1 Majority Item Problem. The dataset we will use is hw3a.txt, which consists of a series of integral numbers.Please treat it as a data stream. For a given value j, let fj.= |{1 ≤ i ≤ n : xi = j}|,...

1 Majority Item Problem.

The dataset we will use is hw3a.txt, which consists of a series of integral numbers.

Please

treat

it

as

a

data

stream.

For

a

given

value

j
,

let

f
j

.
=

|{
1

≤

i

≤

n

:

x
i

=

j
}|
,

i.e.,

the

frequency

of value

j

in the stream.

In the class, we have discussed in detail an algorithm that can output all values

j

satisfying

f
j

> n/k
,

where

k

is a given parameter (known in advance),

and

n

is the size of the data stream

(unknown to the algorithm).

1.1

Please

run

the

algorithm

as

shown

in

the

class

and

output

all

values

j

with

f
j

> n/k

where

k

=

40.

Please attach your codes as part of your solution in addition to the final output.

(30 points)

1.2

Please

run

the

modified

version

of

the

algorithm

after

swapping

the

two

“Ifs”,

which

is

first

to

check

if

there

is

some

empty

box

and

then

check

if

there

is

some

existing

box

storing

the

current

arriving

value.

Compare

the

output

here

with

that

in

1.1.

Is

there

any

difference

between

the

two

outputs?

If

yes,

do

we

miss some feasible candidate of value

j

with

f
j

> n/k
?

(10 points)

2

Reservoir

Sampling.

We

will

use

the

file

hw3b.txt

as

the

data

stream.

For

both

questions

2.1

and

2.2,

please attach your codes as part of your solution in addition to the final outputs.

2.1 In the class, we talk about the original definition of a random sample of a given size without replacement

from a data stream.

Please follow the definition and output a random sample of size 70 without replacement.

(20 points)

2.2

Implement

the

reservoir

sampling

algorithm

discussed

in

class

to

output

a

random

sample

of

size

70

without replacement.

(20 points)

2.3 Run the above two methods each for 1000 times and use the results to calculate the probability of each

number being included in the final sample.

Show that the results of the two methods are approximately the

same.

(20 points)

hw3-3fpl1oy2.pdf

Nov 16, 2022

SOLUTION.PDF

1 Majority Item Problem. The dataset we will use is hw3a.txt, which consists of a series of integral numbers.Please treat it as a data stream. For a given value j, let fj.= |{1 ≤ i ≤ n : xi = j}|,...

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