Randomized algorithm Tutorial 2 Solution for homework 1.
-
Upload
lacey-hover -
Category
Documents
-
view
232 -
download
1
Transcript of Randomized algorithm Tutorial 2 Solution for homework 1.
![Page 1: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/1.jpg)
Randomized algorithm
Tutorial 2Solution for homework 1
![Page 2: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/2.jpg)
Outline
Solutions for homework 1 Negative binomial random variable Paintball game Rope puzzle
![Page 3: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/3.jpg)
Solutions for homework 1
![Page 4: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/4.jpg)
Homework 1-1
: a
: b
: c
: dBox 1 Box 2
step 1
ba
bB
ba
aW
)Pr(
)Pr(
![Page 5: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/5.jpg)
Homework 1-1
Draw a white ball from Box 2 : event A Pr(A)
= Pr(A∩ (W∪B))
= Pr((A∩ W) (∪ A∩ B))= Pr(A∩ W) + Pr(B∩ W)
= Pr(A|W)Pr(W) + Pr(A|B)Pr(B)
![Page 6: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/6.jpg)
Homework 1-1
It is easy to see that
1)db)(c(a
abcac
B)Pr(B)|Pr(A W)Pr(W)|Pr(A
Pr(A)1dc
cB)|Pr(A
1
1W)|Pr(A
dc
c
![Page 7: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/7.jpg)
Homework 1-2
Wi = pick a white ball at the i-th time
3
11
1
PrPr
Pr
112
21
ba
a
ba
a
)(W)|W(W
)W(W
![Page 8: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/8.jpg)
Homework 1-2
22
22
3
1
1-ba
1-a
1-ba
1-a
1-ba
1-a
1-ba
1-a
ba
a
ba
a
ba
a
ba
a
![Page 9: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/9.jpg)
Homework 1-2
2
)13(1
13
13
31)13(
1)1(3
)1()1(3 22
ba
ba
ba
baa
baa
![Page 10: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/10.jpg)
Homework 1-2
ab
ab
ab
aba
aba
2
)13(
13
)13(
3
3)( 22
2
)13(1
2
)13( ba
b
![Page 11: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/11.jpg)
Homework 1-2
1 b maps 1 a only For b=1, 1.36<a<2.36 means a=2
Pr(W1∩W2) = 2/3 * 1/2= 1/3
Then we may try b=2 →a=3 →odd b=4 →a=6 →even
![Page 12: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/12.jpg)
Homework 1-3
By the hint X = sum is even. Y = the number of first die is even Z = the number of second die is odd Pr(X∩Y) = ¼ = ½ * ½ Pr(X∩Z) = ¼ = ½ * ½ Pr(Y∩Z) = ¼ = ½ * ½ Pr(X∩Y∩Z) =0
![Page 13: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/13.jpg)
Homework 1-4
Let C1, C2, …, Ck be all possible min-cut sets.
2
)1(
1)1(
2
1) returns algorithmPr(
)1(
2) returns algorithmPr(
1
nnk
nn
k
C
nnC
k
ii
i
![Page 14: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/14.jpg)
Homework 1-5
i-th pair
![Page 15: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/15.jpg)
Homework 1-5
Indicator variable Pr(Xi) = 1 if the i-th pair are couple.
Pr(Xi) = 0 Otherwise
Linearity of expectation X= Σi=1 to 20Xi
E[X] = Σi=1 to 20E[Xi] = Σi=1 to 201/20 = 1
![Page 16: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/16.jpg)
Homework 1-6
pqqp
pq
pqqppq
qqpp
kYkYkX
YX
k
k
k
kk
k
1
11
)1(
)1()1(
)Pr()|Pr(
)Pr(
![Page 17: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/17.jpg)
Homework 1-6
)1)(1(1
111
)],[min(E][E][E)],[max(E
)1)(1(1
1)],[min(E
1][E
1][E
qpqp
YXYXYX
qpYX
qY
pX
![Page 18: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/18.jpg)
Homework 1-7
To choose the i-th candidate, we need i>m i-th candidate is the best [Event B] The best of the first i-1 candidates should
be in 1 to m. [Event Y]
![Page 19: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/19.jpg)
Homework 1-7
Therefore, the probability that i is the best candidate is
The probability of selecting the best one is
1
1)|Pr()Pr()Pr(
i
m
nBYBE iiii
n
mi
n
mii in
mEE
11 1
1)Pr()Pr(
![Page 20: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/20.jpg)
Homework 1-7
Consider the curve f(x)=1/x. The area under the curve from x = j-1 to x = j is less than 1/(j-1), but the area from x = j-2 to x = j-1 is larger than 1/(j-1).
![Page 21: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/21.jpg)
Homework 1-7
mndxxfj ee
n
m
n
mj
loglog)(1
1
1
)1(log)1(log)(1
1 1
11
mndxxfj ee
n
m
n
mj
![Page 22: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/22.jpg)
Homework 1-7
mnmg
nn
mnmg
mnn
mmg
ee
ee
1)(''
1loglog)('
)log(log)(
→g’(m)=0 when m=n/e
→g’’(m)<0 , g(m) is max when m=n/e
![Page 23: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/23.jpg)
Homework 1-7
e
ne
enne
ennnn
mnm
E
e
ee
ee
1
)(log
))/(loglog(
)loglog(
)Pr(
![Page 24: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/24.jpg)
Homework 1-8 (bonus)
S={±1, ±2, ±3, ±4, ±5, ±6} X={1,-2,3,-4,5,-6} Y={-1,2,-3,4,-5,6} E[X]=-0.5 E[Y]=0.5 E[Z] = -3.5
![Page 25: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/25.jpg)
Negative binomial random variable
![Page 26: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/26.jpg)
Negative binomial random variable
Definition y is said to be negative binomial random
variable with parameter r and p if
For example, we want r heads when flipping a coin.
rkr ppr
kky
)1(1
1)Pr(
![Page 27: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/27.jpg)
Negative binomial random variable
Let z=y-r
kr
kr
ppk
kr
ppr
kr
rky
kz
)1(1
)1(1
1
)Pr(
)Pr(
![Page 28: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/28.jpg)
Negative binomial random variable
Let r=1
Isn’t it familiar?
k
kr
pp
ppk
kr
kz
)1(
)1(1
)Pr(
![Page 29: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/29.jpg)
Negative binomial random variable
Suppose the rule of meichu game has been changed. The one who win three games first would be the champion. Let p be the probability for NTHU to win each game, 0<p<1.
![Page 30: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/30.jpg)
Negative binomial random variable
Let the event
We know
5
3
5
3
)()Pr() winsNTHUPr(k
kk
k APA
5,4,3 }gameth - on the winsNTHU{ kkAk
![Page 31: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/31.jpg)
Negative binomial random variable
where
Hence33
5
3
)1(2
1) winsNTHUPr(
k
k
ppk
33 )1(2
1)gameth on success rd3()Pr(
k
k ppk
kPA
![Page 32: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/32.jpg)
Negative binomial random variable
Given that NTHU has already won the first game. What is the probability of NTHU being the champion?
16
11
16
3
8
2
4
1
2
1
2
1
1
1 224
2
k
k
k
![Page 33: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/33.jpg)
Paintball game
![Page 34: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/34.jpg)
Paintball game
You James Bond, and Robin Hood decide to play a paintball game. The rules are as follows Everyone attack another in an order. Anyone who get “hit” should quit. The survival is the winner. You can choose shoot into air.
![Page 35: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/35.jpg)
Paintball game
James Bond is good at using gun, he can hit with probability 100%.
Robin Hood is a bowman, he can hit with probability 60%.
You are an ordinary person, can only hit with probability 30%.
The order of shot is you, Robin Hood then James Bonds.
![Page 36: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/36.jpg)
Paintball game
What is your best strategy of first shoot?
Please calculate each player’s probability of winning.
![Page 37: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/37.jpg)
Rope puzzle
![Page 38: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/38.jpg)
Rope puzzle
We have three ropes with equal length (Obviously, there are 6 endpoint).
Now we randomly choose 2 endpoints and tied them together. Repeat it until every endpoint are tied.
![Page 39: Randomized algorithm Tutorial 2 Solution for homework 1.](https://reader036.fdocuments.us/reader036/viewer/2022081512/56649cb95503460f949806b6/html5/thumbnails/39.jpg)
Rope puzzle
There can be three cases1.
2.
3.
Which case has higher probability?