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The Fast Fourier Transform and The Fast Polynomial Multiplication Algorithms in Python 3
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from cmath import exp | |
from math import pi | |
# A simple class to simulate n-th root of unity | |
# This class is by no means complete and is implemented | |
# merely for FFT and FPM algorithms | |
class NthRootOfUnity: | |
def __init__(self, n, k = 1): | |
self.k = k | |
self.n = n | |
def __pow__(self, other): | |
if type(other) is int: | |
n = NthRootOfUnity(self.n, self.k * other) | |
return n | |
def __eq__(self, other): | |
if other == 1: | |
return abs(self.n) == abs(self.k) | |
def __mul__(self, other): | |
return exp(2*1j*pi*self.k/self.n)*other | |
def __repr__(self): | |
return str(self.n) + "-th root of unity to the " + str(self.k) | |
@property | |
def th(self): | |
return abs(self.n // self.k) | |
# The Fast Fourier Transform Algorithm | |
# | |
# Input: A, An array of integers of size n representing a polynomial | |
# omega, a root of unity | |
# Output: [A(omega), A(omega^2), ..., A(omega^(n-1))] | |
# Complexity: O(n logn) | |
def FFT(A, omega): | |
if omega == 1: | |
return [sum(A)] | |
o2 = omega**2 | |
C1 = FFT(A[0::2], o2) | |
C2 = FFT(A[1::2], o2) | |
C3 = [None]*omega.th | |
for i in range(omega.th//2): | |
C3[i] = C1[i] + omega**i * C2[i] | |
C3[i+omega.th//2] = C1[i] - omega**i * C2[i] | |
return C3 | |
# The Fast Polynomial Multiplication Algorithm | |
# | |
# Input: A,B, two arrays of integers representing polynomials | |
# their length is in O(n) | |
# Output: Coefficient representation of AB | |
# Complexity: O(n logn) | |
def FPM(A,B): | |
n = 1<<(len(A)+len(B)-2).bit_length() | |
o = NthRootOfUnity(n) | |
AT = FFT(A, o) | |
BT = FFT(B, o) | |
C = [AT[i]*BT[i] for i in range(n)] | |
# nm = (len(A)+len(B)-1) | |
D = [round((a/n).real) for a in FFT(C, o ** -1)] | |
while len(D) > 0 and D[-1] == 0: | |
del D[-1] | |
return D |
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