From af1c6d8e1a822de8d1b3f0d860d9ac7a87fe126c Mon Sep 17 00:00:00 2001
From: Supratim <109270340+sgindeed@users.noreply.github.com>
Date: Tue, 7 Oct 2025 09:46:59 +0530
Subject: [PATCH 1/2] Create karatsuba_multiplication.py

---
 .../karatsuba_multiplication.py               | 76 +++++++++++++++++++
 1 file changed, 76 insertions(+)
 create mode 100644 divide_and_conquer/karatsuba_multiplication.py

diff --git a/divide_and_conquer/karatsuba_multiplication.py b/divide_and_conquer/karatsuba_multiplication.py
new file mode 100644
index 000000000000..9500de47435f
--- /dev/null
+++ b/divide_and_conquer/karatsuba_multiplication.py
@@ -0,0 +1,76 @@
+"""
+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()

From 07d94130948e2b747ef9389e33983a26355e5855 Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
 <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Tue, 7 Oct 2025 04:17:50 +0000
Subject: [PATCH 2/2] [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci
---
 divide_and_conquer/karatsuba_multiplication.py | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/divide_and_conquer/karatsuba_multiplication.py b/divide_and_conquer/karatsuba_multiplication.py
index 9500de47435f..3cdd59692f90 100644
--- a/divide_and_conquer/karatsuba_multiplication.py
+++ b/divide_and_conquer/karatsuba_multiplication.py
@@ -28,6 +28,7 @@
 0
 """
 
+
 def karatsuba_multiply(x: str, y: str) -> int:
     """
     Multiplies two binary strings using the Karatsuba algorithm.
@@ -62,7 +63,7 @@ def karatsuba_multiply(x: str, y: str) -> int:
     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:]
+        bin(int(y_left, 2) + int(y_right, 2))[2:],
     )
 
     # Karatsuba combination formula:
@@ -73,4 +74,5 @@ def karatsuba_multiply(x: str, y: str) -> int:
 
 if __name__ == "__main__":
     import doctest
+
     doctest.testmod()