This document analyzes and compares sorting algorithms including bubble sort, insertion sort, selection sort, shell sort, merge sort, and quick sort. It describes the time complexity of each algorithm using asymptotic analysis and experimental results. Testing was done on different sized data sets from 1-100 values, 1-1000 values, and 1-10000 values. Quick sort performed the best overall, having an average time complexity of O(n log n), while bubble sort performed the worst with O(n^2) time complexity. In conclusion, quick sort is the most efficient sorting method, while bubble sort is the least efficient.

1. STUDY & ANALYSIS OF
SORTING METHODS
By
Aditya Y. Vichare
Department Of Mathematics

2. CONTENTS
1. System Information
2. Algorithm Analysis
3. Sorting Algorithm
4. Observations
5. Conclusion
6. Credits And Bibliography

3. SYSTEM INFORMATION
▸ Operating System : Linux
▸ Distro : Ubuntu 16.0
▸ Type : 64 - Bit
▸ Processor : Intel® Core™2 CPU 4400 @ 2.00GHz × 2
▸ Chipset : Intel® G41
▸ RAM : 2GB
▸ Avg. Temperature of Cores : 55-75 C
SOFTWARE AND MODULE INFORMATION
▸ Language : Python 3.4
▸ Modules : Matplotlib , Time

4. ALGORITHM ANALYSIS
▸A Priori Analysis −
● This is a theoretical analysis of an algorithm.
●Efficiency of an algorithm is measured by assuming that all other
factors,
● e.g. processor speed, are constant and have no effect on the
implementation.
▸A Posterior Analysis −
●This is an empirical analysis of an algorithm.
●The selected algorithm is implemented using programming language.

5. ALGORITHM COMPLEXITY
▸ Time Complexity
Amount of time required by the algorithm to run to completion.
▸ Space Complexity
Amount of memory space required by the algorithm in its life cycle.
EXPERIMENTAL STUDIES
▸ Study the runtime of algorithm
▸ Executing it on various test inputs
▸ Recording of time spent during each execution.
ASYMPTOTIC ANALYSIS
Defining the Mathematical boundaries of the function.

6. SELECTED SORTING METHODS
Merge
Sort
Sorting
Methods
Quick
Sort
Shell
Sort
Insertion
Sort
Bubble Sort
Selection
Sort

7. BUBBLE SORT
▸ Repeatedly pass through the array
▸ Swaps adjacent elements that are out of order

8. INSERTION SORT
▸ Assume first element as
sorted.
▸ Maintain sorted sub list by
adding element.
▸ Insert each new item in
sub list w.r.t. value

9. ▹Find the smallest / largest
element in the array
▹Exchange it with the
element in the first position /
last position
▹Find the second smallest /
largest element and
exchange it with the element
in the second position
▹Continue until the array is
sorted
SELECTION SORT

10. ▸ Divide list into smaller sub list of equal interval h
▸ Sort them by Insertion Sort
▸ Repeat until whole list get sorted
SHELL SORT

11. MERGE SORT
▸ If it is only one element, then its already sorted.
▸ Divide the list recursively into two halves until it can no more get
divided
▸ Merge the smaller list into sorted order

12. QUICK SORT
▸ Make the right most index value pivot
▸ Partition the array using pivot value
▸ Quicksort(left partition)
▸ Quicksort(Right partition)

13. OBSERVATIONS
OF EXPERIMENTAL ANALYSIS
3 Categories
2,214 Total values
6 Algorithms
PROPERTIES :

14. COMPARISON TABLE OF TIME COMPLEXITY
USING ASYMPTOTIC ANALYSIS
TIME
SORTING METHODS AVERAGE CASE BEST CASE WORST CASE
Bubble Sort O(n^2) O(n^2) O(n^2)
Selection Sort O(n^2) O(n^2) O(n^2)
Insertion Sort O(n^2) O(n) O(n^2)
Shell Sort O(n*log n) O(n*log^2 n) O(n*log^2 n)
Merge Sort O(n*log n) O(n*log n) O(n*log n)
Quick Sort O(n*log n) O(n*log n) O(n^2)

15. CATEGORY : 1- 100
▸ Almost Double time taken by
Bubble Sort than rest.
▸ Selection Sort & Insertion Sort
Shows same Behavior.
Bubble Sort, Selection Sort , Insertion Sort
Selection Sort,
Insertion Sort
Bubble Sort

16. CATEGORY : 1- 100 Quick Sort, Merge Sort , Shell Sort
Merge Sort
Quick Sort

17. CATEGORY : 1- 100
▸ Bubble Sort
All Methods
▸ Quick Sort

18. CATEGORY : 1-1000 Bubble Sort, Selection Sort , Insertion Sort
Selection Sort
Bubble Sort

19. CATEGORY : 1-1000 Quick Sort, Merge Sort , Shell Sort
Selection Sort
Quick Sort

20. CATEGORY : 1-1000
Quick Sort
All Methods
Bubble Sort

21. CATEGORY : 1-10000
Selection Sort
Bubble Sort
Bubble Sort, Selection Sort , Insertion Sort

22. CATEGORY : 1-10000
Quick Sort
Shell Sort
Quick Sort, Merge Sort , Shell Sort

23. CATEGORY : 1-10000
Quick Sort, Merge Sort,
Shell Sort
Bubble Sort
All Methods

24. Bubble Sort, Insertion Sort,
Selection Sort O(n^2)
▸ Bubble sort is least efficient sorting
method among all.
▸ Bubble Sort and Insertion Sort are
for small size array sorting.
▸ Selection sort is more efficient than
other two O(n^2) methods.
CONCLUSION
Shell Sort, Merge Sort, Quick Sort
O(n log n)
▸ Shell Sort is least efficient among
all O(n log n) methods.
▸ Merge Sort requires double
memory allocation.
▸ Quick Sort is most efficient method
among six.

25. SPEED OF SORTING METHODS IN DESCENDING ORDER
1. Quick Sort
2. Merge Sort
3. Shell Sort
4. Selection Sort
5. Insertion Sort
6. Bubble Sort
CONCLUSION

26. CREDITS AND BIBLIOGRAPHY
Special thanks to Dr. Rajesh Raut of Department of Mathematics.
▸ Beginning Python, JAMES PAYNE.
▸ Data Structures And Algorithm in Python Goodrich, Michael T., Tamassia.
▸ Geekforgeeks.org
▸ Algolist.com
▸ Matplotlib.org

27. THANK YOU!
Any questions?
You can contact me at vichareaditya123@gmail.com
By Aditya Y. Vichare.
adityavichare.wordpress.com