  
// Function to find the maximum '&' value
// of K elements in subsequence
int findMaxiumAnd(int a[], int n, int k)
{
    int last = 31;
    // Temporary arrays
    vector<int> temp1, temp2;
  
    // Initially temp = arr
    for (int i = 0; i < n; i++) {
        temp2.push_back(a[i]);
    }
  
    // Iterate till we have >=K elements
    while ((int)temp2.size() >= k) {
  
        // Temp array
        temp1 = temp2;
  
        // Find new temp array if
        // K elements are there
        temp2 = findSubset(temp2, last, k);
    }
  
    // Find the & value
    int ans = temp1[0];
    for (int i = 0; i < k; i++)
        ans = ans & temp1[i];
  
    return ans;
}
  
// Driver Code
int main()
{
    int a[] = { 255, 127, 31, 5, 24, 37, 15 };
    int n = sizeof(a) / sizeof(a[0]);
    int k = 4;
  
    cout << findMaxiumAnd(a, n, k);
}

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Java

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// Java program to find the sum of
// the addition of all possible subsets.
import java.util.*;
class GFG
{
static int last;
  
// Function to perform step-1
static Vector<Integer> 
       findSubset(Vector<Integer> temp, int k)
{
    Vector<Integer> ans = new Vector<Integer>();
  
    // Iterate from left till 0
    // till we get a bit set of K numbers
    for (int i = last; i >= 0; i--)
    {
        int cnt = 0;
  
        // Count the numbers whose
        // i-th bit is set
        for (Integer it : temp)
        {
            int bit = it & (1 << i);
            if (bit > 0)
                cnt++;
        }
  
        // If the array has numbers>=k
        // whose i-th bit is set
        if (cnt >= k) 
        {
            for (Integer it : temp) 
            {
                int bit = it & (1 << i);
                if (bit > 0)
                    ans.add(it);
            }
  
            // Update last
            last = i - 1;
  
            // Return the new set of numbers
            return ans;
        }
    }
    return ans;
}
  
// Function to find the maximum '&' value
// of K elements in subsequence
static int findMaxiumAnd(int a[], int n, int k)
{
    last = 31;
      
    // Temporary arrays
    Vector<Integer> temp1 = new Vector<Integer>();
    Vector<Integer> temp2 = new Vector<Integer>();;
  
    // Initially temp = arr
    for (int i = 0; i < n; i++) 
    {
        temp2.add(a[i]);
    }
  
    // Iterate till we have >=K elements
    while ((int)temp2.size() >= k) 
    {
  
        // Temp array
        temp1 = temp2;
  
        // Find new temp array if
        // K elements are there
        temp2 = findSubset(temp2, k);
    }
  
    // Find the & value
    int ans = temp1.get(0);
    for (int i = 0; i < k; i++)
        ans = ans & temp1.get(i);
  
    return ans;
}
  
// Driver Code
public static void main(String[] args)
{
    int a[] = { 255, 127, 31, 5, 24, 37, 15 };
    int n = a.length;
    int k = 4;
  
    System.out.println(findMaxiumAnd(a, n, k));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 program to find the sum of
# the addition of all possible subsets.
last = 31
  
# Function to perform step-1
def findSubset(temp, k):
    global last
    ans = []
  
    # Iterate from left till 0
    # till we get a bit set of K numbers
    for i in range(last, -1, -1):
        cnt = 0
  
        # Count the numbers whose
        # i-th bit is set
        for it in temp:
            bit = it & (1 << i)
            if (bit > 0):
                cnt += 1
  
        # If the array has numbers>=k
        # whose i-th bit is set
        if (cnt >= k):
            for it in temp:
                bit = it & (1 << i)
                if (bit > 0):
                    ans.append(it)
  
            # Update last
            last = i - 1
  
            # Return the new set of numbers
            return ans
  
    return ans
  
# Function to find the maximum '&' value
# of K elements in subsequence
def findMaxiumAnd(a, n, k):
    global last
  
    # Temporary arrays
    temp1, temp2, = [], []
  
    # Initially temp = arr
    for i in range(n):
        temp2.append(a[i])
  
    # Iterate till we have >=K elements
    while len(temp2) >= k:
  
        # Temp array
        temp1 = temp2
  
        # Find new temp array if
        # K elements are there
        temp2 = findSubset(temp2, k)
  
    # Find the & value
    ans = temp1[0]
    for i in range(k):
        ans = ans & temp1[i]
  
    return ans
  
# Driver Code
a = [255, 127, 31, 5, 24, 37, 15]
n = len(a)
k = 4
  
print(findMaxiumAnd(a, n, k))
  
# This code is contributed by Mohit Kumar

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C#

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// C# program to find the sum of
// the addition of all possible subsets.
using System;
using System.Collections.Generic;
  
class GFG
{
static int last;
  
// Function to perform step-1
static List<int>findSubset(List<int> temp, int k)
{
    List<int> ans = new List<int>();
  
    // Iterate from left till 0
    // till we get a bit set of K numbers
    for (int i = last; i >= 0; i--)
    {
        int cnt = 0;
  
        // Count the numbers whose
        // i-th bit is set
        foreach (int it in temp)
        {
            int bit = it & (1 << i);
            if (bit > 0)
                cnt++;
        }
  
        // If the array has numbers>=k
        // whose i-th bit is set
        if (cnt >= k) 
        {
            foreach (int it in temp) 
            {
                int bit = it & (1 << i);
                if (bit > 0)
                    ans.Add(it);
            }
  
            // Update last
            last = i - 1;
  
            // Return the new set of numbers
            return ans;
        }
    }
    return ans;
}
  
// Function to find the maximum '&' value
// of K elements in subsequence
static int findMaxiumAnd(int []a, int n, int k)
{
    last = 31;
      
    // Temporary arrays
    List<int> temp1 = new List<int>();
    List<int> temp2 = new List<int>();;
  
    // Initially temp = arr
    for (int i = 0; i < n; i++) 
    {
        temp2.Add(a[i]);
    }
  
    // Iterate till we have >=K elements
    while ((int)temp2.Count >= k) 
    {
  
        // Temp array
        temp1 = temp2;
  
        // Find new temp array if
        // K elements are there
        temp2 = findSubset(temp2, k);
    }
  
    // Find the & value
    int ans = temp1[0];
    for (int i = 0; i < k; i++)
        ans = ans & temp1[i];
  
    return ans;
}
  
// Driver Code
public static void Main(String[] args)
{
    int []a = { 255, 127, 31, 5, 24, 37, 15 };
    int n = a.Length;
    int k = 4;
  
    Console.WriteLine(findMaxiumAnd(a, n, k));
}
}
  
// This code is contributed by Rajput-Ji

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Output:

24

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