1
Name:
Class:
Muhammad Saim ul haq
BSSE(3A)
Roll Number:
21-Arid-3807
Subject:
Data Structure And Algorithm
2
Name:
Class:
Kamran Arshad
BSSE(3A)
Roll Number:
21-Arid-3799
Subject:
Data Structure And Algorithm
3
Name:
Ayesha Ameen
Class:
BSSE(3A)
Roll Number:
21-Arid-3794
Subject:
Data Structure And Algorithm
4
MERGE SORT
The Merge Sort algorithm is a sorting algorithm
that is based on the Divide and
Conquer paradigm. In this algorithm, the array is
initially divided into two equal halves and then
they are combined in a sorted manner.
5
How Merge Sort Works?
• Merge sort first divides the array into equal halves
and then combines them in a sorted manner.
• To understand merge sort, we take an unsorted
array as the following −
6
We know that merge sort first
divides the whole array iteratively
into equal halves unless the
atomic values are achieved. We
see here that an array of 8 items
is divided into two arrays of size
4.
7
• This does not change the sequence of appearance of items in the
original. Now we divide these two arrays into halves.
• We further divide these arrays and we achieve atomic value
which can no more be divided.
• Now, we combine them in exactly the same manner
as they were broken down. Please note the color
codes given to these lists.
8
• In the next iteration of the combining phase, we
compare lists of two data values, and merge them into
a list of found data values placing all in a sorted
order.
• After the final merging, the list should look like this −
9
• Merge sort keeps on dividing the list into equal
halves until it can no more be divided. By definition, if
it is only one element in the list, it is sorted. Then,
merge sort combines the smaller sorted lists keeping
the new list sorted too.
10
INSERTION SORT
INTRODUCTION:
 Insertion sort is a sorting algorithm that places an unsorted element at its
suitable place in each iteration.
 It works similar to the way you sort playing cards in your hands.
 Insertion sort is efficient for small data values.
11
Working of Insertion Sort
Suppose we need to sort the following array.
The first element in the array is assumed to be sorted. Take the second
element and store it separately in (KEY)
Compare (KEY) with the first element. Ifthe first element is greater than (key),
then (KEY)
is placed in front of the first element.
12
Now, the first two elements are sorted.
Take the third element and compare it with the elements on the left of it.
Placed it just behind the element smaller than it.
Ifthere is no element smaller than it, then place it at the beginning of the
array.
13
Similarly, place every unsorted element at its correct position.
Shell sort
Shell sort is a generalized version of the
insertion sort algorithm. It first sorts
elements that are far apart from each
other and suc c essively reduces the
interval between the elements to be
sorted. The interval between the elements
is reduced based on the sequence used.
We keep reducing the value of the
interval until it becomes 1.
Sort the Data Set by Shell Sorting
Method
70
80
15
30
60
20
First We have to find the Gap
0
1
2
3
4
5
70
80
15
30
60
20
Formula :
Interval/G ap= Floor(n/2)
n= Length= 6
Put the Value in Formula :
Interval/G ap= Floor(6/2) =3
0
1
2
3
4
5
70
80
15
30
60
20
70
30
80
60
20
15
Pass 1:
(0,3) =
30
40
15
70
60
20
(1,4) =
30
60
15
70
80
20
(2,5) =
30
60
15
70
80
20
18
30
60
15
70
80
20
Find the Again Gap Value by Using
the Previous Gap Value
• Interval/Gap=Floor(n/2)
=Floor(3/2)
Pass 2:
=Floor(1.5)=1
30
60
30
60
15
15
30
60
15
60
30
15
60
15
20
15
30
30
70
80
20
70
80
20
80
20
80
20
70
70
70
60
80
70
20
80
20
Bubble Sorting
Bubble sort is a simple sorting algorithm. This sorting
algorithm is comparison-based algorithm in which each
pair of adjacent elements is compared and the elements
are swapped if they are not in order. This algorithm is not
suitable for large data sets as its average and worst case
complexity are of Ο(n2) where n is the number of items.
21
How Bubble Sort Works?
We take an unsorted array for our example. Bubble sort
takes Ο(n2) time so we're keeping it short and precise.
14
33
27
35
10
Bubble sort starts with very first two elements, comparing
them to check which one is greater.
14
33
27
35
10
22
In this case, value 33 is greater than 14, so it is already in sorted locations. Next, we compare
33 with 27.
14
33
27
35
10
We find that 27 is smaller than 33 and these two values must be swapped.
14
27
33
35
10
Next we compare 33 and 35. We find that both are in already sorted
positions.
14
27
33
35
10
23
Then we move to the next two values, 35 and 10. We know then that 10 is smaller
35. Hence they are not sorted.
14
27
33
35
10
We swap these values. We find that we have reached the end of the array. After one
iteration, the array should look like this
14
27
33
10
35
To be precise, we are now showing how an array should look like after each
iteration. After the second iteration, it should look like this
14
27
10
33
35
24
Notice that after each iteration, at least one value moves at the end.
14
10
27
33
35
And when there's no swap required, bubble sorts learns that an array is completely
sorted.
10
14
27
33
35
25
Selection Sort
• The selection sort algorithm sorts an array by repeatedly finding
the minimum element from the unsorted part and putting it at the
beginning.
• In every iteration of the selection sort, the minimum element
from the unsorted subarray is picked and moved to the sorted
subarrays.
26
Selection Sort
• The selection sort algorithm sorts an array by repeatedly finding
the minimum element from the unsorted part and putting it at the
beginning.
• In every iteration of the selection sort, the minimum element
from the unsorted subarray is picked and moved to the sorted
subarrays.
27
How Selection Sort work?
Consider the following array as an example.
7
6
3
4
1
For the first position in the sorted list, 7 in first position is stored
presently, we search the whole list and find 1 is the lowest value.
7
6
3
4
1
28
How Selection Sort work?
Consider the following array as an example.
7
6
3
4
1
For the first position in the sorted list, 7 in first position is stored
presently, we search the whole list and find 1 is the lowest value.
7
6
3
4
1
29
So we replace 7 with 1.After one iteration 10, which happens to the
minimum value in the list, appears in the first position of the sorted list.
1
6
3
4
7
For the second position, where 6 is residing, we start scanning the rest of the
list in a linear manner.
1
6
3
4
7
We find that 3 is the second lowest value in the list and it should appear at the second
place. We swap these values.
30
So we replace 7 with 1.After one iteration 10, which happens to the
minimum value in the list, appears in the first position of the sorted list.
1
6
3
4
7
For the second position, where 6 is residing, we start scanning the rest of the
list in a linear manner.
1
6
3
4
7
We find that 3 is the second lowest value in the list and it should appear at the second
place. We swap these values.
31
1
3
6
4
7
After two iterations, two least values are positioned at the
beginning in a sorted manner.
1
3
6
4
7
The same process is applied to the rest of the entire in the
array.
1
3
4
6
7
32
1
3
6
4
7
After two iterations, two least values are positioned at the
beginning in a sorted manner.
1
3
6
4
7
The same process is applied to the rest of the entire in the
array.
1
3
4
6
7