I'm trying to understand the Karatsuba multiplication algorithm. I've written the following code:
def karatsuba_multiply(x, y):
# split x and y
len_x = len(str(x))
len_y = len(str(y))
if len_x == 1 or len_y == 1:
return x*y
n = max(len_x, len_y)
n_half = 10**(n // 2)
a = x // n_half
b = x % n_half
c = y // n_half
d = y % n_half
ac = karatsuba_multiply(a, c)
bd = karatsuba_multiply(b, d)
ad_plus_bc = karatsuba_multiply((a+b), (c+d)) - ac - bd
return (10**n * ac) + (n_half * ad_plus_bc) + bd
This test case does not work:
print(karatsuba_multiply(1234, 5678)) ## returns 11686652, should be 7006652
But if I use the following code from this answer, the test case produces the correct answer:
def karat(x,y):
if len(str(x)) == 1 or len(str(y)) == 1:
return x*y
else:
m = max(len(str(x)),len(str(y)))
m2 = m // 2
a = x // 10**(m2)
b = x % 10**(m2)
c = y // 10**(m2)
d = y % 10**(m2)
z0 = karat(b,d)
z1 = karat((a+b),(c+d))
z2 = karat(a,c)
return (z2 * 10**(2*m2)) + ((z1 - z2 - z0) * 10**(m2)) + (z0)
Both functions look like they're doing the same thing. Why doesn't mine work?
It seems that in with kerat_multiply implementation you can't use the correct formula for the last return.
In the original kerat implementation the value m2 = m // 2 is multiplied by 2 in the last return (z2 * 10**(2*m2)) + ((z1 - z2 - z0) * 10**(m2)) + (z0) (2*m2)
So you i think you need either to add a new variable as below where n2 == n // 2 so that you can multiply it by 2 in the last return, or use the original implementation.
Hoping it helps :)
EDIT: This is explain by the fact that 2 * n // 2 is different from 2 * (n // 2)
n = max(len_x, len_y)
n_half = 10**(n // 2)
n2 = n // 2
a = x // n_half
b = x % n_half
c = y // n_half
d = y % n_half
ac = karatsuba_multiply(a, c)
bd = karatsuba_multiply(b, d)
ad_plus_bc = karatsuba_multiply((a + b), (c + d)) - ac - bd
return (10**(2 * n2) * ac) + (n_half * (ad_plus_bc)) + bd