Karatsuba algorithm - Error in my implementation

Question

I'm having trouble to implement the karatsuba algorithm in python. I'm working with lists in base 2 (the MSB is at the end in the list). The implementation given to me is this:

Input: 2 bit-numbers of bit length n
Output: their product a*b

function Karatsuba(a, b) 
     if(n == 1) return a*b
     else 
          a1, a0, leftmost(n/2)(rounded up), rightmost(n/2)(rounded down) bits of a
          b1, b0, leftmost(n/2)(rounded up), rightmost(n/2)(rounded down) bits of b

     s1 = Karatsuba(a1, b1)
     s2 = Karatsuba(a0, b0)
     s3 = Karatsuba(a1 + a0, b1 + b0)

     return s1 * 2^n + (s3 - s1 - s2) * 2^(n/2) + s2

and this my python implementation:

def karatsuba(A, B):
    if(len(A) == 1 or len(B) == 1):
        return Multiply(A, B)

    n = max(len(A), len(B))
    m = n / 2

    print "Karatsuba call"

    print "A", A, "\n"
    print "B", B, "\n"

    lowA = A[:m]
    highA = A[m:]

    lowB = B[:m]
    highB = B[m:]

    print "highA", highA, "\n"
    print "lowA", lowA, "\n"

    print "highB", highB, "\n"
    print "lowB", lowB, "\n"

    s1 = karatsuba(highA, highB)
    s2 = karatsuba(lowA, lowB)
    s3 = karatsuba(Add(highA, lowA), Add(highB, lowB))

    f1 = Multiply(s1, pow2(n))
    f2 = Multiply(Sub(Sub(s3, s1), s2), pow2(m))
    return Add(f1, Add(f2, s2))

However running with the input (remember the MSB is the rightmost bit):

A [0, 1, 1] 
B [0, 1, 1]

I get Product Karatsuba [0, 0, 0, 1, 0, 0, 1, 0] 72 but it should output [0, 0, 1, 0, 0, 1] 36 . The functions Add, Substract, pow2 and Multiply are working, I have tested them separately. If it helps here's the full output with the print statements:

Karatsuba call
A [0, 1, 1] 

B [0, 1, 1] 

highA [1, 1] 

lowA [0] 

highB [1, 1] 

lowB [0] 

Karatsuba call
A [1, 1] 

B [1, 1] 

highA [1] 

lowA [1] 

highB [1] 

lowB [1] 

Karatsuba call
A [0, 1] 

B [0, 1] 

highA [1] 

lowA [0] 

highB [1] 

lowB [0] 

Karatsuba call
A [1, 1] 

B [1, 1] 

highA [1] 

lowA [1] 

highB [1] 

lowB [1] 

Karatsuba call
A [0, 1] 

B [0, 1] 

highA [1] 

lowA [0] 

highB [1] 

lowB [0]

I'm searching for hours, and I have no more idea where my error is. Can somebody help me? Thanks

Are you sure that your arithmetic functions are giving correct results? If so, then how come there is a zero in the MSB of your final result? shouldn't it always be a 1 (except if the value is zero)? — samgak, Feb 25 '15 at 09:32
@samgak yes they were given to me; I just add to implement pow2n which is correct (I can show it if you want). I don't know why there is this extra 0. — user2336315, Feb 25 '15 at 13:24
yes please post it. Your print statement output looks correct to me, so I suspect it's something other than the code you have posted. Also what logs do you get if you assign Add(f1, Add(f2, s2)) to a variable and print it out before returning from the function? — samgak, Feb 25 '15 at 13:46
@samgak I tried to compute this with pen and pencil, with the algorithm description. I get as final result 9*2^3 which is indeed 72. So either I misunderstand the algorithm, or it's an error in the description I gave (which would be weird because it's my lesson slide). Could it be possible? — user2336315, Feb 25 '15 at 14:54

score 2 · Accepted Answer · answered Feb 25 '15 at 17:44

2

The error is this one:

f1 = Multiply(s1, pow2(n))

It should be:

f1 = Multiply(s1, pow2(2*m))

Indeed, (a1*2^m+a0)*(b1*2^m+b0)=(a1*b1)*2^(2m) + (a0*b1+a1*b0)*2^m + (a0*b0)

If n > (2*m), that is for an odd n, then you are doing things incorrectly...

answered Feb 25 '15 at 17:44

aka.nice

9,100
1
28
40

Oh! Many thanks! I guess I did the same error when performing the algo on my paper! Now it works :) – user2336315 Feb 25 '15 at 18:13
So to clarify; the pseudo code is a bit misleading right? It says it should be s1 * 2^n but in reality it's s1 ^ 2*(n/2) and due to integer division it does not produce the same result right? I though I could make the same assumption but apparently I was wrong.... – user2336315 Feb 25 '15 at 18:15
Afterthoughts, this sounds like similar to http://stackoverflow.com/questions/17531042/karatsuba-algorithm-without-biginteger-usage/22182739#22182739 – aka.nice Feb 25 '15 at 18:47

Karatsuba algorithm - Error in my implementation

1 Answers1