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CSE 421 Algorithms. Richard Anderson Lecture 15 Fast Fourier Transform. FFT, Convolution and Polynomial Multiplication. FFT: O(n log n) algorithm Evaluate a polynomial of degree n at n points in O(n log n) time Polynomial Multiplication: O(n log n) time. Complex Analysis.

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Cse 421 algorithms

CSE 421Algorithms

Richard Anderson

Lecture 15

Fast Fourier Transform


Fft convolution and polynomial multiplication
FFT, Convolution and Polynomial Multiplication

  • FFT: O(n log n) algorithm

    • Evaluate a polynomial of degree n at n points in O(n log n) time

  • Polynomial Multiplication: O(n log n) time


Complex analysis
Complex Analysis

  • Polar coordinates: reqi

  • eqi = cos q + i sin q

  • a is an nth root of unity if an = 1

  • Square roots of unity: +1, -1

  • Fourth roots of unity: +1, -1, i, -i

    • Eighth roots of unity: +1, -1, i, -i, b + ib, b - ib, -b + ib, -b - ib where b = sqrt(2)


E 2 p ki n
e2pki/n

  • e2pi = 1

  • epi = -1

  • nth roots of unity: e2pki/n for k = 0 …n-1

  • Notation: wk,n = e2pki/n

  • Interesting fact:

    1 + wk,n + w2k,n + w3k,n + . . . + wn-1k,n = 0 for k != 0


Fft overview
FFT Overview

  • Polynomial interpolation

    • Given n+1 points (xi,yi), there is a unique polynomial P of degree at most n which satisfies P(xi) = yi


Polynomial multiplication
Polynomial Multiplication

n-1 degree polynomials

A(x) = a0 + a1x + a2x2 + … +an-1xn-1,

B(x) = b0 + b1x + b2x2 + …+ bn-1xn-1

C(x) = A(x)B(x)

C(x)=c0+c1x + c2x2 + … + c2n-2x2n-2

p1, p2, . . ., p2n

A(p1), A(p2), . . ., A(p2n)

B(p1), B(p2), . . ., B(p2n)

C(p1), C(p2), . . ., C(p2n)

C(pi) = A(pi)B(pi)


FFT

  • Polynomial A(x) = a0 + a1x + . . . + an-1xn-1

  • Compute A(wj,n) for j = 0, . . ., n-1

  • For simplicity, n is a power of 2


Useful trick
Useful trick

A(x) = a0 + a1x + a2x2 + a3x3 +. . . + an-1xn-1

Aeven(x) = a0 + a2x + a4x2 + . . . + an-2x(n-2)/2

Aodd(x) = a1+ a3x + a5x2 + …+ an-1x(n-2)/2

Show: A(x) = Aeven(x2) + x Aodd(x2)


Lemma w 2 j 2n w j n
Lemma: w2j,2n = wj,n

Squares of 2nth roots of unity are nth roots of unity

wj,2n = e2pji/2n

The detail of j >= n is being ignored – don’t mention

unless it comes up.


Fft algorithm
FFT Algorithm

// Evaluate the 2n-1th degree polynomial A at

// w0,2n, w1,2n, w2,2n, . . ., w2n-1,2n

FFT(A, 2n)

Recursively compute FFT(Aeven, n)

Recursively compute FFT(Aodd, n)

for j = 0 to 2n-1

A(wj,2n) = Aeven(w2j,2n) + wj,2nAodd(w2j,2n)


Polynomial multiplication1
Polynomial Multiplication

  • n-1th degree polynomials A and B

  • Evaluate A and B at w0,2n, w1,2n, . . ., w2n-1,2n

  • Compute C(wj,2n) for j = 0 to 2n -1

  • We know the value of a 2n-2th degree polynomial at 2n points – this determines a unique polynomial, we just need to determine the coefficients


Now the magic happens
Now the magic happens . . .

C(x) = c0 + c1x + c2x2 + … + c2n-1x2n-1

(we want to compute the ci’s)

Let dj = C(wj,2n)

D(x) = d0 + d1x + d2x2 + … + d2n-1x2n-1

Evaluate D(x) at the 2nth roots of unity

D(wj,2n) = [see text for details] = 2nc2n-j


Polynomial interpolation
Polynomial Interpolation

  • Build polynomial from the values of C at the 2nth roots of unity

  • Evaluate this polynomial at the 2nth roots of unity


Dynamic programming
Dynamic Programming

  • Weighted Interval Scheduling

  • Given a collection of intervals I1,…,In with weights w1,…,wn, choose a maximum weight set of non-overlapping intervals

Sorted by finish times

4

6

3

5

7

6


Recursive algorithm
Recursive Algorithm

Intervals sorted by finish time

p[i] is the index of the last interval which finishes before i starts

1

2

3

4

5

6

7

8


Optimality condition
Optimality Condition

  • Opt[j] is the maximum weight independent set of intervals I1, I2, . . ., Ij

Opt[ j ] = max( Opt[ j – 1], wj + Opt[ p [ j ] ])


Algorithm
Algorithm

MaxValue(j) =

if j = 0 return 0

else

return max( MaxValue(j-1), wj + MaxValue(p[ j ]))


Run time
Run time

  • What is the worst case run time of MaxValue

  • Design a worst case input


A better algorithm
A better algorithm

M[ j ] initialized to -1 before the first recursive call for all j

MaxValue(j) =

if j = 0 return 0;

else if M[ j ] != -1 return M[ j ];

else

M[ j ] = max(MaxValue(j-1),wj + MaxValue(p[ j ]));

return M[ j ];


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