# Karatsuba Multiplication Algorithm – Python Code

Motivation for this blog post

I’ve enrolled in Stanford Professor Tim Roughgarden’s Coursera MOOC on the design and analysis of algorithms, and while he covers the theory and intuition behind the algorithms in a surprising amount of detail, we’re left to implement them in a programming language of our choice.

And I’m ging to post Python code for all the algorithms covered during the course!

The Karatsuba Multiplication Algorithm

Karatsuba’s algorithm reduces the multiplication of two n-digit numbers to at most $n^{\log_23}\approx n^{1.585}$ single-digit multiplications in general (and exactly $n^{\log_23}$ when n is a power of 2). Although the familiar grade school algorithm for multiplying numbers is how we work through multiplication in our day-to-day lives, it’s slower ( $\Theta(n^2)\,\!$) in comparison, but only on a computer, of course!

Here’s how the grade school algorithm looks:
(The following slides have been taken from Tim Roughgarden’s notes. They serve as a good illustration. I hope he doesn’t mind my sharing them.) …and this is how Karatsuba Multiplication works on the same problem:  A More General Treatment

Let $x$ and $y$ be represented as $n$-digit strings in some base $B$. For any positive integer $m$ less than $n$, one can write the two given numbers as $x = x_1B^m + x_0$ $y = y_1B^m + y_0$,

where $x_0$ and $y_0$ are less than $B^m$. The product is then $xy = (x_1B^m + x_0)(y_1B^m + y_0)$ $xy = z_2B^{2m} + z_1B^m + z_0$

where $z_2 = x_1y_1$ $z_1 = x_1y_0 + x_0y_1$ $z_0 = x_0y_0$

These formulae require four multiplications, and were known to Charles Babbage. Karatsuba observed that $xy$ can be computed in only three multiplications, at the cost of a few extra additions. With $z_0$ and $z_2$ as before we can calculate $z_1 = (x_1 + x_0)(y_1 + y_0) - z_2 - z_0$

which holds since $z_1 = x_1y_0 + x_0y_1$ $z_1 = (x_1 + x_0)(y_1 + y_0) - x_1y_1 - x_0y_0$

A more efficient implementation of Karatsuba multiplication can be set as $xy = (b^2 + b)x_1y_1 - b(x_1 - x_0)(y_1 - y_0) + (b + 1)x_0y_0$, where $b = B^m$.

### Example

To compute the product of 12345 and 6789, choose B = 10 and m = 3. Then we decompose the input operands using the resulting base (Bm = 1000), as:

12345 = 12 · 1000 + 345
6789 = 6 · 1000 + 789

Only three multiplications, which operate on smaller integers, are used to compute three partial results:

z2 = 12 × 6 = 72
z0 = 345 × 789 = 272205
z1 = (12 + 345) × (6 + 789) − z2z0 = 357 × 795 − 72 − 272205 = 283815 − 72 − 272205 = 11538

We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base 1000 like for the input operands):

result = z2 · B2m + z1 · Bm + z0, i.e.
result = 72 · 10002 + 11538 · 1000 + 272205 = 83810205.

Pseudocode and Python code

This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 procedure karatsuba(num1, num2) if (num1 < 10) or (num2 < 10) return num1*num2 /* calculates the size of the numbers */ m = max(size_base10(num1), size_base10(num2)) m2 = m/2 /* split the digit sequences about the middle */ high1, low1 = split_at(num1, m2) high2, low2 = split_at(num2, m2) /* 3 calls made to numbers approximately half the size */ z0 = karatsuba(low1,low2) z1 = karatsuba((low1+high1),(low2+high2)) z2 = karatsuba(high1,high2) return (z2*10^(2*m2))+((z1-z2-z0)*10^(m2))+(z0)

This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 def karatsuba(x,y): """Function to multiply 2 numbers in a more efficient manner than the grade school algorithm""" if len(str(x)) == 1 or len(str(y)) == 1: return x*y else: n = max(len(str(x)),len(str(y))) nby2 = n / 2 a = x / 10**(nby2) b = x % 10**(nby2) c = y / 10**(nby2) d = y % 10**(nby2) ac = karatsuba(a,c) bd = karatsuba(b,d) ad_plus_bc = karatsuba(a+b,c+d) - ac - bd # this little trick, writing n as 2*nby2 takes care of both even and odd n prod = ac * 10**(2*nby2) + (ad_plus_bc * 10**nby2) + bd return prod
view raw karatsuba.py hosted with ❤ by GitHub

## 25 thoughts on “Karatsuba Multiplication Algorithm – Python Code”

1. Murat Serdar Alperen says:

Hi, things like getting input as integers and using functions may be a trouble for beginners, so here is a “running off the shelf” version: https://repl.it/CBno

Like

2. Patrick Dennis says:

Thank you so much for the “writing n as 2*nby2” tip. Without it, just using 10**n * (a * c), the algorithm works for all four-digit numbers (as far as I can tell), but fails on about one eight-digit problem out of eight, more with higher numbers of digits. I had found that the failures were related to getting an odd number of digits with a split, since leading zeros are dropped (45670123 splits as 4567 and 123), and had been trying to fix it by going to strings and padding out leading zeros. That actually reduced the failure rate, but fails still occurred. Your simple fix removed the whole issue!

Like

3. varenya says:

Hi,
I found your implementation amusing because of the usage of / and % operators as they take more time than the actual problem
Did you test your results for large size inputs?
I believe those operators should be replaced by string concatenation for an efficient implementation

Liked by 1 person

4. bhaskarbalachander says:

Hi, thanks for the 2*nby2 tip! However i’m a noob and still don’t understand how it works. Is it because python remembers that for odd numbers, the nby2 is actually a float and accounts for that when doing the multiplication?

Thanks!

Like

5. Ten O'Clock Scholar says:

Didn’t work for me! I copy-pasted verbatim and got RecursionError, maximum recursion depth exceeded with getting the str of an object.

Like

6. Gauri Shankar Badola says:

I wanted to read your whole code, but the starting of the code has the biggest flaw.

if len(str(x)) == 1 or len(str(y)) == 1:
return x*y

what if x is 10, and y is 1 you still cant do 101.. you have to 1001 recursively.
Unnecessary use of ** and %. Do you know ** works internally?

Like

• Miguel says:

“your program should restrict itself to multiplying only pairs of single-digit numbers.”

with that IF statement you are failing to do so… lets say you multiply 7 x 213213321….(large number). Your code will do directly the multiplication without using the recursive method. (That multiplication will be done by whatever implementation python uses … perhaps its karatsuba too :D)

What you can do is (lets say x has 1 digit and y has >1 digit):
– check that we are in that case
– you cannot split x, so you have to split y: 1 part with all digits but last one, and 2nd part with last digit
– set a to 0 so you can use your f(a,b,c,d) katsuba formula when a is 0. if the 1 digit number is y, then c=0.
– do this recursively until you arrive to 1digit by 1digit number multiplications

Btw your code fails, has recursion errors. You shouldnt split numbers by half of the max(len(x),len(y)). what happens if x=123 and y=12345678… how do you split x into 4 digits?…you should use min instead of max. so you split x into [1,23] and y [123456,78] (or [12,3] and y [1234568,8]

Like

7. Kanu Patel says:

Thank you for the python code. This will help me multiply big numbers in short time.

Like

8. sakethv321 says:

This is my first time visit here. From the tons of comments on your articles.I guess I am not only one having all the enjoyment right here! ExcelR Data Science Courses

Like

9. Anna says:

I do not know why I got a wrong result when I have reproduced the code on my test case, it fails, anyone knows where the problem is?

Like

10. video says:

BBQ Grill/Fire Pit in 2001 Years, Our Factory Can Provide Cheap Price and Good Quality Product Around the World, Currently, Main Market is Germany/Poland/UK/US, Quality Inspection Standard is in Accordance with France/Germany. Our products look so cool and have hundreds of styles and sizes for you to choose. In this way, you can meet different requirements in different places. I believe there will be styles that you love!

Like

11. Donald says:

Thank you for the code. I was able to build this website using python.

Like

12. Asma says:

Hi waht should be the implementation of karatsuba for base 2 or any other base?

Like

13. George Iskander says:

The code need some modification to used // instead of /

Like

14. Bhavya says:

Thank you for the great tips. also learn Data Science Course in Jersey City using python and r programming.

Like

15. iot course in bangalore says:

I want to leave a little comment to support and wish you the best of luck .we wish you the best of luck in all your blogging endeavor’s.

Like

16. nearlearns says:

Thank you for Sharing best Useful information !!!!

Like

17. Skillslash says:

I like your post. I appreciate your blogs because they are really good. Please go to this website for the Data Science Course: For Data Science Course in Bangalore Data Science course in Bangalore. These courses are wonderful for professionalism.

Like

18. data science says:

Amazingly by and large very interesting post. I was looking for such an information and thoroughly enjoyed examining this one.
Keep posting.

Like