feat: add Karatsuba multiplication algorithm in divide_and_conquer folder modified

divide_and_conquer/karatsuba_multiplication.py

@@ -0,0 +1,61 @@
+"""
+Karatsuba Multiplication Algorithm
+----------------------------------
+Given two binary strings that represent the value of two integers,
+find the product of the two strings using the Karatsuba algorithm.
+
+Example:
+    "1100" (12) x "1010" (10) = 120
+"""
+
+
+def karatsuba_multiply(bin_num1: str, bin_num2: str) -> int:
+    """
+    Multiply two binary strings using the Karatsuba algorithm.
+
+    Args:
+        bin_num1 (str): The first binary string.
+        bin_num2 (str): The second binary string.
+
+    Returns:
+        int: The product of the two binary strings as an integer.
+
+    Examples:
+        >>> karatsuba_multiply("1100", "1010")
+        120
+        >>> karatsuba_multiply("11", "11")
+        9
+        >>> karatsuba_multiply("101", "10")
+        10
+    """
+    # Convert binary strings to integers
+    x = int(bin_num1, 2)
+    y = int(bin_num2, 2)
+
+    # Base case for recursion
+    if x < 2 or y < 2:
+        return x * y
+
+    # Calculate the size of the numbers
+    n = max(len(bin_num1), len(bin_num2))
+    half = n // 2
+
+    # Split the binary strings
+    x_high = x >> half
+    x_low = x & ((1 << half) - 1)
+    y_high = y >> half
+    y_low = y & ((1 << half) - 1)
+
+    # Recursive multiplications
+    a = karatsuba_multiply(bin(x_high)[2:], bin(y_high)[2:])
+    d = karatsuba_multiply(bin(x_low)[2:], bin(y_low)[2:])
+    e = karatsuba_multiply(bin(x_high + x_low)[2:], bin(y_high + y_low)[2:]) - a - d
+
+    # Combine results
+    return (a << (2 * half)) + (e << half) + d
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()