Lec 13 BblSORT Mul

7/30/2019 Lec 13 BblSORT Mul

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Fall-2012

Lecture#13

Data Structures and Algorithms

CSE 246


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Multiplication of large integer

Quratulain 2

The Karatsuba Ofman algorithm provides a

striking example of how the Divide and

Conquer technique can achieve an

asymptotic speedup over an ancient

algorithm.

The school classroom method of multiplying

two n-digit integers requires O(n2) digitoperations.

He show that a simple recursive algorithm

solves the problem in (n) digit operations,


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Karatsuba Algorithm

Quratulain 3

X=4837

Y=5813

X and Y are 4-digit number. Divide numbers in to half until

you get number of digit one.A=48, B=37 , C=58, D=13

Therefore

X*Y= (10n/2 A+B) (10n/2 C+D)

By multiplying

=10n/2 A 10n/2 C + 10n/2 AD+ 10n/2 BC +BD

= 10n/2 AC + 10n/2 (AD+BC)+BD only 4 multiplication


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KARATSUBA Algorithm

Quratulain 4

= 10n/2 AC + 10n/2(AD+BC)+BD (1)

Now replace (AD+BC) with only one

multiplication

(A+B)(C+D)=AC+AD+BC+BD =this inludes

(AD+BC)

Thus,(AD+BC)=(A+B)(C+D)-AC-BD

Finally, replace (AD+BC) in eq(1) , it become

= 10n/2 AC + 10n/2 ((A+B)(C+D)-AC-BD)+BD


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Algorithm

Quratulain 5

multiply(1113,1

412)

Call recursivelyuntil number is

of digit one.

Multiply(X , Y){

Size=max(X.length,Y.length)

If (Size==1)

return X*Y;

A=11 ,B=13, C=14, D=12

AC=multiply(A,C)

BD=multiply(B,D)

ApB=Add(A,B)CpD=Add(B,D)

Term=multiply(ApB,CpD)

G=Term-AC-BD

Newsize=Size/2Tem =Add BD AddZero G Newsiz


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Quratulain 6

X=1213 , Y=1412

Size=4

A=12 ,B=13, C=14, D=12

AC=multiply(A,C)

BD=multiply(B,D)

ApB=Add(A,B)

CpD=Add(B,D)Term=multiply(ApB,CpD)

G=Term-AC-BD

Newsize=2

Temp=Add(BD,AddZero(G,newsize

))

Return

Add(Temp,AddZero(Temp,size))

X=12 , Y=14

size=2

A=1 ,B=2, C=1, D=4

AC=multiply(A,C)

BD=multiply(B,D)

ApB=Add(A,B)

CpD=Add(B,D)

Term=multiply(ApB,CpD)G=Term-AC-BD

Newsize=1


))

Return

Add(Temp,AddZero(AC,size))

X=25 , Y=26

size=4

A=12 ,B=13, C=14, D=11

AC=multiply(A,C)

BD=multiply(B,D)

ApB=Add(A,B)

CpD=Add(B,D)

Term=multiply(ApB,CpD)

G=Term-AC-BDNewsize=2


))

Return

Add(Temp,AddZero(Temp,size))

X=13 , Y=12

size=2

A=1 ,B=3, C=1, D=2

AC=multiply(A,C)

BD=multiply(B,D)

ApB=Add(A,B)

CpD=Add(B,D)

Term=multiply(ApB,CpD)G=Term-AC-BD

Newsize=1


))

Return

Add(Temp,AddZero(AC,size))

X=1 , Y=1

Size=1

Return

X*Y

X=2 , Y=4

Size=1

Return

X*Y

Return 1+2

Return 1+4

X=3 , Y=5

Size=1

Return

X*Y

Return

8+60

Return

68+100

Call

Ret 69

X=1 , Y=1

Size=1

Return

X*Y

X=3, Y=2

Size=1Return

X*Y

Return 1+3

Return 1+2

X=4 , Y=3

Size=1

Return

X*Y

Return

6+50

Return

56+100

Visual representation of function call

Do it . And compute final answer


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Elementary Sorting

Methods

Lecture #13


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Why study elementary methods?

They provide context in which we can learnterminology and basic mechanisms for sortingalgorithms

These simple methods are actually more effectivethan the more powerful general-purposemethods in many applications of sorting

Third, several of the simple methods extend to

better general-purpose methods or are useful inimproving the efficiency of more sophisticatedmethods.

4/28/2013 Muhammad Usman Arif 8


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Time

Elementary methods we discuss take time N2

to sort N randomly arranged items.

These methods might be fast than most of the

sophisticated algorithms for smaller files.

Not suitable for large files.



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External or Internal?

If the file to be sorted will fit into memory,

then the sorting method is called internal

Internal sort can access any item easily

Sorting files from tape or disk is called

external sorting

External sort must access items sequentially, or at

least in large blocks



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The Bubble Sort

Algorithm

The Bubble Sort compares adjacent elements in alist, and swaps them if they are not in order.Each pair of adjacent elements is compared andswapped until the largest element bubbles tothe bottom. Repeat this process each timestopping one indexed element less, until you

compare only the first two elements in the list.We know that the array is sorted after both of thenested loops have finished.



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A Bubble Sort Example


6

5

43

2

1

Compare

We start by comparing the first

two elements in the List.


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5

6

43

2

1

Swap


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5

6

43

2

1

Compare


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5

4

63

2

1

Swap


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5

4

63

2

1

Compare


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5

4

36

2

1

Swap


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5

4

36

2

1

Compare


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5

4

32

6

1

Swap


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5

4

32

6

1Compare


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5

4

32

1

6Swap

As you can see, the largest

number has bubbled down to

the bottom of the List after the

first pass through the List.


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5

4

32

1

6

Compare

For our second pass throughthe List, we start by

comparing these first two

elements in the List.


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4

5

32

1

6

Swap


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4

5

32

1

6

Compare


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4

3

52

1

6

Swap


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4

3

52

1

6

Compare


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4

3

25

1

6

Swap


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4

3

25

1

6

Compare


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4

3

21

5

6

Swap

At the end of the second pass, we

stop at element number n - 1,

because the largest element in the

List is already in the last position.


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4

3

21

5

6

Compare

We start with the first twoelements again at the beginning

of the third pass.


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3

4

21

5

6

Swap


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3

4

21

5

6

Compare


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3

2

41

5

6

Swap


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3

2

41

5

6

Compare


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3

2

14

5

6

Swap

At the end of the third pass, we stop

comparing and swapping at element

number n - 2.


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3

2

14

5

6

Compare

The beginning of the fourth pass...


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2

3

14

5

6

Swap


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2

3

14

5

6

Compare


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2

1

34

5

6

Swap

The end of the fourth pass

stops at element number n - 3.


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2

1

34

5

6

Compare

The beginning of the fifth pass...


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1

2

34

5

6

Swap

The last pass compares onlythe first two elements of the

List. After this comparison

and possible swap, the

smallest element has

bubbled to the top.


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What Swapping Means

6

5

43

2

1

TEMP

Place the first element into the

Temporary Variable.

6


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What Swapping Means

5

5

43

2

1

TEMP

Replace the first element with

the second element.

6


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What Swapping Means

5

6

43

2

1

TEMP

Replace the second element

with the Temporary Variable.

6


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Java Code For Bubble Sort



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Java Code for Bubble sort


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Big - O Notation

Big - O notation is used to describe the efficiency of a

search or sort. The actual time necessary to

complete the sort varies according to the speed of

your system. Big - O notation is an approximatemathematical formula to determine how many

operations are necessary to perform the search or

sort. The Big - O notation for the Bubble Sort is

O(n2), because it takes approximately n2 passes tosort the elements.


B bbl t Si l t f ll b t


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Bubble sort. Simplest of all, but

also slowest

Worst case performance

Best case performance

Makes passes through a sequence of items.

On each pass it compares adjacent elements

in pairs and swaps them if they are out of

order.

Lec 13 BblSORT Mul

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