Maximize the largest number K such that bitwise and of K till N is 0

Given an integer N, the task is to find the maximum value of K such that N & (N-1) & (N-2) & … & (K) = 0. Here & represents bitwise AND operator.

Example:

Input: N = 5
Output: 3
Explanation: The value of the expression 5 & 4 & 3 = 0. any value greater than 3 (example 4) will not satisfy
our given condition

Input: N =17
Output: 15

Naive approach: The brute force approach is to start from N-1 and perform bitwise AND operation till we get 0.

Below is the implementation of the above approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find maximum value of k
// which makes bitwise AND zero.
int findMaxK(int N)
{
    // Take k = N initially
    int K = N;

    // Start traversing from N-1 till 0
    for (int i = N - 1; i >= 0; i--) {
        K &= i;

        // Whenever we get AND as 0
        // we stop and return
        if (K == 0) {
            return i;
        }
    }
    return 0;
}

// Driver Code
int main()
{
    int N = 5;
    cout << findMaxK(N);
}

Java

// Java program for the above approach

import java.io.*;

class GFG {
  // Function to find maximum value of k
// which makes bitwise AND zero.
static int findMaxK(int N)
{
    // Take k = N initially
    int K = N;

    // Start traversing from N-1 till 0
    for (int i = N - 1; i >= 0; i--) {
        K &= i;

        // Whenever we get AND as 0
        // we stop and return
        if (K == 0) {
            return i;
        }
    }
    return 0;
}
// Driver Code
    public static void main (String[] args) {
       int N = 5;
        System.out.println(findMaxK(N));
    }
}
// This code is contributed by Potta Lokesh

Python3

# Python 3 program for above approach

# Function to find maximum value of k
# which makes bitwise AND zero.
def findMaxK(N):
    # Take k = N initially
    K = N

    # Start traversing from N-1 till 0
    i = N-1
    while(i >= 0):
        K &= i

        # Whenever we get AND as 0
        # we stop and return
        if (K == 0):
            return i
        i -= 1
    return 0

# Driver Code
if __name__ == '__main__':
    N = 5
    print(findMaxK(N))

    # This code is contributed by ipg2016107.

C#

// C# program for the above approach
using System;

class GFG
{

  // Function to find maximum value of k
  // which makes bitwise AND zero.
  static int findMaxK(int N)
  {

    // Take k = N initially
    int K = N;

    // Start traversing from N-1 till 0
    for (int i = N - 1; i >= 0; i--) {
      K &= i;

      // Whenever we get AND as 0
      // we stop and return
      if (K == 0) {
        return i;
      }
    }
    return 0;
  }

  // Driver Code
  public static void Main (String[] args)
  {
    int N = 5;
    Console.Write(findMaxK(N));
  }
}

// This code is contributed by shivanisinghss2110

Javascript

  <script>
        // JavaScript Program to implement
        // the above approach

  // Function to find maximum value of k
// which makes bitwise AND zero.
function findMaxK(N)
{

    // Take k = N initially
    let K = N;

    // Start traversing from N-1 till 0
    for (let i = N - 1; i >= 0; i--) {
        K &= i;

        // Whenever we get AND as 0
        // we stop and return
        if (K == 0) {
            return i;
        }
    }
    return 0;
}

 // Driver Code
      let N = 5;
      document.write(findMaxK(N));

// This code is contributed by sanjoy_62.
    </script>

Output

3

Time complexity: O(N)
Auxiliary Space: O(1)

Efficient approach: By some observation, it can be seen that the answer is always equal to the highest power of 2, which is less than or equal to (N-1). So finally, the answer is always equal to 2^K -1, where K is some value.

Below is the implementation of the above approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find maximum value of k
// which makes bitwise AND zero.
int findMaxK(int N)
{
    // Finding the power less than N
    int p = log2(N);
    return pow(2, p);
}

// Driver Code
int main()
{
    int N = 5;
    cout << findMaxK(N) - 1 << endl;
    return 0;
}

Java

/*package whatever //do not write package name here */
import java.io.*;

class GFG {

    // Function to find maximum value of k
    // which makes bitwise AND zero.
    static int findMaxK(int N)
    {
        // Finding the power less than N
        int p = (int)(Math.log(N) / Math.log(2));
        return (int)Math.pow(2, p);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 5;
        System.out.println(findMaxK(N) - 1);
    }
}

// This code is contributed by maddler.

Python3

import math

# Function to find maximum value of k
# which makes bitwise AND zero.
def findMaxK(N):

    # Finding the power less than N
    p = math.log(N) // math.log(2);
    return int(pow(2, p));

# Driver Code
N = 5;
print(findMaxK(N) - 1);

# This code is contributed by _saurabh_jaiswal

C#

/*package whatever //do not write package name here */
using System;

class GFG
{

    // Function to find maximum value of k
    // which makes bitwise AND zero.
    static int findMaxK(int N)
    {

        // Finding the power less than N
        int p = (int)(Math.Log(N) / Math.Log(2));
        return (int)Math.Pow(2, p);
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int N = 5;
        Console.Write(findMaxK(N) - 1);
    }
}

// This code is contributed by shivanisinghss2110

Javascript

<script>
/*package whatever //do not write package name here */

// Function to find maximum value of k
// which makes bitwise AND zero.
function findMaxK(N)
{
// Finding the power less than N
var p = Math.log(N) / Math.log(2);
return parseInt(Math.pow(2, p));
}

    // Driver Code
var N = 5;
document.write(findMaxK(N) - 1);

// This code is contributed by 29AjayKumar
</script>

Output

3

Time complexity: O(log N)
Auxiliary Space: O(1)