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Overview of Algorithms for k-th Smallest Element and Probability Concepts

This document provides a detailed outline of key topics including the k-th smallest element selection using the modified quicksort algorithm, as well as principles of probability, independence, and expected value calculations. It includes examples and necessary proofs such as the linearity of expectation and discussions on finite probability spaces. Furthermore, it highlights the polynomial degree theory and root determination, enriching understanding and enhancing problem-solving skills in algorithms and statistics. Study for HW4 due on Sept. 28 and quiz preparation for Oct. 4.

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Overview of Algorithms for k-th Smallest Element and Probability Concepts

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  1. Administrative Sep. 27 (today) – HW4 due Sep. 28 8am – problem session Oct. 2 Oct. 4 – QUIZ #2 (pages 45-79 of DPV)

  2. Recap algorithm for k-select with O(n) worst-case running time modification of quick-sort which has O(n.log n) worst-case running time randomized k-select GOAL: O(n) expected running-time

  3. Finding the k-th smallest element Select(k,A[c..d]) x=random element from A[c..d] Split(A[c..d],x) >x x j j k  k-th smallest on left j<k  (k-j)-th smallest on right

  4. Finite probability space set  (sample space) function P:  R+ (probability distribution)  P(x) = 1 x elements of  are called atomic events subsets of  are called events probability of an event A is  P(x) P(A)= xA

  5. Examples A B C Are A,B independent ?Are A,C independent ? Are B,C independent ? Is it true that P(ABC)=P(A)P(B)P(C)?

  6. Examples Events A,B,C are pairwise independent but not (fully) independent A B C Are A,B independent ?Are A,C independent ? Are B,C independent ? Is it true that P(ABC)=P(A)P(B)P(C)?

  7. Full independence Events A1,…,An are (fully) independent If for every subset S[n]:={1,2,…,n} P ( Ai ) =  P(Ai) iS iS

  8. Random variable set  (sample space) function P:  R+ (probability distribution)  P(x) = 1 x A random variable is a function Y :   R The expected value of Y is E[X] :=  P(x)* Y(x) x

  9. Examples Roll two dice. Let S be their sum. If S=7 then player A gives player B $6 otherwise player B gives player A $1 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

  10. Examples Roll two dice. Let S be their sum. If S=7 then player A gives player B $6 otherwise player B gives player A $1 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 -1 , -1,-1 ,-1, -1, 6 ,-1 ,-1 , -1 , -1 , -1 Y: Expected income for B E[Y] = 6*(1/6)-1*(5/6)= 1/6

  11. Linearity of expectation LEMMA: E[X + Y] = E[X] + E[Y] More generally: E[X1+ X2+ … + Xn] = E[X1] + E[X2]+…+E[Xn]

  12. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. Let n be the number of people in the class. For what n is the game advantageous for me?

  13. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X1 = -9 if player 1 gets his card back 1 otherwise E[X1] = ?

  14. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X1 = -9 if player 1 gets his card back 1 otherwise E[X1] = -9/n + 1*(n-1)/n

  15. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X1 = -9 if player 1 gets his card back 1 otherwise X2 = -9 if player 2 gets his card back 1 otherwise E[X1+…+Xn] = E[X1]+…+E[Xn] = n ( -9/n + 1*(n-1)/n ) = n – 10.

  16. Do you expect to see the expected value? • with probability ½ • 3 with probability ½ X= E[X] =

  17. Expected number of coin-tosses until HEADS? H ½ TH ¼ TTH 1/8 TTTH 1/16 TTTTH 1/32 ....

  18. Expected number of coin-tosses until HEADS?  n.2-n = 2 n=1 Expected number of dice-throws until you get “6” ?

  19. Finding the k-th smallest element Select(k,A[c..d]) x=random element from A[c..d] Split(A[c..d],x) >x x j j k  k-th smallest on left j<k  (k-j)-th smallest on right

  20. FFT

  21. Polynomials Polynomial of degree d p(x) = a0 + a1 x + ... + ad xd

  22. Multiplying polynomials Polynomial of degree d p(x) = a0 + a1 x + ... + ad xd Polynomial of degree d’ q(x) = b0 + b1 x + ... + bd’ xd’ p(x)q(x) = (a0b0) + (a0b1 + a1b0) x + .... + (adbd’) xd+d’

  23. Polynomials Polynomial of degree d p(x) = a0 + a1 x + ... + ad xd THEOREM: A non-zero polynomial of degree d has at most d roots. COROLLARY: A polynomial of degree d is determined by its value on d+1 points.

  24. COROLLARY: A polynomial of degree d is determined by its value on d+1 points. Find a polynomial p of degree d such that p(a0) = 1 p(a1) = 0 .... p(ad) = 0

  25. COROLLARY: A polynomial of degree d is determined by its value on d+1 points. Find a polynomial p of degree d such that p(a0) = 1 p(a1) = 0 .... p(ad) = 0 (x-a1)(x-a2)...(x-ad) (a0-a1)(a0-a2)...(a0-ad)

  26. Representing polynomial of degree d the coefficient representation d+1 coefficients evaluation interpolation the value representation evaluation on d+1 points

  27. Evaluation on multiple points p(x) = 7 + x + 5x2 + 3x3 + 6x4 + 2x5 p(z) = 7 + z + 5z2 + 3z3 + 6z4 + 2z5 p(-z) = 7 – z + 5z2 – 3z3 + 6z4 – 2z5 p(x) = (7+5x2 + 6x4) + x(1+3x2 + 2x4) p(x) = pe(x2) + x po(x2) p(-x) = pe(x2) – x po(x2)

  28. Evaluation on multiple points p(x) = a0 + a1 x + a2 x2 + ... + ad xd p(x) = pe(x2) + x po(x2) p(-x) = pe(x2) – x po(x2) To evaluate p(x) on -x1,x1,-x2,x2,...,-xn,xn we only evaluate pe(x) and po(x) on x12,...,xn2

  29. Evaluation on multiple points To evaluate p(x) on -x1,x1,-x2,x2,...,-xn,xn we only evaluate pe(x) and po(x) on x12,...,xn2 To evaluate pe(x) on x12,...,xn2 we only evaluate pe(x) on ?

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

  31. FFT (a0,a1,...,an-1,) (s0,...,sn/2-1)= FFT(a0,a2,...,an-2,2) (z0,...,zn/2-1) = FFT(a1,a3,...,an-1,2) s0 + z0 s1 +  z1 s2 + 2 z2 .... s0 – z0 s1 -  z1 s2 - 2 z2 ....

  32. Evaluation of a polynomial viewed as vector mutiplication 1 x x2 . . xd (a0,a1,a2,...,ad)

  33. Evaluation of a polynomial on multiple points 1 xn xn2 . . xnd 1 x1 x12 . . x1d 1 x2 x22 . . x2d (a0,a1,a2,...,ad) . . . Vandermonde matrix

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