feat: add Karatsuba multiplication algorithm in divide_and_conquer folder

divide_and_conquer/karatsuba_multiplication.py

@@ -0,0 +1,78 @@
+"""
+Performs multiplication of two binary strings using the Karatsuba algorithm.
+
+Given two binary strings of equal length n, the goal is to compute
+their product as integers.
+
+For example:
+"1100" (12) × "1010" (10) = 120
+
+Karatsuba's algorithm reduces the multiplication of two n-bit numbers
+to at most three multiplications of n/2-bit numbers. It achieves
+a time complexity of O(n^log₂3) ≈ O(n^1.585), which is faster
+than the naive O(n²) approach.
+
+References:
+https://en.wikipedia.org/wiki/Karatsuba_algorithm
+
+Example:
+>>> karatsuba_multiply("1100", "1010")
+120
+>>> karatsuba_multiply("10", "11")
+6
+>>> karatsuba_multiply("101", "111")
+35
+>>> karatsuba_multiply("0", "0")
+0
+>>> karatsuba_multiply("1", "0")
+0
+"""
+
+
+def karatsuba_multiply(x: str, y: str) -> int:
+    """
+    Multiplies two binary strings using the Karatsuba algorithm.
+
+    Both input strings should be of equal length.
+
+    >>> karatsuba_multiply("1100", "1010")
+    120
+    >>> karatsuba_multiply("11", "10")
+    6
+    >>> karatsuba_multiply("1111", "1111")
+    225
+    """
+    n = max(len(x), len(y))
+
+    # Pad the shorter string with leading zeros
+    x = x.zfill(n)
+    y = y.zfill(n)
+
+    # Base case: single bit multiplication
+    if n == 1:
+        return int(x) * int(y)
+
+    mid = n // 2
+
+    # Split the binary strings into left and right halves
+    x_left, x_right = x[:mid], x[mid:]
+    y_left, y_right = y[:mid], y[mid:]
+
+    # Recursively compute partial products
+    p1 = karatsuba_multiply(x_left, y_left)
+    p2 = karatsuba_multiply(x_right, y_right)
+    p3 = karatsuba_multiply(
+        bin(int(x_left, 2) + int(x_right, 2))[2:],
+        bin(int(y_left, 2) + int(y_right, 2))[2:],
+    )
+
+    # Karatsuba combination formula:
+    result = (p1 << (2 * (n - mid))) + ((p3 - p1 - p2) << (n - mid)) + p2
+
+    return result
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()