Algorithms Lecture 4: Sorting Algorithms I

Analysis and Design of Algorithms
Sorting Algorithms I

Sorting Algorithms
Bubble Sort
Selection Sort
Insertion Sort

 Sorting Algorithm is an algorithm made up of a series of instructions
that takes an array as input, and outputs a sorted array.
 There are many sorting algorithms, such as:
 Selection Sort, Bubble Sort, Insertion Sort, Merge Sort,
Heap Sort, QuickSort, Radix Sort, Counting Sort, Bucket
Sort, ShellSort, Comb Sort, Pigeonhole Sort, Cycle Sort

Bubble Sort

Bubble Sort is the simplest sorting algorithm
that works by repeatedly swapping the
adjacent elements if they are in wrong
order.

Algorithm:
 Step1: Compare each pair of adjacent elements in the list
 Step2: Swap two element if necessary
 Step3: Repeat this process for all the elements until the
entire array is sorted

 Example 1 Assume the following Array:
5 1 4 2

 First Iteration:
 Compare
5 1 4 2

j

j+1

 Swap
1 5 4 2

j

j+1

 Compare
1 5 4 2

j

j+1

 Swap
1 4 5 2

j

j+1

 Compare
1 4 5 2

j

j+1

 Swap
1 4 2 5

j

j+1

1 4 2 5

 Second Iteration:
 Compare
1 4 2 5

j

j+1

 Second Iteration:
 Swap
1 2 4 5

j

j+1

1 2 4 5

 Third Iteration:
 Compare
1 2 4 5

j

j+1

 Array is now sorted
1 2 3 4

5 4 2 1 3
4 5 2 1 3
4 2 5 1 3
4 2 1 5 3
4 2 1 3 5
4 2 1 3 5
2 4 1 3 5
2 1 4 3 5
2 1 3 4 5
2 1 3 4 5
1 2 3 4 5
1 2 3 4 5
 Example 2:

 What is the output of bubble sort after the 1st iteration given the
following sequence of numbers: 13 2 9 4 18 45 37 63
a) 2 4 9 13 18 37 45 63
b) 2 9 4 13 18 37 45 63
c) 13 2 4 9 18 45 37 63
d) 2 4 9 13 18 45 37 63

 Python Code

 Time Complexity: O(n2) as there are two nested loops
 Example of worst case
5 4 3 2 1

Selection Sort

The selection sort algorithm sorts an array by
repeatedly finding the minimum element
(considering ascending order) from unsorted
part and putting it at the beginning.

Algorithm:
 Step1: Find the minimum value in the list
 Step2: Swap it with the value in the current position
 Step3: Repeat this process for all the elements until the
entire array is sorted

 Example 1 Assume the following Array:
8 12 5 9 2

 Compare
8 12 5 9 2

i

j

min

 Move
8 12 5 9 2

j

i

min

 Compare
8 12 5 9 2

i

min

j

 Move
8 12 5 9 2

i

j

min

 Smallest
8 12 5 9 2

i

min

 Swap
8 12 5 9 2

min

i

 Sorted
 Un Sorted
2 12 5 9 8

Sorted

Un Sorted

 Compare
2 12 5 9 8

i

min

j

Sorted

 Move
2 12 5 9 8

i

min

j

Sorted

 Compare
2 12 5 9 8

Sorted

i

j

min

 Smallest
2 12 5 9 8

Sorted

i

min

 Swap
2 12 5 9 8

Sorted

i

min

 Sorted
 Un Sorted
2 5 12 9 8

Sorted

Un Sorted

 Compare
2 5 12 9 8

Sorted

i

j

min

 Move
2 5 12 9 8

Sorted

i

j

min

 Compare
2 5 12 9 8

Sorted

i

min

j

 Move
2 5 12 9 8

Sorted

i

min

j

 Smallest
2 5 12 9 8

Sorted

i

min

 Swap
2 5 12 9 8

Sorted

i

min

 Sorted
 Un Sorted
2 5 8 9 12

Sorted

Un Sorted

 Compare
2 5 8 9 12

Sorted

i

min

j

 Sorted
 Un Sorted
2 5 8 9 12

Sorted

i

min

2 5 8 9 12

Sorted

12 10 16 11 9 7 Example 2:
7 10 16 11 9 12
7 9 16 11 10 12
7 9 10 11 16 12
7 9 10 11 16 12
7 9 10 11 12 16
12 10 16 11 9 7

 What is the output of selection sort after the 2nd iteration given
the following sequence of numbers: 13 2 9 4 18 45 37 63
a) 2 4 9 13 18 37 45 63
b) 2 9 4 13 18 37 45 63
c) 13 2 4 9 18 45 37 63
d) 2 4 9 13 18 45 37 63

 Time Complexity: O(n2) as there are two nested loops
2 3 4 5 1

Insertion Sort

Insertion sort is a simple sorting
algorithm that works the way we sort
playing cards in our hands.

 Algorithm:
 Step1: Compare each pair of adjacent elements in the list
 Step2: Insert element into the sorted list, until it occupies correct
position.
 Step3: Swap two element if necessary
 Step4: Repeat this process for all the elements until the entire
array is sorted

 Assume the following Array:
5 1 4 2

 Compare
 Store=
5 1 4 2

i

j

j+1
1

 Move
 Store=
5 4 2

i

j

j+1
1

 Move
 Store=
1 5 4 2

i

j+1

 Compare
 Store=
1 5 4 2

i

j
4

j+1

 Move
 Store=
1 5 2

i

j
4

j+1

 Compare
 Store=
1 5 2

i

j
4

j+1

 Move
 Store=
1 4 5 2

i

j

j+1

 Compare
 Store=
1 4 5 2

i

j
2

j+1

 Move
 Store=
1 4 5

i

j
2

j+1

 Compare
 Store=
1 4 5

i

j
2

j+1

 Compare
 Store=
1 2 4 5

i

j

j+1

1 2 4 5

5 1 8 3 9 2
 Example 2:
1 5 8 3 9 2
1 5 8 3 9 2
1 3 5 8 9 2
1 3 5 8 9 2
1 2 3 5 8 9
5 1 8 3 9 2

 What is the output of insertion sort after the 1st iteration given the
a) 3 7 5 1 9 8 4 6
b) 1 3 7 5 9 8 4 6
c) 3 4 1 5 6 8 7 9
d) 1 3 4 5 6 7 8 9

 What is the output of insertion sort after the 2nd iteration given the
a) 3 5 7 1 9 8 4 6
b) 1 3 7 5 9 8 4 6
c) 3 4 1 5 6 8 7 9
d) 1 3 4 5 6 7 8 9

 Time Complexity: O(n2)
5 4 3 2 1

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Algorithms Lecture 4: Sorting Algorithms I

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Algorithms Lecture 4: Sorting Algorithms I