1. Administrative Oct. 2 Oct. 4 – QUIZ #2 (pages 45-79 of DPV)

2. Polynomials p(x) = a 0 + a 1 x +... + a d x d polynomial of degree d

3. Representing polynomial of degree d (d+1 coefficients) (evaluation on d+1 points) the coefficient representation the value representation evaluationinterpolation

4. Evaluation p(x) = a 0 + a 1 x +... + a d x d polynomial of degree d

5. Evaluation p(x) = a 0 + a 1 x +... + a d x d polynomial of degree d (Horner’s rule) a 0 +x(a 1 +x(a 2 +... ))) R  0 for i from d to 0 do R  R*x + a i

6. Interpolation a polynomial p of degree d such that p(a 0 ) = 1 p(a 1 ) = 0.... p(a d ) = 0 (x-a 1 )(x-a 2 )...(x-a d ) (a 0 -a 1 )(a 0 -a 2 )...(a 0 -a d ) p 1 = (1/2) x 2 – (1/2) x p 2 = - x 2 + 1 p 3 = (1/2) x 2 + (1/2) x

7. Interpolation 1) compute p = (x-a 0 )...(x-a d ) 2) compute p i = p / (x-a i ) for i=0,...,d r i = p i / p i (a i ) 3) compute q = c 0 r 0 +... + c d r d Claim: q(a i ) = c i for i=0,...,d

8. Polynomials evaluation on 1 point O(d) evaluation on d points O(d 2 ) interpolation O(d 2 )

9. MAIN GOAL: Multiplying polynomials p(x) = a 0 + a 1 x +... + a d x d Polynomial of degree d q(x) = b 0 + b 1 x +... + b d’ x d’ Polynomial of degree d’ p(x)q(x) = (a 0 b 0 ) + (a 0 b 1 + a 1 b 0 ) x +.... + (a d b d’ ) x d+d’

10. p,q evaluate on >2d points p,q in evaluation representation multiply in O(d) time pq in evaluation representation pq interpolate

11. Evaluation on multiple points p(x) = 7 + x + 5x 2 + 3x 3 + 6x 4 + 2x 5 p(z) = 7 + z + 5z 2 + 3z 3 + 6z 4 + 2z 5 p(-z) = 7 – z + 5z 2 – 3z 3 + 6z 4 – 2z 5 p(x) = (7+5x 2 + 6x 4 ) + x(1+3x 2 + 2x 4 ) p( x) = p e (x 2 ) + x p o (x 2 ) p(-x) = p e (x 2 ) – x p o (x 2 )

12. Evaluation on multiple points p(x) = a 0 + a 1 x + a 2 x 2 +... + a d x d p(x) = p e (x 2 ) + x p o (x 2 ) p(-x) = p e (x 2 ) – x p o (x 2 ) To evaluate p(x) on -x 1,x 1,-x 2,x 2,...,-x n,x n we only evaluate p e (x) and p o (x) on x 1 2,...,x n 2

13. Evaluation on multiple points To evaluate p(x) on -x 1,x 1,-x 2,x 2,...,-x n,x n we only evaluate p e (x) and p o (x) on x 1 2,...,x n 2 To evaluate p e (x) on x 1 2,...,x n 2 we only evaluate p e (x) on ?

14. n-th roots of unity 2  ik/n e  k  n = 1  k.  l =  k+l  0 +  1 +... +  n-1 = 0 FACT 1: FACT 2: FACT 3: FACT 4:  k = -  k+n/2

15. FFT (a 0,a 1,...,a n-1,  ) (s 0,...,s n/2-1 )= FFT(a 0,a 2,...,a n-2,  2 ) (z 0,...,z n/2-1 ) = FFT(a 1,a 3,...,a n-1,  2 ) s 0 + z 0 s 1 +  z 1 s 2 +  2 z 2.... s 0 – z 0 s 1 -  z 1 s 2 -  2 z 2....

16. Evaluation of a polynomial viewed as vector mutiplication (a 0,a 1,a 2,...,a d ) 1xx2..xd1xx2..xd

17. Evaluation of a polynomial on multiple points (a 0,a 1,a 2,...,a d ) 1x1x12..x1d1x1x12..x1d 1x2x22..x2d1x2x22..x2d 1xnxn2..xnd1xnxn2..xnd... Vandermonde matrix

18. Fourier transform (a 0,a 1,a 2,...,a d ) 111..1111..1 1  1  2.  d-1 1  d-1  2(d-1).  (d-1)... FT - matrix  = e 2  i / d 2

19. Inverse fourier transform  = e 2  i / d 111..1111..1 1  1  2.  d-1 1  d-1  2(d-1).  (d-1)... 2 111..1111..1 1  -1  -2.  1-d 1  1-d  2(1-d).  -(d-1)... 2

20. Fourier transform – matrix view 0 0 FnFn FnFn I I D -D 1 1 1 1... D=diag(1, ,...,  d-1 ) d=2 n

21. String matching string = x 1,....,x n pattern = p 1,...,p k  (x i – p i ) 2 i=1 k Occurs on position j ? TEST =

22. String matching string = x 1,....,x n pattern = p 1,...,p k  (x i – p i ) 2 i=1 k  (x i+j-1 – p i ) 2 i=1 k Occurs on position j ? TEST = TEST j =

23. String matching with “don’t cares”  (x i+j-1 – p i ) 2 i=1 k Occurs on position j ? TEST j = p j = 0 DON’T CARE  p j (x i+j-1 – p i ) 2 i=1 k TEST j =