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# Java Program to find Square Root of a number using Binary Search

Square Root of a number is an integer value when multiplied by itself, gives the original number. In this article, we are going to write a java program to find the square root of a number using binary search. Finding square root of a number is one of the application of the binary search algorithm. We will discuss in detail how we calculate the square root using binary search in this article.

Examples

Input: 64 Output: 8

As, the square root of 64 is 8, the output is 8.

Input: 16 Output: 4

As, the square root of 16 is 4, the output is 4.

## Binary Search

Binary search is an algorithm used to find an element i.e., key in a sorted array. Binary algorithm works as below −

Let us say that array is ‘arr’. Sort the array in ascending or descending order.

Initialize low = 0 and high = n-1 (n = number of elements) and calculate middle as middle = low + (high-low)/2. If arr[middle] == key then return middle i.e., middle index of array.

Else if the key value is less than the arr[middle] element set the high index as middle index-1 or if the key value is more than middle element set the low index as middle index+1

Continue the binary search until the element that needs to be find is found.

If low is greater than high than return false directly as key is not present in the array ‘arr’.

### Example to find Key using Binary Search

**Problem** − Given a sorted array of integers arr = [1, 3, 5, 7, 9, 11], find the index of the element i.e., key = 7 using binary search.

### Solution

Initialize the low = 0 and the high= 5 (the last index of the array).

The first iteration of the while loop gives us the middle index mid = low+ (high-low)/2

mid = 0+(5-0)/2 = 2

The value of arr[mid] is 5, which is less than the key value 7. So, we update the low= mid+1 = 3.

The second iteration of the while loop gives us the middle index mid = 4 by using the low+ (high-low)/2.

The value of arr[mid] is 9, which is greater than the key value 7. So, we update the high= 3 (mid - 1).

The third iteration of the while loop gives us the middle index mid = 3.

The arr[mid] is 7, which is equal to the key value. So, we return the middle index, which is 3.

Thus, the index of the key = 7 in the given array is 3, which we found using the binary search algorithm.

### Algorithm to find the Square Root using Binary Search

Consider a number ‘n’ and initialise low=0 and right= n (given number).

Find mid value of low and high using mid = low + (high-low)/2.

find the value of mid * mid, if mid * mid == n then return mid value.

If mid value is less than n then low=mid+1 else high =mid-1

Repeat from steps 2 to 4 until we find the value.

### Example 1: Using Binary Search

In this example, we create a custom class ‘BinarySearchSqrt’ and implement the binary search code for finding the square root of a number code in ‘sqrt’ function. Now, create the custom Class object and initialise a variable called ‘number’ with an integer number and using the class object class the ‘sqrt’ function is called and thus the desired output is displayed.

//Java Program to find Square root of a number using Binary Search import java.util.*; class BinarySearchSqrt { public int sqrt(int number) { int low = 0; int high = number; while (low <= high) { int mid = (low + high) / 2; int square = mid * mid; if (square == number) { return mid; } else if (square < number) { low = mid + 1; } else { high = mid - 1; } } return 0; } } public class Main { public static void main(String[] args) { int n = 64; BinarySearchSqrt Obj = new BinarySearchSqrt(); int result= Obj.sqrt(n); System.out.println("Square root of " + n + " = " + result); } }

### Output

Square root of 64 = 8

Time Complexity: O(NlogN) Auxiliary Space: O(1)

### Example 2: Using Binary Search and Math.pow()

In the below example, we created a custom class ‘BinarySearchSqrt’ and implemented the binary search code for finding the square root of a number code in ‘sqrt’ function. The ‘sqrt’ function uses the in-built function ‘Math.pow()’ to calculate the square of a number .Now, create the custom Class object and initialise a variable called ‘number’ with an integer number and using the class object class the ‘sqrt’function is called and thus the desired output is displayed.

//Java Program to find Square root of a number using Binary Search and Math.pow() import java.util.*; class BinarySearchSqrt { public int sqrt(int number) { int low = 0; int high = number; while (low <= high) { int mid = (low + high) / 2; if (Math.pow(mid,2) == number) { return mid; } else if (Math.pow(mid,2) < number) { low = mid + 1; } else { high = mid - 1; } } return 0; } } public class Main { public static void main(String[] args) { int n = 64; BinarySearchSqrt Obj = new BinarySearchSqrt(); int result= Obj.sqrt(n); System.out.println("Square root of " + n + " = " + result); } }

### Output

Square root of 64 = 8

Time Complexity: O(NlogN) Auxiliary Space: O(1)

Thus, in this article we have discussed how to find the square root of a number using Binary Search Algorithm in Java.