Add Karatsuba Multiplication Algorithm Implementation

divide_and_conquer/karatsuba_multiplication_algorithm.py

@@ -0,0 +1,70 @@
+"""
+Description:
+    Karatsuba Multiplication Algorithm is an efficient way to multiply two large numbers
+    by reducing the multiplication operations using divide and conquer approach.
+
+Functionality:
+    The algorithm breaks down the multiplication of two n-digit numbers into
+    at most 3n^log2(3) (approximately 3n^1.585) single-digit multiplications in general.
+
+Complexity:
+    The time complexity of Karatsuba Multiplication Algorithm is O(n^log2(3)),
+    which is better than the traditional O(n^2) complexity for large n.
+
+Example:
+    >>> karatsuba_multiplication(1234, 5678)
+    7006652
+    >>> karatsuba_multiplication(123, 456)
+    56088
+    >>> karatsuba_multiplication(12345, 6789)
+    83810205
+"""
+
+
+def karatsuba_multiplication(x, y):
+    """
+    Multiply two integers using Karatsuba's algorithm.
+
+    Parameters:
+    x (int): First integer to multiply.
+    y (int): Second integer to multiply.
+
+    Returns:
+    int: The product of x and y.
+
+    >>> karatsuba_multiplication(1234, 5678)
+    7006652
+    >>> karatsuba_multiplication(123, 456)
+    56088
+    >>> karatsuba_multiplication(0, 12345)
+    0
+    """
+    # Base case for recursion
+    if x < 10 or y < 10:
+        return x * y
+
+    # Calculates the size of the numbers.
+    n = max(len(str(x)), len(str(y)))
+    half = n // 2
+
+    # Split the digit sequences in the middle.
+    a, b = divmod(x, 10**half)
+    c, d = divmod(y, 10**half)
+
+    # 3 calls made to numbers approximately half the size.
+    ac = karatsuba_multiplication(a, c)
+    bd = karatsuba_multiplication(b, d)
+    ad_plus_bc = karatsuba_multiplication(a + b, c + d) - ac - bd
+
+    # Performs the multiplication.
+    return ac * 10 ** (2 * half) + (ad_plus_bc * 10**half) + bd
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    x = 1234
+    y = 5678
+    print(f"The product of {x} and {y} is {karatsuba_multiplication(x, y)}")