Square Root Sorting Algorithm

Square Root Sorting Algorithm
Dissertation submitted in partial fulfillment of the requirement
for the degree of
Master of Technology
In
Computer Science & Engineering
Under the Supervision of
MS Nisha Gautam
Assistant Professor
By
Mir Omranudin Abhar
17001651
Department of Computer Science and Engineering
APG (Alakh Prakash Goyal) SHIMLA UNIVERSITY
Shoghi-Mehli By-Pass Road Shimla-171009
Himachal Pradesh, India

i
Declaration
I’m Mir Omranudin Abhar, declare that the dissertation work with the title “Square
Root Sorting Algorithm” is my original work carried out under the guidance of
“Ms. Nisha Gataum”. I have learned and followed all the rules and regulations
provided by the university while writing this dissertation. I have tested this
dissertation for plagiarism which is under 23 percent in similarity. This dissertation
work is being submitted in the fulfillment of the requirement for the degree of Master
of Technology in Computer Science and Engineering at APG (Alakh Prakash Goyal)
Shimla University, Shimla (HP) for the academic session 2017-2019.
This research work has not been submitted partially or fully to any other University or
Institute for the award of this or any other degree or diploma.
Mir Omranudin Abhar: 10-07-2019
Enrollment Number: 17001651
Department of CSE
APG (Alakh Prakash Goyal) Shimla University
Shimla-171009 (HP)
………………. …………………. ……………………….
Nisah Gataum Nisah Gataum External Examiner
Assistant Professor HOD
Department of CSE Department of CSE

ii
Completion Certificate
This is to certify that the thesis entitled “Square Root Sorting Algorithm” is being
submitted by Mir Omranudin Abhar in fulfillment for the Award of MASTERS OF
TECHNOLOGY in (Computer Science Engineering) to the APG (Alakh Prakash
Goyal) Shimla University. Thesis has been corrected as per the evaluation reports
dated 10-07-2019 and all the necessary changes / modifications have been
inserted/incorporated in the thesis.
Ms. Nisha Gautam
Head of Department
Computer Science & Engineering
e0112@agu.edu.in
Date: 10-07-2019

iii
Certificate
This is to certify that dissertation titled “Square root sort algorithm”, submitted by
Mir Omranudin Ahbar in partial fulfillment for the award of degree of Master of
Technology in Computer Science & Engineering to APG (Alakh Prakash Goyal)
Shimla University, Shimla-171009 (HP) has been made under my supervision.
This report has not been submitted partially or fully to any other University or
Institute for the award of this or any other degree or diploma.
Date: 10-07-2019 Signature
Ms. Nisha Gataum
HOD

iv
Acknowledgement
First of all, I would like to thank the most merciful and the most gracious Allah who
teaches and shows us the way of happiness of this and next word by the agency of his
messenger the owner of honor Mohammad (peace be open him) and give command for
learning the knowledge. I am grateful from the all lectures especially my supervisor
MS.NISHA GAUTAM for his valuable contributions, patience and guidance through this
study. I am grateful to my parents for their hidden help in terms of prayers and faith in
me, which actually empowered me to fulfill this task. And last, I am grateful to all my
entourage, classmates and companions who helped me on the journey of university career
and made it easier. I would also like to thank my entourage, classmates and companions
especially from my partner Eng. Ehsan Bayat who helped me on the journey of university
career and made it easier.
Date: 10-07-2019 Signature:
Mir Omranudin Abhar

v
Abstract
Sorting is nothing but arrangement of a set of data in some order, so it will make
it easier and faster to locate items in a sorted list rather than unsorted and sorting
algorithms helps us to sort an array for later searching or writing out to an ordered
file. Different methods are used to sort the data in ascending or descending orders.
So far there are many algorithms that have been used for sorting like: Bubble sort,
insertion sort, selection sort, quick sort, merge sort, heap sort etc. Each of these
algorithms are using according to the list of elements for specific uses and they
have specific space and time complexity.
In this thesis we want to introduce and evaluate a new sorting algorithm. Those
algorithms who provides better time and space complexity are better algorithms
as compare to the other existing ones. In this thesis we are introducing a new
algorithm that has better time complexity and space complexity than some other
previous sorting algorithms and it is preferably for long lists. The base of working
of this algorithm is depend on the square root operation, so we call this algorithm
as square root algorithm.

vi
Table of Contents
Declaration......................................................................................................................i
Completion Certificate...................................................................................................ii
Certificate......................................................................................................................iii
Acknowledgement ........................................................................................................iv
Abstract..........................................................................................................................v
Table of Contents..........................................................................................................vi
List of Figures.............................................................................................................viii
List of Table..................................................................................................................ix
Abbreviation ..................................................................................................................x
CHAPTER 1: INTRODUCTION
1.1 Introduction.....................................................................................................1
1.2 Data Structure and Algorithm .........................................................................3
1.2.1 Data Structure..............................................................................................3
1.2.2 Relation between Data Structure and Algorithm ........................................5
1.2.3 Algorithm.....................................................................................................5
1.2.4 Implementation of Algorithm......................................................................6
1.2.5 Analysis of Algorithm .................................................................................6
1.2.6 Analysis of Program ....................................................................................7
1.2.7 Complexity of Algorithm ............................................................................7
1.3 Sorting Algorithm.........................................................................................12
1.3.1 Sorting .......................................................................................................13
1.3.2 Sorting Methods........................................................................................14
1.4 Need of Study.....................................................................................................19
1.5 Organization of thesis.........................................................................................19
CHAPTER 2: LITERATURE SURVEY
2.1 Literature Survey..............................................................................................20
2.2 Conclusion of literature review..........................................................................23
CHAPTER 3: PROBLEM STATEMENT & OBJECTIVE
3.1 Problem Statement .............................................................................................25
3.2 Objectives...........................................................................................................25
CHAPTER 4: Proposed Methodology

vii
4.1 Proposed Methodology ......................................................................................26
4.2 Analysis of Square Root Sorting Algorithm ......................................................26
4.3 Algorithm Design...............................................................................................28
4.3.1 Flow Chart ...................................................................................................28
4.3.2 Algorithms...................................................................................................30
4.4 Complexity Analysis..........................................................................................30
4.5 Development & Implementation........................................................................32
4.5.1 JAVA...........................................................................................................32
4.5.2 Design........................................................................................................33
CHAPTER 5: RESULTS AND DISCUSSION
5.1 Introduction........................................................................................................35
5.2 Compare SR Sorting Algorithm with other Sorting Algorithm (Merge Sort) ...35
CHAPTER 6: CONCLUSION AND FUTURE SCOPE
6.1 Conclusion..........................................................................................................39
6.2 Future Work .......................................................................................................39
References....................................................................................................................40
Appendix A: Source Code ...........................................................................................42
Accepted Papers ............................................................................................................46

viii
List of Figures
No Title Page No
Figure 1.1 Sorting Algorithms 2
Figure 1.2 Categories of Data Structures 4
Figure 1.3 Representation Algorithms 5
Figure 1.4 Rate of growth of common computing time function 8
Figure 1.5 Asymptotic Notation 10
Figure 1.6 Sorting Algorithms 15
Figure 4.1 Flow Chart SRS - 1 28
Figure 4.2 Flow Chart SRS - 2 29
Figure 4.3 SRS Best Case 31
Figure 4.4 SRS Average Case 31
Figure 4.5 SRS Worst Case 32
Figure 4.6 NetBeans IDE 33
Figure 4.7 Structure of SR Sorting 33
Figure 5.1 Compare Merge Sort with SR- Sort at the best case 36
Figure 5.2 Compare Merge Sort with SR- Sort at the Average case 36
Figure 5.3 Compare Merge Sort with SR- Sort at the worst case 37
Figure 5.4 Steps of average case of 10 random steps 38

ix
List of Table
No Title Page No
Table 1.1 Rate of growth of common computing time function 8
Table 1.2 Bubble Sort 16
Table 1.3 Insertion Sort 16
Table 1.4 Selection Sort 17
Table 1.5 Heap Sort 17
Table 1.6 Quick Sort 17
Table 1.7 Shell Sort 18
Table 1.8 Merge Sort 19
Table 5.1 Compare SR- Sort with other Sorting Algorithm 37

x
Abbreviation
No Short Form Full Form
1 DS Data structure
2 SRS Square root sorting
3 ALG Algorithm
4 OSSA Optimize section sorting Algorithm
5 GPU Graphics processing unit
6 CPU Center processing unit

1
CHAPTER 1
Introduction
This part describes an overview of whole this thesis which provides an
overarching theme of this thesis to reader.
1.1 Introduction
In the field of Computers, data management is the most crucial job. So, we need a
field that helps us in managing data which is known as data structure, A data
structure is a data organization, management and storage format that enables
effective access and modification [2]. In other words, Data structure is a
collection of data values, the relationships among them, and the functions or
operations that can be applied to the data [24].
Various operations are deploying in data structure:
1. Insertion: Insertion means addition of a new data component in a data
structure.
2. Deletion: Deletion means removal of a data component from a data
structure if it is found.
3. Searching: Searching includes searching for the specified data component
in a data structure.
4. Traversal: Traversal of a data structure means processing all the data
components present in it.
5. Sorting: Arranging data components of a data structure in a specified order
is called sorting.
6. Merging: Combining elements of two similar data structures to form a new
data structure of the same type, is called merging. [42]
Sorting is one of the most important and valuable parts in the data structure,
Sorting is nothing but arrangement of a set of data in some order or a process
that rearranges the records in a file into a sequence that is sorted on any key
shown in figure 1.1 sorting algorithm. Different methods are used to sort the data
in ascending or descending orders. The different sorting methods can be
divided into two categories. They are
 Internal sorting: We can define internal sorting as a sorting of data items in
the main memory.

2
 External sorting: External sorting is defined as a sorting when the data to be
sorted is so large that some of the data is present in the memory and some
is kept in auxiliary memory.
Arrays (sets of items of the same type of stored in contiguous memory space) are
stored in the fast, high-speed random access internal store on computers, whereas
files are stored on the slower but more spacious External store. [11]. the algorithm
that we introduce in this thesis is related to internal sorting algorithm. Some
common internal sorting algorithms include: Bubble Sort, Insertion Sort, Quick
Sort, Heap Sort, Radix Sort, Selection Sort, etc. Every sorting algorithm has some
advantages and disadvantages. The “Performance Analysis of Sorting
Algorithms” deals with the most commonly used internal sorting algorithms and
evaluates their performance. To sort a list of components, First of all we analyzed
the given problem i.e. the given problem is of which type (small numbers, large
values). The time complexity may vary depending on the sorting algorithm used.
Each sorting algorithm follows a unique method to sort an array of numbers either
by ascending or descending order [42].
The complexity of an algorithm is a function f (n) which measures the time and
space use by an algorithm in terms of input size n, in computer science, the
complexity of an algorithm is a way to classify how effective an algorithm is. As
compared to alternative ones. The emphasis is on how execution time increases
with the data set to be processed. The computational complexity and efficient
implementation of the algorithm are important in computing, and this depends on
suitable data structures [40].

3
1.2 Data Structure and Algorithm
1.2.1 Data Structure
In computer science, with the help of an appropriate data structure the computer
system will perform its task more efficiently because the data structure
influencing the ability of the computer to store and retrieve data from any location
in its memory, A data structure is an individual way of storing and organizing data
in a computer so that it can be used efficiently. Different kinds of data structures
are suited to different kind of applications, and some arc highly specialized to
specific tasks, Data structures are used in almost every program or software
system. Data structures provides a means to manage huge amounts of data
efficiently, such as large databases and internet indexing services. Usually,
effective data structures are a key to designing efficient algorithms, some official
design methods and programming languages emphasize data structures, rather
than algorithms, as the key organizing factor in software design [13, 12].
Data structure deals with the study of how data is organized in the computer’s
main memory and how it maintains logical relationship between individual
elements of data and also, how efficiently the data can be retrieved and
manipulated. A data structure is a study of organizing all the data elements that
consider not only the elements stored but also their relationship with each other.
Data structures are the building blocks of a program and hence, the selection of a
particular data structure addresses the following two things. [7]
Data structures are generally based on the ability of a computer to fetch and
store data at any place in its memory, specified by an address - a bit string that can
be itself store in memory and manipulated by the program thus the record and
array data structures are based on computing the addresses of data items with
arithmetic operations, while the linked data structures are based on storing
addresses of data items within the structure itself, Many data structures use both
principles, sometimes combined in non-trivial ways. The implementation of a data
structure usually needs writing a set of procedures that create and manipulate
instances of that structure. The effective of a data structures cannot be analyze
separately from those operations, this observations motivates the theoretical
concept of an abstract data type a data structure that is defined indirectly by the
operations that may be performed on it, and the mathematical properties of

4
those operations (including their space and time cost) [13].
• Data structure should be rich enough in structure for reflecting real
relationship existing between data.
• A structure must be simple such that we can process data effectively
whenever needed.
Classification: Data structure is broadly classified into two categories shown in
figure 1.2 categories of data structures.
1. Primitive data structures
2. Non-primitive data structures
1.2.1.1 Primitive data structures
These data structures are basic structures and are manipulated/ operated directly
by machine instructions. We will present storage representations for these data
structures for a variety of machines. The integer’s floating-point numbers (real
numbers), character constants, string constants, pointers etc. Are some of the
primitive data structures? In C language, these primitive data structures are
defined using data types such as int, float char and double. But you are already
aware of these representations in computer main memory [7].
1.2.1.2 Non-Primitive data structures
These data structures are more sophisticated and are derived from the primitive
structures. But, these cannot be manipulated/operated directly by machine

5
instructions. A non-primitive data structure emphasize on structuring of a set of
homogeneous (same data type) or heterogeneous (different data types) data
elements. Arrays, structures, stacks queues linked lists, files etc. are some of the
non-primitive data structures [7].
1.2.2 Relation between Data Structure and Algorithm
Implementation of data structures can be done with the help of programs. To write
any program we need an algorithm. Algorithm is nothing but collection of
instructions which has to be executed in step by step manner. And data structure
tells us the way to organize the data. Algorithm and data structure together give
the implementation of data structures. To write any program we have to select
proper algorithm and the data structure. If we choose improper data structure then
the algorithm cannot work effectively. Similarly if we choose improper algorithm
then we cannot utilize the data structure effectively. Thus there is a strong
relationship between data structure and algorithm. As data structure can be very
well understood with the help of a programminglanguage [2].
Data Structure + Algorithm = Programs
1.2.3 Algorithm
Algorithm is a procedure or formula for solving a problem. The word derives
from the name of the mathematician, Mohammed ibn-Musa al-Khwarizmi, who
was part of the royal court in Baghdad and who lived from about 780 to 850. Al-
Khwarizmi’s work is the likely source for the word algebra as well [14, 18].
An algorithm is a step by step method of solving a problem [16]. It is commonly
used for data processing, calculation and other related computer and mathematical
operations [4]. Once an algorithm is given for a problem and decided to be
correct, an important step is to determine how much resources, in term of time or

6
space, the algorithm will require. An algorithm that solves a problem but
requires a year is hardly of any use shown in figure 1.3 representation Algorithm
[41].
1.2.4 Implementation of Algorithm
To develop any program we should first select a proper data structure, and then we
should develop an algorithm for implementing the given problem with the help of
a data structure which we have chosen. Before actual implementation of the
program, designing a program is very important step. Suppose, if we want to build
a house then we do not directly start constructing the house. In fact we consult an
architect, we put our ideas and suggestions, accordingly he draws a plan of the
house, and he discusses it with us. If we have some suggestion, the architect notes
it down and makes the necessary changes accordingly in the plan. This process
continues till we are satisfied. Finally the blue print of house gets ready. Once
design process is over actual construction activity starts. Now it becomes very
easy and systematic for construction of desired house. In this example, you will
find that all designing is just a paper work and at that instance if we want some
changes to be done then those can be easily carried out on the paper. After a
satisfactory design the construction activities start. Same is a program
development process.
If we could follow same kind of approach while developing the program then we
can call it as Software development life cycle which involves several steps as -
a. Feasibility study
b. Requirement analysis and problem specification
c. Design
d. Coding
e. Debugging
f. Testing and
g. Maintenance
1.2.5 Analysis of Algorithm
Analysis of algorithm means to determine the amount of resources (such as time
and storage) required to execute it, most algorithms are designed to works with
inputs of arbitrary length. Usually the efficiency or current time of an algorithm is

7
stated as a function relating the input length to the number of steps (time
complexity) or storage locations (space complexity) [10, 22].
Algorithm analysis is an important part of a widespread computational
complexity theory, which provides theoretical estimates for the resources needed
by any algorithm which solves a given computational problem. These estimates
provide an insight into reasonable directions of search for effective algorithms. In
theoretical analysis of algorithm it is common to evaluate their complexity in the
asymptotic sense, i.e., to estimate the complexity function for arbitrarily large
input [10, 6].
1.2.6 Analysis of Program
The analysis of the program does not mean simply working of the program but to
check whether for all possible situations program works or not. The analysis also
involves working of the program efficiently. Efficiency, means
• The program requires less amount of storage space.
• The program get executed in very less amount of time.
The time and space are the factors which determine the efficiency of the program.
Time required for execution of the program cannot be computed in terms of
seconds because of the following factors -
 The hardware of the machine.
 The amount of time required by each machine instruction.
 The amount of time required by the compilers to execute the
instruction.
 The instruction set.
Hence we will assume that time required by the program to execute means the
total number of times the statements get executed. This is known as frequency
count [2].
1.2.7 Complexity of Algorithm
The complexity of an algorithm is a function f (n) which measures the time and
space use by an algorithm in terms of input size n, in computer science, the
complexity of an algorithm is a way to classify how efficient an algorithm is,
compared to alternative ones, the emphasize is on how execution time
increases with the data set to be processed.

8
The computational complexity and efficient implementation of the algorithm are
important in computing, and this depends on suitable data structures [29, 3]
If we have two algorithms that perform same task, and the first one has a
computing time of O (n) and the second of O(n2), then we will usually prefer the
first one. The reason is that as n increases the time required for the execution of
second algorithm is far more than the time required for the execution of first. We
will study various values for computing function for the constant values. The
graph given below will indicate the rate of growth of common computing time
functions [2] shown in figure 1.4 Rate of growth of common computing time
function.
Notice how the times O (n) and O (n log n) grow much more slowly than the
others, for large data sets algorithms with a complexity greater than O (n log n)
are often impractical. The very slow algorithm will be the one who is having the
time complexity as 2n [2] shown in table 1.1 Rate of growth of common
computing time function.
𝒏 𝐥𝐨𝐠 𝟐 𝒏 𝒏 𝐥𝐨𝐠 𝟐 𝒏 𝒏 𝟐
𝒏 𝟑
𝟐 𝒏
1 0 0 1 1 2
2 1 9 4 8 4
3 2 8 16 64 16
8 3 24 64 512 256
16 4 64 256 4096 65536
32 5 160 1024 32768 2147483648
1.2.7.1 Asymptotic Notation
Asymptotic notation describes the behavior of the time or space complexity for
large instance characteristics [21]. To select the best algorithm, we need to check
Table 1.1: Rate of growth of common computing time function

9
efficiency of each algorithm. The efficiency can be measured by calculating time
complexity of each algorithm, asymptotic notation is a shorthand way to represent
the time complexity. Use asymptotic notations we can give time complexity as
“fastest possible”, “slowest possible” or “average time”. Various notations such as
Ω, θ and O used are called asymptotic notions [1].
1.2.7.1.1 Theta Notation : Theta notation denoted as ’θ’ is a method of
representing running time between upper bound and lower bound.
Explanation: Let, f (n) and g (n) be two non-negative functions. There exists a
positive constant C1 and C2 such that C1g (n) ≤ f (n) ≤ C2g (n) and f (n) is theta
of g of n [1, 2] shown in figure1.5 Figure Asymptotic Notation (a).
1.2.7.1.2 Big Oh Notation: Big Oh notation means by ’O’ is a method of
representing the upper bound of algorithm’s running time. Using large oh notation
we can give longest amount of time taken by the algorithm to complete.
Explanation: Let, f (n) and g (n) are two non-negative functions. And if there
exists an integer no. and constant C such that C > 0 and for all integers n > n0, f
(n) ≤ c ∗ g(n) , then f (n) is big oh of g(n). It is also denoted as “f (n) = O (g (n))”
[1,2] shown in figure1.5 Asymptotic Notation (b).
1.2.7.1.3 Omega Notation: Omega notation denoted as Ω is a method of
representing the lower bound of algorithm’s running time. Using omega notation
we canmeansshortest amount of time taken by algorithm to complete.
Explanation: Let, f (n) and g(n) are two non-negative functions and if there exists
constant C and integer no. such that C > 0 and n > no then f (n) > C ∗ g(n) i.e.
f(n) is omega of g of n. This is denoted as f (n) = Ω (g (n))[1, 2] shown in figure1.5
Asymptotic Notation (c).
(a) : Theta Notation (b) : Big O Notation (c) : Omega Notation

10
1.2.7.2 Space Complexity
Another useful measure of an algorithm is the amount of storage space it needs.
The space complexity of an algorithm can be computed by considering the data
and their sizes. Again we are concerned with those data items which demand for
maximum storage space. A similar notation ’O’ is used to denote the space
complexity of an algorithm. When computing for storage requirement we assume
each data element needs one unit of storage space. While as the aggregate data
items such as arrays will need n units of storage space n is the number of
elements in an array. This assumption again is independent of the machines on
which the algorithms are to be executed [2].
To calculate the space complexity we use two factors: constant and instance
characteristics, the space requirement S (p) can be set as
S(p) = C + Sp
Where C is a constant i.e. fixed part and it mean the space of inputs and outputs,
this space is an amount of space taken by instruction, variables and identifiers.
Sp is a space dependent upon instance characteristics. This is a variable part
whose space requirement depends on particular problem instance [1].
Constant Complexity: O (1)
A constant tasks run time won’t change no matter what the input value is,
considera function that prints a value in an array.
No matter which components value you’re asking the function to print, only
one step is required. So we can say the function runs in O (1) time; its run-time
does not increase. Its order of magnitude is always 1[19, 5].
Linear Complexity: O (n)
A linear tasks run time will vary depending on its input value, if you ask a function to
print all the items in a 10- component array, it will require less steps to complete than it
would a 10,000 element array. This is said to run at O (n); its run time increases at an
order of magnitude proportional to n [19, 5].

11
Quadratic Complexity: O (N 2)
A quadratic task requires a number of steps equal to the square of its input value.
Let’s look at a function that takes an array and N as its input values where N is
the number of values in the array. If I use a nested loop both of which use N as its
limit condition, and I ask the function to print the arrays contents, the function
will perform N rounds, each round printing N lines for a total of O (N 2) print
steps. Let’s look at that practically. Assume the index length N of an array is 10,
if the function prints the contents of its array in a nested-loop, it will perform 10
rounds, each round printing 10 lines for a total of 100 print steps. This is said to
run in O (N 2) time; its total run time increases at an order of magnitude
proportional to 𝑁2
[19, 5].
Exponential: O (2n)
O (2n) is just one example of exponential growth (among O (3n), O (4n), etc.).
Time complexity at an exponential rate means that with each step the function
performs, its subsequent step will take longer by an order of magnitude
equivalent to a factor of N, For instance, with a function whose step-time doubles
with each sub sequent step, it is said to have a complexity of O (2n). A function
whose step-time triples with each iteration is said to have a complexity of O (3n)
and so on [19, 5].
Logarithmic Complexity: O (log n)
This is the type of algorithm that makes calculation blazingly fast, Instead of in-
creasing the time it takes to perform each subsequent step, the time is decreased at
magnitude inversely proportional to N.

12
Let’s say we want to search a database for a particular number. In the data set
follow, we want to search 20 numbers for the number 100, in this example,
searching by this 20 numbers is a non-issue. But imagine were dealing with data
sets that store millions of users profile information, searching through each index
value from starting to end would be ridiculously inefficient. Especially if it had to
be done multiple times.
A logarithmic algorithm that performs a binary search looks through only half of
an increasingly smaller data set per step, assume we have an ascending ordered set
of numbers. The algorithm starts by searching half of the entire data set. If it
doesn’t find the number, it discards the set just checked and then searches half of
the remaining set of numbers.
Round printing 10 lines for a total of 100 print steps. This is said to run in O (N 2)
time; its total run time increases at an order of magnitude proportional to N 2.
As illustrated above, each round of searching consists of a smaller data set than
the previous, decreasing the time each subsequent round is performed. This makes
log n algorithms very scalable [19, 5].
1.3 Sorting Algorithm
Introduction-In the previous section, we described the Data Structure, Algorithm,
Complexity, and the relation between the data structure and Algorithm.
Examined the complexity of the algorithm.

13
In this section we will discuss about existing sorting algorithms and examining
their complexities.
1.3.1 Sorting
Sorting is any process of organizing items in some sequence and/or in different
sets, and accordingly, it has two common, yet distinct meanings:
• Ordering: organizing items of the same kind, class, nature, etc. in some
ordered sequence,
• Categorizing: grouping and labeling items with equivalent properties
together (by sorts).
In computer science, a sorting algorithm is an algorithm that puts components of a
list in a some order, the most-used order are numerical order and lexicographical
order, beneficial sorting is important for improve the use of other algorithms
(such as search and merge algorithms) that require sorted lists to work correctly; it
is also often useful for cannibalizing data and for producing human-readable
output 20] shown in figure 1.6 Sorting Algorithm.
Every sorting algorithm has some advantages and disadvantages, the
“Performance analysis of sorting algorithms” deals and analyze the most generally
used internal sorting algorithms and evaluate their performance. To sort a list of
components, First of all we analyzed the given problem i.e. the assumed problem
is of which type (small numbers, large values). The time complexity may vary
depended upon the sorting algorithm used, each sorting algorithm follows a
unique method to sort an array of numbers either by ascending or descending
order [42].
Many computer scientists discuss sorting to be the most fundamental problem in
the study of algorithms. There are several reasons [43]:
1. Sometimes an application inherently requirements to sort information. For
example, in order to prepare client statements, banks need to sort checks by
check number.
2. Algorithms often use sorting as a key sub routine. For example, a program
that render graphical objects that are layer on top of each other might have
to sort the objects according to an ”above” relation so that it can draw

14
these objects from bottom to top, we shall see numerous algorithms in this
text that use sorting as a sub routine.
3. There is a wide variety of sorting algorithms, and they use a rich set of
techniques. In fact, many important methods used throughout algorithm
scheme are represented in the body of sorting algorithms that have been
developed over the years, in this way, sorting is also a problem of historical
concern.
4. Sorting is a problem for which we can prove a nontrivial lower bound. Our
best upper bound match the lower bound asymptotically, and so we know
that our sorting algorithms are asymptotically optimal. Also, we can use
the lower bound for sorting to prove lower bound for certain other
problems.
5. Many engineering issues come to the fore when implementing sorting
algorithms. The fastest sorting program for a particular situation may
belong on many factors, such as prior knowledge about the keys and
satellite data, the memory hierarchy (caches and virtual memory) of the
host computer and the software environment many of these issues are best
dealt with at the algorithmic level, rather than by “tweaking” the code.
1.3.2 Sorting Methods
The function of sorting or ordering a list of objects according to some linear order

15
is so fundamental that it is ubiquitous in engineering applications in all
disciplines. There are two broad categories of sorting methods [25]:
1. Internal sorting takes place in the main memory, where we can take
advantage of the random access nature of the main memory;
 Quick Sort
 Heap Sort
 Bubble Sort
 Insertion Sort
 Selection Sort
 Shell Sort and etc.
2. External sorting is necessary when the number and size of objects are
prohibitive to be accommodated in the main memory
 Merge Sort
 Radix Sort
 Polyphase Sort and etc.
1.3.2.1 Bubble Sort
Bubble Sort is a simple algorithm which is used to sort a given set of n components
provided in form of an array with n number of components. Bubble Sort
compares all the elements one by one and sort them based on their values. If the
given array has to be sorted in ascending order, then bubble sort will start by
matching the first element of the array with the second element, if the first
element is greater than the second component, it will swap both the elements,
and then move on to compare the second and the third element, and so on. If we
have total n components, then we need to repeat this process for n-1 times.
It is known as bubble sort, because with every complete iteration the largest
component in the given array, bubbles up towards the last place or the highest
index, just like a water bubble rises up to the water surface. Sorting takes place
by stepping through all the elements one-by-one and matching it with the
adjacent component and swapping them if required.
Following are the Time and Space complexity for the Bubble Sort algorithm [8].

16
Table 1.2: Bubble Sort
Worst Case Average Case Best Case
Time Complexity n2 n2
n
Space Complexity 1 1 1
1.3.2.2 Insertion Sort
Insertion sort is based on the idea that one component from the inputs components
is consumed in each iteration to find its correct position i.e., the position to which
it belongs in a sorted array.
It iterates the input components by growing the sorted array at each iteration. It
compares the current component with the largest value in the sorted array. If
the current element is greater, then it leaves the element in its place and moves
on to the next component else it finds its correct position in the sorted array and
moves it to that position. This is done by shifting all the components, which are
larger than the current element, in the sorted array to one position ahead [15].
Following are the Time and Space complexity for the Insertion Sort algorithm [8].
Time Complexity n2 n2
n
1.3.2.3 Selection Sort
Selection sort is conceptually the easy sorting algorithm. This algorithm will first
find the smallest component in the array and swap it with the component in the
first position, then it will find the second smallest component and swap it with the
component in the second position, and it will keep on doing this until the entire
array is sorted. It is called selection sort because it repeatedly selects the next-
smallest component and swaps it into the right place.
Following are the Time and Space complexity for the Selection sort algorithm [8].
Time Complexity n2 n2 n2
Table 1.3: Insertion Sort
Table 1.4: Selection Sort

17
1.3.2.4 Heap Sort
Heap Sort is one of the best sorting techniques being in-place and with no
quadratic worst-case current time. Heap sort involves building a heap data
structures from the given array and then utilizing the heap to sort the array.
You must be wondering, how converting an array of numbers into a heap data
structures will help in sorting the array. To understand this, let’s start by
understanding what a heap is.
Following are the Time and Space complexity for the Heap Sort algorithm [8].
Time Complexity 𝑛 log 𝑛 𝑛 log 𝑛 𝑛 log 𝑛
1.3.2.5 Quick Sort
Quick sort is based on the divide-and-conquer approach based on the idea of
choosing one component as a pivot component and partitioning the array around
it such that: Left side of pivot contains all the components that are less than the
pivot component Right side contains all components greater than the pivot.
It reduces the space complexity and removes the use of the auxiliary array that is
used in merge sort. Selecting a random pivot in an array result in an improved
time complexity in most of the cases.
Following are the Time and Space complexity for the Quick Sort algorithm [8].
Time Complexity 𝑛2
𝑛 log 𝑛 𝑛 log 𝑛
Space Complexity 𝑛 log 𝑛 𝑛 log 𝑛 𝑛 log 𝑛
1.3.2.6 Shell Sort
Shell Sort is a generalized version of insertion sorts it is an in place comparison
sort. Shell Sort is also known as diminishing increase sort, it is one of the oldest
sorting algorithms invented by Donald L. Shell (1959.)
This algorithm uses insertion sort on the large interval of components to sort.
Table 1.5: Heap Sort
Table 1.6: Quick Sort

18
2
Then the interval of sorting keeps on decreasing in a sequence until the interval
reaches 1. These intervals are known as gap sequence.
This algorithm work quite efficiently for small and medium size array as its
average time complexity is near to O (n).
Following are the Time and Space complexity for the Shell Sort algorithm [23].
Time Complexity 𝑛 log 𝑛 𝑛 log 𝑛 𝑛
1.3.2.7 Merge Sort
Merge sort is a divide and conquer algorithm based on the idea of break down a
list into multiple sub lists until each sub list consist of a single component and
merging those sub lists in a manner that results into a sorted list.
Idea:
• Divide the un-sorted list into N sub lists, each containing 1 component.
• Take adjacent pairs of two singleton lists and merge them to form a
list of 2 Components. N will now convert into n list of size 2.
• Repeat the process till a single sorted list of obtained.
While comparing two sub lists for merging, the first component of both lists is
taken into consideration. While sorting in ascending order, the component that is
of a lesser value becomes a new element of the sorted list. This procedure is
repeated until both the smaller sub lists are empty and the new combined sub list
comprises all the components of both the sub lists [15].
Following are the Time and Space complexity for the Merge Sort algorithm [8].
Time Complexity 𝑛 𝑙𝑜𝑔 𝑛 𝑛 𝑙𝑜𝑔 𝑛 𝑛 𝑙𝑜𝑔 𝑛
Space Complexity 𝑛 𝑛 𝑛
Table 1.7: Shell Sort
Table 1.8: Merge Sort

19
1.4 Need of Study
One of the greatest challenges these days in the field of computer science is large
data management within the memory. As the memory faces difficulties to arrange
and organize “Large data”, I decided to introduce a way for facing this challenge
and helping other researchers to come up with better and more efficient ways in
the future.
So, I have come across the Idea of new sorting algorithm for “Large Data”.
1.5 Organization of thesis
The thesis is organized as follows:
Chapter 2 - It presents some Literature review about sorting algorithm.
Chapter 3 - It presents Problem statement and objective about our thesis work.
Chapter 4 - It presents propose methodology our sorting algorithm includes:
Analysis of square root sorting algorithm, Deign our algorithm, Complexity
analysis our algorithm and development & implementation our algorithm.
Chapter 5 - It presents results & discussion about our research work.
Finally, the last chapter is devoted to the Conclusion and Feature work.

20
CHAPTER 2
Literature Survey
2.1 Literature Survey
V.P.Kulalvaimozhi et.al, [39] In this paper the authors explained “Performance
analysis of sorting algorithms” deals and analyzed the most commonly used
internal sorting algorithms and evaluate their performances. To sort a list of
components, First of all the given problem is analyzed i.e. the given problem is of
which type (small numbers, large values). The time complexity may vary
depending on the sorting algorithm used. Each sorting algorithm follow a unique
method to sort an array of numbers either by ascending or descending
order. The ultimate goals of this study is to match the various sorting
algorithms and finding out the asymptotic complexity of each sorting
algorithm. This study proposed a methodology for the users to choose an
effective sorting algorithm. Finally, the reader with a particular problem in mind
can choose the best sorting algorithm.
Wang Xiang , [29] This paper explains about time complexity of quick sort
algorithm and makes a comparison between the improved bubble sort and quick
sort through analyzing the first order derivative of the function that is founded to
correlate quick sort with other sorting algorithm, the comparison can promote
programmers make the right decision when they face the choice of sort algorithms
in a variety of circumstances so as to reduce the code size and improve efficiency
of application program and Quick sort algorithm has been widely used in data
processing systems, because of its high efficiency, fast speed, and scientific
structure. Therefore, thorough study based on time complexity of quick sort
algorithms is of great significance. Especially on time complexity aspect, the
comparison of quick sort algorithm and other algorithm is particularly important.
S.-S.Chen et.al, [31] this paper described efficient bubble-sort-based algorithms
for the two- and three-layer non-Manhattan channel routing problems. Based on
the same routing model ,the time complexities of their proposed algorithm, two
previous algorithms Chaudhary's and Chen's for the two-layer (and three-layer)
non-Manhattan channel routings are 𝑂(𝑘𝑛), 𝑂(𝑘𝑛2), and 𝑂(𝑘2𝑛), respectively,

21
where k is the number of sorting passes required and n is the number of two
terminal nets in a channel. To further conduct the performance analysis of the
three bubble-sort based algorithms, they have tested them on a set of examples.
Experimental results indicate that the proposed algorithm requires only 1% more
routing tracks than the optimal Chen's router and the time improvement is over
40% of on the average. Clearly, the time performance of this algorithm is better
than previous algorithms. For future work, they planned to integrate their bubble-
sort router into an over-the-cell (OTC) channel router to reduce the Final channel
height in VLSI chip design.
Indradeep Hayaran et.al, [32] This paper presents a new algorithm for sorting
namely Couple Sort. It is a hybrid sorting algorithm which is influenced from
quick sort and bubble sort techniques which results in a considerably lower time
complexity by eliminating some useless comparisons and the proposed couple
sort algorithm has a worst case complexity of less than n2, which results in lesser
number of computations. It ranges from 𝑂(𝑛 𝑙𝑜𝑔 𝑛) to 𝑂(𝑛2
) as the size of array
increases. Also, the best case is linear time which proves that the proposed couple
sort algorithm is efficient over many other algorithms which require unnecessary
computations for already sorted list. In addition, it is an in-place algorithm, i.e.,
no extra memory space is required. The comparisons are well demonstrated which
show the out performance of the proposed sorting algorithm over other existing
algorithms.
Vignesh R et.al, [35] This paper aims at introducing a new sorting algorithm
which sorts the elements of an array In Place. This algorithm has 𝑂 (𝑛) best case
Time Complexity and 𝑂(𝑛 𝑙𝑜𝑔 𝑛) average and worst case Time Complexity, the
goal had been achieved using Recursive Partitioning combined with In Place
merging to sort a given array, a comparison is made between this particular idea
and other popular implementations. Finally, a conclusion had drawn out and
observed the case where this outperforms other sorting algorithms. The authors
also looked at its shortcomings and list the scope for future improvements that
could be made and the Future improvements can be made to enhance the
performance over larger number of input array. Since they had the minimum and
maximum value of the sub array at any time, instead of starting from the

22
beginning, they could have combined the current logic with an end first search to
reduce the number of iterations. Regarding its stability, as mentioned earlier, this
algorithm can be made stable by increasing the number of pivots but this would
lead to other complications. Any improvement though, however trivial, would be
highly appreciated.
Sultan Ullah et.al, [36] This paper explain the idea of Optimized Selection Sort
Algorithm (OSSA) is based on the already existing selection sort algorithm, with
a difference that old selection sort; sorts one element either smallest or largest in a
single iteration while optimized selection sort, sorts both the elements at the same
time i.e smallest and largest in a single iteration. In this study the authors have
developed a variation of OSSA for two-dimensional array and called it Optimized
Selection Sort Algorithms for Two-Dimensional arrays OSSA2D. The
hypothetical and experimental analysis revealed that the implementation of the
proposed algorithm is easy. The comparison shows that the performance of
OSSA2D is better than OSSA by four times and when compared with old
Selection Sort algorithm the performance is improved by eight times (i.e if OSSA
can sort an array in 100 seconds, OSSA2D can sort it in 24.55 Seconds, and
similarly if Selection Sort takes 100 Seconds then OSSA2D take only 12.22
Seconds). This performance is remarkable when the array size is very large. The
experiential results also demonstrate that the proposed algorithm has much lower
computational complexity than the one dimensional sorting algorithm when the
array size is very large and It is evident from the above results that optimized
selection algorithm for two-dimensional arrays is more efficient than the other
sorting algorithm of the same complexity for the same amount of data. Hereafter
proved their claim to be an efficient algorithm of the order of 𝑂(𝑛2
). It could be
useful algorithm when it needs to solve the problem of sorting huge volume of
data in reasonably easy manner and efficiently.
Liu Shenghui et.al, [33] This paper presents an internal sorting algorithm by
GPU assisted. It consist of two algorithms: a GPU-based internal sorting
algorithms and a CPU-based multi-way merging algorithms, the algorithm
divided the large-scale data into multiple chunks to fit GPU global memory.
Then copy the chunk to the GPU's global memory one by one, and sort them by

23
GPU quicksort algorithm. Then merge these sub-sequences to one sorted
sequence by CPU. They have used the loser tree algorithms to reduce the number
of comparisons when merging. Finally, this algorithm is tested using a variety of
data distribution, the experimental results show that this algorithm improves the
efficiency of large-scale data sorting efficiently.
Neetu Faujdar et.al, [37] this paper described the detailed analysis of bucket sort
has been done. The insertion, count and merge sort algorithms have been used
within the buckets. For testing the algorithms, sorting benchmark has been used.
Three algorithms which are bucket with insertion, bucket with count and bucket
with merge sort have been implemented and compared to each other.
The threshold (𝜏 ) is defined for saving the time as well as space of the
algorithms. Results indicate that, count sort comes out to be more efficient within
the buckets for every type of dataset with respect to the range of key elements.
The range used in this work is from 0 to 65535. But if the range of key element
increases then count sort will be worst in comparison to other sorting algorithms
in both aspects (space, time). Based on Window 7, operating system of 64 bit,
core i5 processor of Intel algorithms are tested which are implemented in C-
language. Borland C++ 5.02 compiler is used for program designing and verified
after running on 2.2 𝐺𝐻𝑧 clock speed.
In the future work, they can to further classify other sorting algorithm like
quick sort based on the number of elements in bucket which will not only make
the working faster of the bucket sort but also will reduce the time.
2.2 Conclusion of literature review
In conclusion, we know that every sorting algorithm has its own specific
properties. Each and every algorithm has its special usage and benefits. Some
algorithms are ideal for organizing and arranging items in the internal memory
and some others for external memory. Additionally some of the algorithms are
appropriate to be used in both internal and external memory.
Sorting algorithms such as bubble sort, selection sort and insertion sort are ideal
for lists with few elements inside internal memory but using them for huge lists
are not effective not efficient as they consume more time.
Merge sort and Quick sort algorithms are comparatively efficient for relatively

24
large lists inside external and internal memories.
One of the algorithms that has ability to implement in either external or internal
memories, is Merge sort.
Although, it will get challenging for Merge sort and Quick sort algorithms if the
list of items getting larger; leading to higher time consumption.

25
CHAPTER 3
Problem Statement & Objective
3.1 Problem Statement
The author have been done many surveys and investigation in the field of data
science and come across with one of the greatest challenge of managing and
organizing huge amount of data with least complexity in terms of time and space
complexity. So, our problem statement is:
To introduce a new sorting algorithm which its complexity is less as compare to
the other previous sorting algorithms.
3.2 Objectives
1. Generated a data set for analysis.
2. Wrote a new algorithm with reduced complexity.
3. Analyzed the complexity of new algorithm with existing algorithms.
.

26
CHAPTER 4
Proposed Methodology
4.1 Proposed Methodology
The algorithm we want to introduce in this thesis is the algorithm used to sorting
elements, so far, many algorithms have been designed in the sorting of elements,
each of which, as we have already mentioned, has its own advantages and
disadvantages, We’ll describe it later in detail. The algorithm we’ve designed is
named square root, because the base operation in this algorithm is depend on
square root operation. This sorting algorithm works better than some of the
previously designed algorithms, especially in the large list of elements.
The differences between the algorithm proposed in this paper with existing
algorithms is that in previous algorithms all elements of the list is processed for
sorting simultaneously which need more time. But with SRS algorithm, the list
first divided into equal parts for further process. Each equal part are going under
sorting process individually, then all parts are combining together which consume
less time for sorting operation.
4.2 Analysis of Square Root Sorting Algorithm
The goal of designing the SRS (Square Root Sorting) algorithm is to reduce the
time complexity of the memory.
We use one of the best sorting algorithms in combining this scheme of sorting
from data items, which if we find better algorithms in the future we can give it a
place. But the current algorithm doesn’t have any problem by using mergesort.
This is how our algorithm works Shown in Figures 4.1, 4.2 Flow chart of this
algorithm.
1. Initially, we divide the list of data using square root operation. We will
have equal numbers of elements in each part except last part, it might
contain less elements than other parts.
2. The second step is to sort each part separately using one of the
available algorithms, which has the least time complexity.
3. The third step is to start comparing data items and swapping.

27
In this step, we will compare 2 parts and perform swapping it required.
The comparison method is such that we compare the first element of the first
part with the first element of the other sections, if the first element of the other
part is larger, we will not perform the comparison with other elements of that
section. Because if the first element of one of the segments is larger, it states that
other elements of this section are also larger than the first element of the first part.
We can find the smallest element in the first swapping operation and put it in the
first index of the array. Similarly, we start the comparison with the second
element of the first part. We still compare it with only the first elements of the
other segments. If one of the first elements of the other part is smaller than the
first element of the first part, we will perform the swap operation. This means
that the second small element is in the second index of array.
Now, if the swap element of that section is larger than the next element, we
will perform the sorting operation with the same algorithm we have selected twice
over. If others are in the sorted order, there is no need to be sorted again.
Similarly, it is time to compare the third element of the first part. We do the same
thing twice before the end.

28
4.3 Algorithm Design
4.3.1 Flow Chart
Figure 4.1: Flow Chart SRS -1

29
Figure 4.2: Flow Chart SRS -2

30
4.3.2 Algorithms
Step 1. 𝑛 ← 𝑙𝑒𝑛𝑔𝑡ℎ of List 𝐴, 𝑟 ← √ 𝑛 ,𝑠 = 0 .
Step 2. Divide the list 𝐴 in 𝑟 part and every part have 𝑟 item in that list.
Step 3. 𝑓𝑜𝑟(𝑝 = 1 𝑡𝑜 𝑟, 𝑝 + +) Part of list 𝐴 individual sort, with the Merge sort or
other Algorithms.
𝑚𝑒𝑟𝑔𝑒(𝑝𝑎𝑟𝑡. 𝑝)
Step 4. The first item will compare with the first item of seconds part.
𝑓𝑜𝑟(𝑝 = 1 𝑡𝑜 𝑟, 𝑝 + +){
𝑓𝑜𝑟(𝑖 = 1 𝑡𝑜 𝑟, 𝑖 + +){
𝑖𝑓(𝑝𝑎𝑟𝑡. (𝑝)[𝑖] ≤ 𝑝𝑎𝑟𝑡. (𝑝 + 1)[𝑠]){
𝑠 + +
}𝑒𝑙𝑠𝑒{
𝐴 ← 𝑝𝑎𝑟𝑡. (𝑖)[1]
𝐵 ← 𝑝𝑎𝑟𝑡. (𝑖 + 1)[𝑠]
𝑝𝑎𝑟𝑡. (𝑖 + 1)[𝑠] ← 𝐴
𝑝𝑎𝑟𝑡. (𝑝)[𝑖] ← 𝐵
𝑚𝑒𝑟𝑔𝑒(𝑝𝑎𝑟𝑡. (𝑖 + 1))
}
}
𝑠 = 0
}
Step 5. Return A
4.4 Complexity Analysis
The complexity of a sorting algorithm measures the running time as a
function of the number of n items to be sorted, each sorting algorithm S will be
made up of the following operations, where A1, A2...An contain the items to be
sorted and B is an auxiliary location.
 Comparison which test whether Ai < Ai or test whether Ai < B.
 Interchange which switch the contents of Ai and Aj or Ai and B.
 Assignment which set B = Ai and then set Aj = B or Aj = Ai.
Generally the complexity function measures only the number of comparison,
since the number of other operations is at most a constant factor of the number of
other operations is at most a constant factor of the number of comparison,
Suppose space is fixed for one algorithm then only run time will be considered
for obtaining the complexity of algorithm. [27, 17]
Complexity analysis of the proposed SR Sorting Algorithm for best case and
worstcase is discussed here with the help of examples.

31
A. Best Case :
In this case the array is already sorted and the terminating conditions is obtained
after traversing the array twice. In this case, the complexity of the SR algorithm is
O(𝑛 log √ 𝑛) shown in figure 4.3 SRS in the Best case.
B. Average Case :
In this case the array data items used the random number, In this case, the
complexity of the SR algorithm is O((𝑛 log √ 𝑛) + 𝑐𝑛)shown in figure 4.4 SRS
in the average case.

32
−
C. Worst Case :
Reverse sorted array which involves every element to be moved to (n 1)th position
from its initial ith position. In this case, the complexity of the SR algorithm is
O((𝑛 log √ 𝑛) + 𝑐𝑛) shown in figure 4.5 SRS in the worst case.
4.5 Development & Implementation
In this part we will discuss the implementation of SR Sorting Algorithm, to
implement each algorithm in one of the programming languages, we have a set of
common principles that exist in any programming language. This principles are:
The data type, the mathematical operations, operation of the conditional, and
loop. We also deal with this programming operation to implement this algorithm.
The programming language that we chose to implement our algorithm is java.
4.5.1 JAVA
The programing language that author chose for implementation of this algorithm

33
is JAVA.
Java is a general purpose and high level programming language that was
repurposed in 1995 to create applications on the World Wide Web. Today, Java is
highly used for creating web and mobile applications.
Because of its platform-Independent nature that means any program that wrote in
it can run easily on any other software and hardware platform, the author
preferred to use this programming language shown in figure 4.6 NetBeans IDE
for java programing language.
.
4.5.2 Design
In an implementation of this algorithm, we have used a class that has five methods
that each method is created for a specific task, in which case we introduce each of
these methods shown in figure 4.7 structure of SRS algorithm.
 main: This method is a method in any programming language that is
necessary because this method is the first method that is executed.

34
 setArray: This is a way to create a list of arrays in three cases, the best
case, average case, worst case, or create an array of cases.
This method has three parameters, the first parameter is the array, and the
second is the size of the array, and the third parameter is created to specify
the case from the array.
 sortAllRow We have created this method for sorting separate parts of
array.
 compareTwoPart: We can create this method so that we can compare
parts of the array.
 printList: We have created this method to show the array before and after
sorting the list..

35
CHAPTER 5
Results and Discussion
5.1 Introduction
After development and implementation of SR Algorithm, We have decided to
compare this algorithm with one of the best existed algorithm that is currently
using for sorting. For this purpose we have chosen the Merge Sort algorithm
because this algorithm is one of the most powerful algorithms in terms of time
complexity, furthermore, Merge Sort can be used in an internal and external
memory.
5.2 Compare SR Sorting Algorithm with other Sorting Algorithm
(Merge Sort)
Which sorting algorithm is the fastest? This question doesn’t have an easy or
unambiguous answer, however. The speed of sorting can depend quite heavily on
the environments where the sorting is done, the type of items that are sorted and
the distribution of these items.
For example, sorting a database which is so big that can-not fit into memory all
at once is quite different from sorting an array of 100 integers, not only will the
implementation of the algorithm be quite different, naturally, but it may even be
that the same algorithm which is fast in one case is slow in the other. Also sorting
an array may be different from sorting a linked list, for example. [9]
In order to better understand the advantages and disadvantages of an algorithm,
there is a need for time that the algorithm should be adapted in different criteria’s
so that the results obtained can be examined in order to understand the
advantages and disadvantages of the algorithm. When we discuss the complexity
of the SR algorithm, we have achieved the results after it was implemented in
Java programming language which indicates that the algorithm performs a
comparative and substitute comparison with previous algorithms. This al- growth
can still be called a technique that replaces any sorting algorithm that exists up to
now or in the future in order to reduce the time complexity of the algorithm and
when we compare merge sort with this algorithm, the results obtained from 100 to
102400 data items are presented in the following way figure 5.1, figure 5.2 and
figure 5.3 compare merge sort with SRS algorithm:

37
The following table illustrates the complexity of some sorting algorithms with SR
sorting algorithms.
Table 5.1 Compare SR- Sort with other Sorting Algorithm
Algorithms Best case Average case Worst case
SR Sort 𝑛 log(√ 𝑛) 𝑛 log(√ 𝑛) + 𝑐𝑛 𝑛 log(√ 𝑛) + 𝑐𝑛
Merge Sort 𝑛 log(𝑛) 𝑛 log(𝑛) 𝑛 log(𝑛)
Bubble Sort 𝑛 𝑛2
𝑛2
Insertion Sort 𝑛 𝑛2
𝑛2
Selection Sort 𝑛2
𝑛2
𝑛2
Following we have been tested the algorithm with java with 10 inputs to check
the average case of the algorithm.
The figure 5.4 shows the 5 steps of the algorithm which is the sorting of random
numbers.

39
CHAPTER 6
Conclusion and future scope
6.1 Conclusion
Each algorithm has its own advantages and disadvantages according to
environment and this algorithm has some disadvantages that will become known
in the future after being implemented in different environments, which requires
time.
In this thesis, we have introduced a new algorithm called Square Root Sorting
Algorithm which uses a new method for sorting data items in the memory, and
compared the time complexity of this algorithm with some of the algorithms
available in the sorting section. The time complexity of this algorithm in best case
is equal to O(𝑛 log √ 𝑛) and in the cases of the worst case and the average case
is equal to O((𝑛 log √ 𝑛) + 𝑐𝑛).
Which is much better than the previous algorithms like merge, insertion, and
selection etc., this algorithm is like a method that one of its parameters could be
one of the best available algorithms, which is a feature of the SR algorithm that
reduces its time complexity that we have explained in our result and discussion.
6.2 Future Work
1. Our future vision is to develop a better version of SR algorithm which will have
half of time complexity comparing to current algorithm in worst case.
2. And we will try to adopt SR algorithm technique in future for using in searching
within big data.

40
References
[1] A.A.Puntambekar. (2007). Advance Data Structures.
[2] A.A.Puntambekar. (2009). Data Structures And Algorithms.
[3] Abdel-hafeez, S. and Gordon-Ross, A. (2014). A Comparison-Free Sorting Algorithm. IEEE Journals.
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41
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42
Appendix A: Source Code
package rootsort;
import java.util.Arrays;
import java.util.Random;
/**
*
@author Mir Omranudin Abhar
*/
public class RootSort {
public static final String RED = "u001B[41m";
public static final String BLACK = "u001B[47m";
public static final String BGREEN = "u001B[42m";
static int com = 0;
static int swapp = 0;
public static void main(String[] args) {
// TODO code application logic here
// list of item
// int array[] = {
//// 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
// 10,9,8,7,6,5,4,3,2,1
// };
////
int size = 10;
int array[] = new int[size];
RootSort obj = new RootSort();
// base,worst, average
obj.setArray(array, size, "average");
// Getting the size of list and getting the Sqaure root
int i = array.length;
int item = (int) Math.ceil(Math.sqrt(i));
int part = (int) Math.ceil((i / item)) + 1;
// Create the object of class for accessing the method of the class
// Show the list before sorted.
System.out.print("nList before sort: ");
obj.printList(array, item);
// This method used for sorting every part of list
obj.sortAllRow(array, part, item);
// This method used for comparing.
obj.compaireTwoPart(array, part, item);
// This loop used for showing the sorted list.
System.out.print("nList after sort: ");
obj.printList(array, item);
// System.out.println("Total Comparing : " + com);
// System.out.println("Total Swapping : " + swapp);

43
}
// This methtod used for comparing.
public void compaireTwoPart(int array2D[], int part, int item) {
int max = 0, min = 0, vc = 0, vr = 0, i = 1, last = 0;
for (int row = 0; row < array2D.length; row += item) {
last = row;
for (int col = row; col < ((col >= array2D.length) ? array2D.length - 1 : row + item); col++) {
for (int rowV = row + item; rowV < array2D.length; rowV += item) {
for (int colV = rowV; colV < ((colV >= array2D.length) ? array2D.length - 1 :
rowV + item); colV++) {
vc = colV;
vr = colV;
com++;
if (array2D[col] > array2D[colV]) {
swapp++;
for (int rowVJ = row + item + item; rowVJ < array2D.length; rowVJ += item) {
for (int colVJ = rowVJ; colVJ < ((colVJ >= array2D.length) ?
array2D.length - 1 : rowVJ + item); colVJ++) {
com++;
if (array2D[vc] > array2D[colVJ]) {
vc = colVJ;
vr = colVJ;
}
break;
}
}
max = array2D[col];
min = array2D[vc];
i++;
int ll = 1;
System.out.println("");
for (int j = 0; j < array2D.length; j++) {
if (j == item * i) {
i++;
}
if (j == col) {
System.out.print(RED + " " + array2D[j] + " " + BLACK);
} else if (j == vc) {
System.out.print(BGREEN + " " + array2D[j] + " " + BLACK);
} else {
System.out.print(" " + array2D[j] + " ");
}
if (ll * item == j + 1) {
System.out.print("u001B[43m" + "|" + BLACK);

44
ll++;
}
}
array2D[vc] = max;
array2D[col] = min;
Arrays.sort(array2D, vr, ((vr + item > array2D.length) ? array2D.length : vr +
item));
break;
} else {
break;
}
}
}
}
}
}
public void printList(int array[], int item) {
int i = 1;
System.out.println("n");
for (int j = 0; j < array.length; j++) {
if (j == item * i) {
i++;
System.out.print("u001B[43m" + "|" + BLACK);
}
System.out.print(BLACK + " " + array[j] + " ");
}
}
public void sortAllRow(int array2D[], int part, int item) {
int i = 1;
int cc = 0;
for (int row = 0; row <= part; row++) {
if (item * i >= array2D.length) {
if ((item * i) - item < array2D.length) {
cc = array2D.length;
} else {
break;
}
} else {
cc = item * i;
}
Arrays.sort(array2D, item * row, (cc));
i++;
}

45
}
public void setArray(int array[], int size, String Case) {
Random rand = new Random();
if (Case.equals("best")) {
// Base Case
for (int set = 0; set < size; set++) {
array[set] = set;
}
} else if (Case.equals("worst")) {
// Worst Case
for (int set = 0; set < size; set++) {
array[set] = size - set;
}
} else {
// Random Case
for (int set = size - 1; set >= 0; set--) {
array[set] = rand.nextInt(10);
}
}
}
}

46
Accepted Papers
1. "A Review Data structure , Algorithms & Analysis", International Journal
of Emerging Technologies and Innovative Research (www.jetir.org),
ISSN:2349-5162, Vol.6, Issue 6, page no.59-64, June-2019, Available :
http://www.jetir.org/papers/JETIR1906791.pdf
2. "Square Root sorting algorithm", International Journal of Emerging
Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162,
Vol.6, Issue 6, page no.358-362, June-2019, Available :
http://www.jetir.org/papers/JETIR1906780.pdf

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