Add Integer Square Root Algorithm (#10949)

* Add Integer Square Root Algorithm

* Update integer_square_root.py

* Update integer_square_root.py

---------

Co-authored-by: Christian Clauss <cclauss@me.com>

maths/integer_square_root.py

@@ -0,0 +1,73 @@
+"""
+Integer Square Root Algorithm -- An efficient method to calculate the square root of a
+non-negative integer 'num' rounded down to the nearest integer. It uses a binary search
+approach to find the integer square root without using any built-in exponent functions
+or operators.
+* https://en.wikipedia.org/wiki/Integer_square_root
+* https://docs.python.org/3/library/math.html#math.isqrt
+Note:
+    - This algorithm is designed for non-negative integers only.
+    - The result is rounded down to the nearest integer.
+    - The algorithm has a time complexity of O(log(x)).
+    - Original algorithm idea based on binary search.
+"""
+
+
+def integer_square_root(num: int) -> int:
+    """
+    Returns the integer square root of a non-negative integer num.
+    Args:
+        num: A non-negative integer.
+    Returns:
+        The integer square root of num.
+    Raises:
+        ValueError: If num is not an integer or is negative.
+    >>> [integer_square_root(i) for i in range(18)]
+    [0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4]
+    >>> integer_square_root(625)
+    25
+    >>> integer_square_root(2_147_483_647)
+    46340
+    >>> from math import isqrt
+    >>> all(integer_square_root(i) == isqrt(i) for i in range(20))
+    True
+    >>> integer_square_root(-1)
+    Traceback (most recent call last):
+        ...
+    ValueError: num must be non-negative integer
+    >>> integer_square_root(1.5)
+    Traceback (most recent call last):
+        ...
+    ValueError: num must be non-negative integer
+    >>> integer_square_root("0")
+    Traceback (most recent call last):
+        ...
+    ValueError: num must be non-negative integer
+    """
+    if not isinstance(num, int) or num < 0:
+        raise ValueError("num must be non-negative integer")
+
+    if num < 2:
+        return num
+
+    left_bound = 0
+    right_bound = num // 2
+
+    while left_bound <= right_bound:
+        mid = left_bound + (right_bound - left_bound) // 2
+        mid_squared = mid * mid
+        if mid_squared == num:
+            return mid
+
+        if mid_squared < num:
+            left_bound = mid + 1
+        else:
+            right_bound = mid - 1
+
+    return right_bound
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()