Find square root of a number using Bit Manipulation

Given a non-negative integer N, the task is to find the square root of N using bitwise operations. If the integer is not the perfect square, return largest integer that is smaller than or equal to square root of N i.e., floor(√N).

Examples:

Input: N = 36
Output: 6
Explanation:  The square root of 36 is 6.

Input: N = 19
Output: 4
Explanation:  The square root of 19 lies in between
4 and 5 so floor of the square root is 4.

Approach: To solve the problem using bitwise operators follow the below idea:

Let's assume that square root of N is X i.e., N ≥ X². 
Let's consider binary representation of X = ( b_m, b_m-1, ....., b₂, b₁, b₀ ) where b_i represents the i^th bit in binary representation of X. Since, the value of each bits can either be 1 or 0, we can represent 
X = ( a_m + a_m-1 + . . . + a₂ + a₁ + a₀ ) where a_i = 2ⁱ or a_i = 0.

Consider an approximate solution:
S_j = ( a_m + a_m-1 + . . . + a_j ) and also let S_j = S_j+1 + 2^j. 
If S_j² ≤ X² ≤ N then j^th bit is set and 2^j is part of the answer. Otherwise, it is 0.

Follow the below illustration for a better understanding.

Illustrations:

N = 36  , result = 0
Binary representation of N: 100100
MSB of N is 5.

Initially result = 0, a = 2⁵ = 32

For 5th bit:
    =>(result + a) = (0 + 32) = 32, and 32 * 32 = 1024 is greater than N (36)
    => update a = a/2 = 32/2 = 16, result = 0

For 4th bit:
    => Now, (result + a) = 16, and 16 * 16 = 256 is greater than N (36)
    => update a = a/2 = 16/2 = 8

For 3rd bit:
    => Now, (result + a) = 8, and 8 * 8 = 64 is greater than N (36)
    => update a = a/2 = 8/2 = 4

For 2nd bit:
    => Now, (result + a) = 4, and 4 * 4 = 16 is less than N (36) so add (a) to result 
    => update a = a/2 = 4/2 = 2, result = 4

For 1st bit:
    => Now, (result + a) = (4+2) =6, and 6 * 6 = 36 is equal to N (36) so add (a) to result
    => update a = a/2 = 2/2 = 1, result = 6

So, the final result = 6.

Based on the above observation in each bit position find the contribution of that bit in the answer and add that value to the final answer. Follow the steps mentioned below to implement the above approach:

• Initialize a variable (say result) to store the final answer.
• Start iterating from the MSB of N:
  • If the square of (result + a_i) is at most N then that bit has contribution in the result [where a_i is 2ⁱ].
  • So add the value a_i in result.
• After iteration is over, the value stored at result is the required answer.

Below is the implementation of the above approach:

// C++ code to implement the approach

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

// Function to find square root
int square_root(int N)
{
    // Find MSB (Most significant Bit) of N
    int msb = (int)(log2(N));

    // (a = 2^msb)
    int a = 1 << msb;
    int result = 0;
    while (a != 0) {

        // Check whether the current value
        // of 'a' can be added or not
        if ((result + a) * (result + a) <= N) {
            result += a;
        }

        // (a = a/2)
        a >>= 1;
    }

    // Return the result
    return result;
}

// Driver Code
int main()
{
    int N = 36;

    // Function call
    cout << square_root(N);
    return 0;
}

// C code to implement the approach

#include <math.h>
#include <stdio.h>

// Function to find square root
int square_root(int N)
{
    // Find MSB(Most significant Bit) of N
    int msb = (int)(log2(N));

    // (a = 2^msb)
    int a = 1 << msb;
    int result = 0;
    while (a != 0) {

        // Check whether the current value
        // of 'a' can be added or not
        if ((result + a) * (result + a) <= N) {
            result += a;
        }

        // (a = a/2)
        a >>= 1;
    }

    // Return the result
    return result;
}

// Driver Code
int main()
{
    int N = 36;

    // Function call
    printf("%d\n", square_root(N));
    return 0;
}

// Java code to implement the approach

import java.io.*;

class GFG {

    // Function to find square root
    static int square_root(int N)
    {
        // Find MSB(Most significant Bit) of N
        int msb = (int)(Math.log(N) / Math.log(2));

        // (a = 2^msb)
        int a = 1 << msb;
        int result = 0;
        while (a != 0) {

            // Check whether the current value
            // of 'a' can be added or not
            if ((result + a) * (result + a) <= N) {
                result += a;
            }

            // (a = a/2)
            a >>= 1;
        }

        // Return the result
        return result;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 36;

        // Function call
        System.out.println(square_root(N));
    }
}

# Python code to implement the approach

import math

# Function to find square root
def square_root (N):
  
    # Find MSB(Most significant Bit) of N
    msb = int(math.log(N, 2))
    
    # (a = 2 ^ msb)
    a = 1 << msb
    result = 0
    while a != 0:
      
        # Check whether the current value 
        # of 'a' can be added or not
        if (result + a) * (result + a) <= N :
            result += a
        
        # (a = a / 2)
        a >>= 1
        
    # Return the result
    return result


# Driver Code
if __name__ == '__main__':
    N = 36
    
    # Function call
    print(square_root(N))

// C# code to implement the approach

using System;

public class GFG{
    
    // Function to find square root
    static int square_root(int N)
    {
        // Find MSB(Most significant Bit) of N
        int msb = (int)(Math.Log(N) / Math.Log(2));

        // (a = 2^msb)
        int a = 1 << msb;
        int result = 0;
        while (a != 0) {

            // Check whether the current value
            // of 'a' can be added or not
            if ((result + a) * (result + a) <= N) {
                result += a;
            }

            // (a = a/2)
            a >>= 1;
        }

        // Return the result
        return result;
    }

    // Driver Code
    public static void Main()
    {
        int N = 36;

        // Function call
        Console.WriteLine(square_root(N));
    }
}

// This code is contributed by Ankthon

function square_root(N)
{
    // Find MSB(Most significant Bit) of N
    let msb = parseInt(Math.log2(N));

    // (a = 2^msb)
    let a = 1 << msb;
    let result = 0;
    while (a != 0) {

        // Check whether the current value
        // of 'a' can be added or not
        if ((result + a) * (result + a) <= N) {
            result += a;
        }

        // (a = a/2)
        a >>= 1;
    }

    // Return the result
    return result;
}

let N = 36;

    // Function call
    let ans = square_root(N);
    console.log(ans);

// This code is contributed by akashish.

6

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