Question

[2 marks]Describe how to multiply two n-degree polynomials together in O(n logn) time, using the Fast Fourier Transform (FFT). You do not need to explain how FFT works – you may treat it as a black box.

In this part we will use the Fast Fourier Transform (FFT) algorithm described in class to multiply multiple polynomials together (not just two).

Suppose you have K polynomials P1, . . . , Pk so that

degree(P1) +···+ degree(PK) =S

(i)[6 marks]Show that you can find the product of these K polynomials in O(K SlogS)time.

Hint: How many points do you need to uniquely determine an S-degree polynomial?

(ii)[12 marks]Show that you can find the product of these K polynomials in O(S logS logK)time.

Hint: consider using divide-and-conquer; a tree which you used in the previous assignment might be helpful here as well. Also, remember that if x, y, z are all positive, then log(x+y)

Answer 1

FFT(a){
n = length(a) // a is the input coefficient vector
if n = 1
  then return a

// wn is principle complex nth root of unity.
wn = e^(2*pi*i/n)
w = 1

// even indexed coefficients
A0 = (a0, a2, ..., an-2 )

// odd indexed coefficients
A1 = (a1, a3, ..., an-1 ) 

y0 = Recursive_FFT(A0) // local array
y1 = Recursive-FFT(A1) // local array

for k = 0 to n/2 - 1

  // y array stores values of the DFT 
  // of given polynomial. 
  do y[k] = y0[k] + w*y1[k]  
     y[k+(n/2)] = y0[k] - w*y1[k]
     w = w*wn
return y
}