The document introduces several sorting techniques, including bubble sort, selection sort, insertion sort, merge sort, and quick sort. It provides examples to illustrate how each algorithm works and the basic steps involved. For each sorting technique, it also discusses the time complexity, such as O(n2) for bubble sort and quicksort in the worst case, and O(n log n) on average for mergesort and quicksort.

1. Introduction to Sorting Techniques
Objectives
In this lesson, you will learn to:
State the different sorting techniques
State the complexities in the different sorting
algorithms
Introduction to Sorting Techniques/Lesson 6/Slide 1 of 40
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2. Introduction to Sorting Techniques
Sorting
Sorting and searching are fundamental operations
performed on data
Sorting refers to operation of arranging data in given
order
Searching refers to finding a location of given item
Algorithm chosen depends on operation to be
performed on data
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3. Introduction to Sorting Techniques
Sorting (Contd..)
Some sorting algorithms in use are:
Bubble sort
Selection sort
Insertion sort
Merge sort
Quick sort
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4. Introduction to Sorting Techniques
Bubble Sort
Take a list of numbers consisting of n elements
This algorithm requires to traverse the list n-1 times
Traversing the entire list once is called a pass
For a list of n numbers there will be n-1 passes
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5. Introduction to Sorting Techniques
Bubble Sort (Contd..)
Step 1:
Compare first two elements and arrange them
Compare second and third element and arrange
them
Compare fourth and fifth and arrange them
Compare n-1 with nth element and arrange them
Step 2:
Repeat step 1 with one less comparison
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6. Introduction to Sorting Techniques
Bubble Sort (Contd..)
Step 3:
Repeat step 1 with two less comparison
Step N:
This step will have only one comparison
Example:
Sort 5, 2, 6, 7, 3 using bubble sort
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7. Introduction to Sorting Techniques
Bubble Sort (Contd..)
Pass 1: Compare first and second element. If X1
X2 interchange as 2, 5, 6, 7, 3
Compare X2 with X3. If X2 X3 interchange
else the list remains as it is
Compare X3 with X4. Since X3 X4, list
remains the same
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8. Introduction to Sorting Techniques
Bubble Sort (Contd..)
Compare X4 with X5. Since X4 X5 interchange
happens
2, 5, 6, 3, 7
At the end of this pass, the largest number is placed
at the last position. Let us see what happens for the
remainder of passes
Pass 2:
2 5 6 3 7
2 5 6 3 7
2 5 3 6 7
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9. Introduction to Sorting Techniques
Bubble Sort (Contd.)
Pass 3 :
2 5 3 6 7
2 3 5 6 7
Pass 4 :
2 3 5 6 7
All the elements are sorted after the fourth pass
Introduction to Sorting Techniques/Lesson 6/Slide 9 of 40
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10. Introduction to Sorting Techniques
Problem Statement 6.D.1
Write a program to store 10 student marks. After
implementing the bubble sorting, display the contents
of sorted array.
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11. Introduction to Sorting Techniques
Selection Sort
The algorithm finds the smallest element and puts it in
first position
Then finds second smallest and puts in the second
position
This is done for n-1 times
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12. Introduction to Sorting Techniques
Selection Sort (Contd..)
Consider an array Num of n elements
Step 1:
Search for location of smallest element
Interchange the contents with the first location in
the array
Step 2:
Search for smallest element in sublist of n-1
elements
Interchange with the contents of the second
location
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13. Introduction to Sorting Techniques
Selection Sort (Contd..)
Step n:
Search for the smaller between n-1 and n and
interchange
Take a variable MIN to contain the current smallest
value
A variable LOC to contain the location of the
smallest value
Traverse the list comparing minimum with other
elements
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14. Introduction to Sorting Techniques
Selection Sort (Contd..)
If MIN = Num[i] then MIN=Num[i] else move to
next location
MIN will contain the smallest value after comparison
with the last element
LOC will contain the location of MIN
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15. Introduction to Sorting Techniques
Selection Sort (Contd..)
This procedure finds the smallest of the elements among
NUM[I], NUM[I+1], . .. , NUM[N]
MINIMUM(NUM,I,N,LOC)
1. Set MIN=NUM[I] and LOC = I
2. Repeat for J = I + 1, I + 2 , . . . , N
If MIN NUM[J]
then set MIN = NUM[J] and LOC = J
3. Return
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16. Introduction to Sorting Techniques
Selection Sort (Contd..)
Algorithm:
This algorithm sorts an array with N elements.
4. Repeat steps 2 and 3 for I=1,2,3,..., N
5. Call MINIMUM(NUM,I,N, LOC)
6. [Interchange Num[I] with NUM[LOC]]
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17. Introduction to Sorting Techniques
Selection Sort (Contd.)
Set TEMP= NUM[I],
NUM[I] = NUM[LOC]
NUM[LOC]=TEMP.
[loop of step 1 ends]
6. Exit
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18. Introduction to Sorting Techniques
Selection Sort (Contd..)
Pass Num[1] Num[2] Num[3] Num[4] Num[5] Num[6]
I=1 LOC=5 35 57 30 76 3 10
I=2 LOC=6 3 57 30 76 35 10
I=3 LOC=3 3 10 30 76 35 57
I=4 LOC=5 3 10 30 76 35 57
I=5 LOC=6 3 10 30 35 76 57
Sorted 3 10 30 35 57 76
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19. Introduction to Sorting Techniques
Insertion Sort
An element is put into its correct position with each
iteration of the main loop, relative to the sub-array that
has already been processed
Inserts each element to its proper location
Frequently used with small number of elements
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20. Introduction to Sorting Techniques
Insertion Sort (Contd..)
The following steps are performed
Pass 1: Num[1] by itself is sorted
Pass 2: Num[2] is inserted either before or after
Num[1] , such that Num[1] and Num[2] are sorted
Pass 3: Num[3] is inserted into its proper place, i.e.
before Num[1], between Num[1] and Num[2], or after
Num[2], such that Num[1], Num[2], Num[3] are sorted
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21. Introduction to Sorting Techniques
Insertion Sort (Contd..)
.... .... ....
.... .... ....
Pass N: Num[n] is inserted into its proper place such
that Num[1], Num[2],... ,Num[n] is sorted
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22. Introduction to Sorting Techniques
Insertion Sort (Contd..)
Pass Num[1] Num[2] Num[3] Num[4] Num[5] Num[6] Num[6]
K=1 -∞ 35 57 30 76 3 10
K=2 -∞ 35 57 30 76 3 10
K=3 -∞ 35 57 30 76 3 10
K=4 -∞ 30 35 57 76 3 10
K=5 -∞ 30 35 57 76 3 10
K=6 -∞ 3 30 35 57 76 10
Sorted -∞ 3 10 30 35 57 76
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23. Introduction to Sorting Techniques
Insertion Sort (Contd..)
Algorithm:
INSERTION (NUM,N)
The algorithm sorts the array NUM with N elements
1. Set Num[0] = -∞ [ Initialize the sentinel element]
2. Repeat steps 3 to 5 for K=2,3,4,... , N
3. Set TEMP=Num[K] and PTR =K-1
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24. Introduction to Sorting Techniques
Insertion Sort (Contd..)
1. Do while TEMP Num[PTR]
a. Set Num[PTR+1] = Num[PTR]
b. Set PTR=PTR – 1
(end of loop)
1. Set Num[PTR+1]=TEMP
(end of loop)
1. Return
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25. Introduction to Sorting Techniques
Problem Statement 6.P.1
Write a function implementing insertion sorting.
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26. Introduction to Sorting Techniques
Merge Sort
Step 1:
Split the list into sublists of equal size
Step 2:
Sort them separately
Step 3:
Merge them together
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27. Introduction to Sorting Techniques
Merge Sort (Contd..)
Example: Let us sort 53, 10,30,76,3,57,24 using merge
sort
Step 1:
Chop them into pairs and sort them
53, 10 30, 76 3, 57 24
Step 2:
Merge the pairs to obtain the following sorted sublists
10,53 30,76 3,57 24
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28. Introduction to Sorting Techniques
Merge Sort (Contd..)
Step 3:
Merge the sorted sublists to obtain a single sorted
array
3, 10, 24, 30, 53, 57, 76
The original array is now sorted
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29. Introduction to Sorting Techniques
Quick Sort
We arrange the list in some logical order
Algorithm is divide and conquer type
Sorting is reduced to sorting two smaller subsets
Reduction step finds the final position of one number
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30. Introduction to Sorting Techniques
Quick Sort (Contd..)
Step1:
Take first element as key element
Scan right to left for an element less than key
element
Interchange positions of the elements
Step 2:
Scan right to left for element greater than key
element
Interchange positions of the elements
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31. Introduction to Sorting Techniques
Quick Sort (Contd..)
Step 3:
Continue step 1 and step 2 till
® all element in right are greater than key
® all element in left are less than key
The key element will be placed in its correct position
Repeat reduction step for each sub list with two or
more elements
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32. Introduction to Sorting Techniques
Quick Sort (Contd..)
Example: 35 57 30 76 3 10
In this example we take the number 35 as the key
STEP 1: 10 57 30 76 3 35
STEP 2: 10 35 30 76 3 57
STEP 3: 10 3 30 76 35 57
STEP 3: 10 3 30 35 76 57
35 is correctly placed in its final position
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33. Introduction to Sorting Techniques
Problem Statement 6.D.2
Write a program to store 10 numbers in an array.
After implementing quick sorting, display the contents
of the sorted array.
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34. Introduction to Sorting Techniques
Complexities
For bubble sort
The number of comparisons
f(n) = (n-1) + (n-2) + (n-3) +...+ 2 + 1
= n(n-1)
2
= n2 + O(n) = O(n2)
2
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35. Introduction to Sorting Techniques
Complexities (Contd..)
For quick sort
Worst case (when the list is already sorted)
f(n) = n + (n-1) + (n-2) +...+2 + 1
= n(n-1) = n2 + O(n) = O(n2)
2 2
Average case
® In kth level finds the location of 2K-1 elements
f(n) = O(n log n)
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36. Introduction to Sorting Techniques
Complexities (Contd..)
For selection sort
Number of comparisons depends on original order of
elements
There are n-1 comparisons in pass1
There are n-2 comparisons in pass 2
f(n) = (n-1) + (n-2) + (n-3) +...+ 2 + 1
= n(n-1) = n2 + O(n) = O(n2)
2 2
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37. Introduction to Sorting Techniques
Complexities (Contd..)
For insertion sort
Worst case occurs when the array is in the reverse
order
Inner loop executes maximum number K-1 of
comparisons
f(n) = 1+ 2 + . . . + (n-1) = n(n-1) = O(n2)
2
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38. Introduction to Sorting Techniques
Complexities (Contd..)
Average case
There will be approximately (K-1)/2 comparisons in
the inner loop
f(n) = 1+ 2+ 3+ . . .+ (n-1) = n(n-1) = O(n2)
2 2 2 2 4
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39. Introduction to Sorting Techniques
Summary
In this lesson, you learned that:
Different types of sorting are:
Bubble sort
Selection sort
Insertion sort
Merge sort
Quick sort
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40. Introduction to Sorting Techniques
Summary (Contd..)
The time required to execute bubble sort algorithm is
proportional to n2, where n is the number of inputs
The number of comparisons in a selection sort is
independent of the original order of the elements
The insertion sort is a very slow algorithm when the
number of elements in the array is very large
The Quick sort algorithm has a worst case running
time of order n2/2
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