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Administrative
Sep. 27 (today) – HW4 due
Sep. 28 8am – problem session
Oct. 2
Oct. 4 – QUIZ #2
(pages 45-79 of DPV)
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
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
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(A)=
 P(x)
xA
Examples
A
C
B
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)?
Examples
C
Events A,B,C are
pairwise independent
A but not
(fully) independent
B
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)?
Full independence
Events A1,…,An are (fully) independent
If for every subset S[n]:={1,2,…,n}
P(
 A ) =  P(A )
iS
i
iS
i
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
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
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
Y: -1 , -1,-1 ,-1, -1, 6 ,-1 ,-1 , -1 , -1 , -1
Expected income for B
E[Y] = 6*(1/6)-1*(5/6)= 1/6
Linearity of expectation
LEMMA:
E[X + Y] = E[X] + E[Y]
More generally:
E[X1+ X2+ … + Xn] = E[X1] + E[X2]+…+E[Xn]
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?
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] = ?
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
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
1
X2 = -9
1
if player 1 gets his card back
otherwise
if player 2 gets his card back
otherwise
E[X1+…+Xn] = E[X1]+…+E[Xn] =
n ( -9/n + 1*(n-1)/n ) = n – 10.
Do you expect to see the
expected value?
X=
1 with probability ½
3 with probability ½
E[X] =
Expected number of coin-tosses
until HEADS?
H
TH
TTH
TTTH
TTTTH
....
½
¼
1/8
1/16
1/32
Expected number of coin-tosses
until HEADS?

 n.2-n = 2
n=1
Expected number of dice-throws
until you get “6” ?
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
FFT
Polynomials
Polynomial of degree d
p(x) = a0 + a1 x + ... + ad xd
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’
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.
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
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
(x-a
)(x-a
)...(x-a
)
1
2
d
p(a1) = 0
....
(a0-a1)(a0-a2)...(a0-ad)
p(ad) = 0
Representing polynomial of degree d
the coefficient representation
d+1 coefficients
evaluation
interpolation
the value representation
evaluation on d+1 points
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)
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
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 ?
n-th roots of unity
e
FACT 1:
n = 1
FACT 2:
k . l = k+l
FACT 3:
0 + 1 + ... + n-1 = 0
FACT 4:
k = -k+n/2
2ik/n
= k
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
....
Evaluation of a polynomial
viewed as vector mutiplication
(a0,a1,a2,...,ad)
1
x
x2
.
.
xd
Evaluation of a polynomial
on multiple points
(a0,a1,a2,...,ad)
1
x1
x12
.
.
x1d
1
x2
x22
.
.
.
.
.
x2d
1
xn
xn2
.
.
xnd
Vandermonde matrix