MERGE SORT

ANKIT S. CHITNAVIS

MERGE SORT :- All sorting method based on merging can be divided into two broad categories :-

1. Internal merge sort 2. External merge sort.

In internal merge sort , the lists under sorting are small and assumed to be stored in the high speed primary memory. There are two type of Internal merge sort :-

I. Simple merge sort.II. Two-way merge sort.

In external merge sort, deals with very large lists of elements such that size of primary memory is not adequate to accommodate the entire lists in it.

There are two type of External merge sort :- I. Balanced two-way merge sort.II. Multi-way merge sort.

MERGE SORT

INTERNALMERGE SORT

EXTERNALMERGE SORT

SIMPLE MERGE SORT

TWO -WAY MERGE

SORT

BALANCED TWO-WAYMERGE

SORT

MULTI -WAY MERGE

SORT

SIMPLE MERGE SORT :-

The simple merge sort (or merge sort) technique closely follows the divide –and-conqure paradigm.

let a list of n elements to be sorted with l and r being the position of leftmost and rightmost element in the list.

The three tasks in this divided-and-conqure technique are as followed :-

1. Divide :- Partition the list midway ,that is ,at[ (l+r)/2] into two sub lists with (n/2) elements in each, if n is even or [n/2] and [n/2]-1 element if n is odd .

2. Conquer :- Sort the two lists recursively using the merge sort.

3. Combine :- Merge the sort sub listed to obtain the sorted output.

Div

ide

Con

qure

Com

bine

..........................................................................

...................................

...................................

...................................

...................................

Sorted List

l r

[(l+r)/2]

Sort this left-part by merge sortSort this right-part by merge sort

Merge the two list

Fig. Divide-and-conqure strategy in the merge sort

Merge

Merge

Merge

Merge

Merge Sort

Merge Sort

Merge Sort

Merge Sort

Merge Sort

Merge SortDivide

DivideDivide

Divide

Divide

6 2 4 5 1 3

6 2 4 5 1 3

6 2 4

6 2

6 2

2 6

2 4 6

5 1 3

5 1

5 1

1 5

1 3 5

1 2 3 4 5 6

4 3

Merge Sort

Merge Sort

Merge Sort

EX :-

I/P ->I/P ->

O/P ->

Algorithm Merge sort

Input :- An array A[l…r] where l and r are the lower and upper index of A.Output :- Array a[l…r] with all element arranged in ascending order .

Steps: 1. if(r<=l) then2. Return3. else

4. Mid=[(l+r)/2]5. MergeSort(A[l…mid])6. MergeSort(A[mid+1…r])7. Merge(A,L,mid,r)8. Endif9. Stop

Two-Way Merge Sort :-

The two-way merge sort is based on the principle ‘burns the candle at both ends’ in manner similar to the scanning procedure.

In the two-way merge sorting, we examine the input list from both the ends:

Left and Right and moving toward the middle.

i j k l

qP

Scan is doneScan is done

Ascending run at right

Ascending run at left

Stored merge sequneces

Stored merge sequneces

(a) Source list

(b) Destination list

EX :-

44 99

57 63 77 55 88 22 96 33 1

1 66A

A

B

B

Input List->

Auxiliary List ->

<- output List

(a) After Pass 1

(b) After Pass 2

(c) After Pass 3

11 22

33 44 55 57 63 66 77 88 9

6 99

44 66

99 22 55 88 96 77 63 57 3

3 11

A

B 11 33

44 57 63 66 77 96 99 88 5

5 22

44 66

99 22 55 88 96 77 63 57 3

3 11

11 33

44 57 63 66 77 96 99 88 5

5 22

3

1 1

11 1

11 1

1

2 2

22

2

3 3

BALANCED TOW-WAY MERGE SORT :-

The balanced two-way merge is based on combining two ascending runs into a single ascending run.

The merging-procedure can be applied to more than two runs at each time. That why it is two way merging.

In external balanced two-way merge all intial intial runs of equal length possibly not the last run.

This why the ‘ balanced’ tag is used in this technique.

Example :-

Run :- A sorted segment of a file, which is termed as an ascending run or simply run.

MULTI-WAY MEGRE SORT :-

In multi-way merge sort , this procedure is extended to m-way merging , where m (m>2) runs are combined into a single run.

In the multi-way merge sort , m input runs are stored on m tapes.

Initially all element are in ascending order on all these tapes.

In multi-way merge , we merge them together , that is ,we look at the first element of each run and select the smallest element.

This smallest element tranferred to an output tape.

This process is repeated till all runs are fully examined.

Example :-

11 44 66 88

10 20 50 90

15 35 55 75

22 60 65 99

11 44 66 88

10 20 50 90

15 35 55 75

22 60 65 99

10

T1

T2

T3

T4

T1

T2

T3

T4

4 input runs on tape

4 input runs on tape

Output Tape

Output Tape

T

T

(a) Initial Status

(b) After step 1

11 44 66 88

10 20 50 90

15 35 55 75

22 60 65 99

10

11

11 44 66 88

10 20 50 90

15 35 55 75

22 60 65 99

10

11

15

20

22

35

44

50

55

60

65

66

75

88

90

99

T1

T2

T3

T4T1

T2

T3

T4

. . . . . . . . . . . . . . . . . . . . . . . . .

4 input runs on tape

4 input runs on tape

Output Tape

Output Tape

T

T

(c) After step 2

(b) After step 16

Fig :- illustration of the multi-way merge sort.

THANK

YOU