Find smallest value of K such that bitwise AND of numbers in range [N, N-K] is 0

Given an integer N, the task is to find the smallest number K such that bitwise AND of all the numbers in range [N, N-K] is 0, i.e. N & (N – 1) & (N – 2) &… (N – K) = 0.

Examples:

Input: N = 17

Output: 2

Explanation:

17&16 = 16

16&15 = 0

Since, we need to find the smallest K, So we stop here.

K = N – 15 = 17 – 15 = 2

Input: N = 4

Output: 1

Explanation:

4&3 = 0

Since, we need to find the smallest k, So we stop here.

Naive Approach: The simplest approach to solve the problem is to start with given number and take bitwise AND with one number less than the current number till the cumulative bitwise AND is not equal to 0. Follow the steps below to solve this problem:

• Declare a variable cummAnd that stores the commutative bitwise AND and initialize it with n.
• Declare a variable i and initialize it with n – 1.
• Run a loop while cummAnd not equal to 0:
  • cummAnd = cummAnd & i.
  • Decrement i by 1.
• Finally, print n – i.

Below is the implementation of the following approach:

C++

// C++ program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0

#include <iostream>
using namespace std;

// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
int findSmallestNumK(int n)
{
    int cummAnd = n;

    int i = n - 1;
    // Since, we need the largest no,
    // we start from n itself, till 0
    while (cummAnd != 0) {

        cummAnd = cummAnd & i;
        if (cummAnd == 0) {
            return i;
        }

        i--;
    }

    return -1;
}

// Driver Code
int main()
{

    int N = 17;
    int lastNum = findSmallestNumK(N);
    int K = lastNum == -1 ? lastNum : N - lastNum;
    cout << K << "\n";
    return 0;
}

Java

// Java program for the above approach
import java.io.*;

class GFG
{

    // Function is to find the largest no which gives the
    // sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
    public static int findSmallestNumK(int n)
    {
        int cummAnd = n;
        int i = n - 1;

        // Since, we need the largest no,
        // we start from n itself, till 0
        while (cummAnd != 0) {

            cummAnd = cummAnd & i;
            if (cummAnd == 0) {
                return i;
            }

            i--;
        }

        return -1;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 17;
        int lastNum = findSmallestNumK(N);
        int K = lastNum == -1 ? lastNum : N - lastNum;
        System.out.println(K);

    }
}

// This code is contributed by Potta Lokesh

Python3

# Python3 program to find smallest value of K
# such that bitwise AND of numbers
# in range [N, N-K] is 0

# Function is to find the largest no which gives the
# sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
def findSmallestNumK(n):

    cummAnd = n
    i = n - 1

    # Since, we need the largest no,
    # we start from n itself, till 0
    while (cummAnd != 0):
        cummAnd = cummAnd & i

        if (cummAnd == 0):
            return i

        i -= 1

    return -1

# Driver Code
if __name__ == '__main__':

    N = 17
    lastNum = findSmallestNumK(N);
    K = lastNum if lastNum == -1 else N - lastNum

    print(K)

# This code is contributed by ipg2016107

C#

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

class GFG{

// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
public static int findSmallestNumK(int n)
{
    int cummAnd = n;
    int i = n - 1;

    // Since, we need the largest no,
    // we start from n itself, till 0
    while (cummAnd != 0)
    {
        cummAnd = cummAnd & i;
        if (cummAnd == 0)
        {
            return i;
        }
        i--;
    }
    return -1;
}

// Driver code
static void Main()
{
    int N = 17;
    int lastNum = findSmallestNumK(N);
    int K = lastNum == -1 ? lastNum : N - lastNum;

    Console.WriteLine(K);
}
}

// This code is contributed by abhinavjain194

Javascript

<script>
       // JavaScript program for the above approach

       // Function is to find the largest no which gives the
       // sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
       function findSmallestNumK(n)
       {
           let cummAnd = n;

           let i = n - 1;

           // Since, we need the largest no,
           // we start from n itself, till 0
           while (cummAnd != 0) {

               cummAnd = cummAnd & i;
               if (cummAnd == 0) {
                   return i;
               }

               i--;
           }

           return -1;
       }

       // Driver Code
       let N = 17;
       let lastNum = findSmallestNumK(N);
       let K = lastNum == -1 ? lastNum : N - lastNum;
       document.write(K + "<br>");

 // This code is contributed by Potta Lokesh
   </script>

Output

2

Time Complexity: O(N)

Auxiliary Space: O(1)

Efficient Approach: This problem can be solved by using the properties of bitwise AND operator i.e If both the bits are set then only the result will be non-zero. So, we have to find the largest power of 2, which is less than or equal to N(let say X). Follow the steps below to solve this problem:

• The log2(n) function gives the power of 2, which is equal to n.
• Since, its return type is double. So we use floor function to get the largest power of 2, which is less than or equal to n and store it into X.
• X = (1 << X) – 1.
• Finally, print N – X.

Below is the implementation of the following approach:

C++

// C++ program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0

#include <bits/stdc++.h>
using namespace std;

// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
int findSmallestNumK(int n)
{

    // find largest power of 2
    // less than or equal to n
    int larPowOfTwo = floor(log2(n));

    larPowOfTwo = 1 << larPowOfTwo;

    return larPowOfTwo - 1;
}

// Driver Code
int main()
{

    int N = 17;
    int lastNum = findSmallestNumK(N);
    int K = lastNum == -1 ? lastNum : N - lastNum;
    cout << K << "\n";
    return 0;
}

C

// C program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0

#include <math.h> //for log2
#include <stdio.h>

// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
int findSmallestNumK(int n)
{

  // find largest power of 2
  // less than or equal to n
  int larPowOfTwo = floor(log2(n));

  larPowOfTwo = 1 << larPowOfTwo;

  return larPowOfTwo - 1;
}

// Driver Code
int main()
{

  int N = 17;
  int lastNum = findSmallestNumK(N);
  int K = lastNum == -1 ? lastNum : N - lastNum;
  printf("%d\n", K);
  return 0;
}

// This code is contributed by phalashi.

Java

// Java program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0
import java.io.*;

class GFG {

    // Function is to find the largest no which gives the
    // sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
    static int findSmallestNumK(int n)
    {

        // find largest power of 2
        // less than or equal to n
        int larPowOfTwo
            = (int)Math.floor(Math.log(n) / Math.log(2));

        larPowOfTwo = 1 << larPowOfTwo;

        return larPowOfTwo - 1;
    }

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

        int N = 17;
        int lastNum = findSmallestNumK(N);
        int K = lastNum == -1 ? lastNum : N - lastNum;
        System.out.println(K);
    }
}

// This code is contributed by rishavmahato348.

Python3

# Python program to find smallest value of K
# such that bitwise AND of numbers
# in range [N, N-K] is 0
# Function is to find the largest no which gives the
# sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
import math

def findSmallestNumK( n):


    # find largest power of 2
    # less than or equal to n
    larPowOfTwo = math.floor(math.log2(n))

    larPowOfTwo = 1 << larPowOfTwo

    return larPowOfTwo - 1


# Driver Code
N = 17
lastNum = findSmallestNumK(N)
K = lastNum if (lastNum == -1 ) else N - lastNum
print(K)

# this code is contributed by shivanisinghss2110

C#

// C# program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0
using System;

class GFG{

// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
static int findSmallestNumK(int n)
{

    // Find largest power of 2
    // less than or equal to n
    int larPowOfTwo = (int)Math.Floor(Math.Log(n) /
                                      Math.Log(2));

    larPowOfTwo = 1 << larPowOfTwo;

    return larPowOfTwo - 1;
}

// Driver Code
public static void Main()
{
    int N = 17;
    int lastNum = findSmallestNumK(N);
    int K = lastNum == -1 ? lastNum : N - lastNum;

    Console.Write(K);
}
}

// This code is contributed by subhammahato348

Javascript

<script>

// JavaScript program to find smallest value of K
// such that bitwise AND of numbers
// in range [N, N-K] is 0
// Function is to find the largest no which gives the
// sequence n & (n - 1) & (n - 2)&.....&(n - k) = 0.
function findSmallestNumK(n)
    {

        // find largest power of 2
        // less than or equal to n
        var larPowOfTwo
            = Math.floor(Math.log(n) / Math.log(2));

        larPowOfTwo = 1 << larPowOfTwo;

        return larPowOfTwo - 1;
    }

// Driver Code
    var N = 17;
    var lastNum = findSmallestNumK(N);
    var K = lastNum == -1 ? lastNum : N - lastNum;
    document.write(K);

// This code is contributed by shivanisinghss2110

</script>

Output

2

Time Complexity: O(log N)

Space Complexity: O(1)