ANALYSIS ON SEARCHING AND SORTING ALGORITHMS · A number of searching algorithms have been developed like that sequential search, binary search and hashing etc. Every searching algorithm

ANALYSIS ON SEARCHING AND SORTING ALGORITHMS

Anjana Kamana, Melissa Mary Smith, Nikhitha Beecharaju,

Mr. Rajasekhar Sastry, Dr. B V Ramana Murthy and Mr. C Kishor Kumar Reddy.

Stanley College of Engineering and Technology for Women, Hyderabad

[email protected],@[email protected],[email protected],

[email protected],[email protected],[email protected]

ABSTRACT

In this survey paper, the authors present various types of sorting and searching algorithms. Sorting and

Searching are two basic operations in computer science. Sorting techniques are analyzed based on time

complexity and space complexity. Searching Algorithms are designed to check for element or retrieve an

element from any data structure where it is stored. Searching algorithms are based on two categories

sequential search and interval search. Binary search works on sorted set of elements only. And is also

found out that it is suitable for midsized data. Linear search sequentially checks each element of the list

until it finds an element that matches the required value. Hash search is best for large data items and

exponential search can be used for infinite set of elements. This paper provides a analysis on searching

and sorting algorithms.

Keywords: Searching, Sorting, Algorithm, Time and Space complexity

1. INTRODUCTION

An algorithm is a process or set of rules to be followed in calculations or other problem-solving

operations, especially by computer. Sorting means arranging of data in given order such that increment or

decrement. Searching means finding out the location or finding out an element of a given item in a

collection of item. Many data structures are used to store information but Arrays, linked lists and tree are

fundamental data structures used to sorting and searching operation. Searching element is any type of a

numerical data, alphabet, and character data. A number of searching algorithms have been developed like

that sequential search, binary search and hashing etc. Every searching algorithm depends on the specific

problem, property of data and algorithm complexity. If the data is kept in some sorted order then

searching becomes easy. Implementation of sorting algorithm can be simply classified into internal and

external sorting. Internal sorting, where values are compared and rearranged within the system, may be

efficient in case of small data sets. However, when the records to be sorted are too large to be processed

by the system internally, external sorting algorithms are implemented. This method, also known as file

sorting, increases the parallelism of the process, since data is sorted in other nodes and finally merged at

the local node.

1.1. TIME COMPLEXITY

JASC: Journal of Applied Science and Computations

Volume VI, Issue I, January/2019

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mailto:[email protected],[email protected],[email protected]

Time complexity makes it easy to estimate the run time of a program. Performing an accurate

calculation in a program’s operation time is a very labor-intensive process (it depends on the compiler and

the type of computer or speed of the processor).Therefore, it will not make an accurate measurement, just

a measurement of a certain order of magnitude. It can be viewed as the maximum number of primitive

operations that a program may execute. Regular operations are single additions, multiplications,

assignments etc. It leave some operations uncounted and concentrate on those that are performed the

largest number of times. Such operations are called as dominant. The number of dominant operations

depends upon the specific input data. An algorithm's performance time varies with different inputs of the

same size, one commonly uses the worst-case time complexity of an algorithm, denoted as T(n), which is

defined as the maximum amount of time taken on an input of size (n). Less common and the usually

specified explicitly is the measure of average case complexity. Time complexities are classified according

to the nature of the function T(n). For instance an algorithm with T(n) = O (n) is called a linear time

algorithm, and an algorithm with log T(n) =Ω(nα) for some constant α ≥1 is said to be an exponential

time algorithm.

1.2. SPACE COMPLEXITY

Space complexity is the amount of memory which is used by the algorithm (including the input

values to the algorithm) to execute and produce the result.To calculate the space complexity, it should

know the memory required to store different data type value. Sometimes Auxiliary Space is confused with

Space Complexity. But Auxiliary Space is the extra space or the temporary space used by the algorithm

during its execution.

Space Complexity = Auxiliary Space + Input space (1)

In Space Complexity of any algorithm, it usually considers only Data Space and it neglects the

Instruction Space and Environmental Stack. Big O particularly describes the worst-case scenario, and can

be used to describe the execution time required or the space used (e.g. in memory or on disk) by an

algorithm. Space complexity of O (1) means that the space required by an algorithm to process data is

constant, it does not grow with the size of the data on which the algorithm is operating. A space

complexity of O (N) describes an algorithm whose performance will grow linearly and in direct

proportion to the size of the input data set.

2. LITERATURE SURVEY

Thomas Niemann in his research work on sorting and searching algorithms stated about internal

and external sort, compared different sorting and searching algorithms on basics of time, space and

simplicity [1]. Brad Miller and David Ranumanalysed different sorting and searching techniques.[2].

Debadrita Roy and Arnab Kundu in their review summarized three searching algorithmsand also included

searching concept, algorithm, coding & time complexity [3]. Ajay Kumar, Bharat Kumar, Chirag Dawar

and Dinesh Bajaj explained about different sorting techniques their behavior for different inputs. The

research reveals that the Insertion sort is best for small data items and, Merge sort and quick sort is used

for large data sets [4]. Amy CsizmarDalal gave a brief description on sorting and searching techniques.

He explained about different searching and sorting techniques, algorithms and implementation of the

techniques with examples [5]. The website describes about Selection sort and binary search. It provides



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complexity analysis of binary search and selection sort. It gives the detailed working procedure of binary

search and selection sort algorithms [6]. Sorting is a process of rearrangement a list of elements to the

correct order since handling the elements in a certain order more efficient than handling randomize

elements [7]. FahriyeGemciFurat explained and compared insertion and selection sort. Implementation

and running time of them is also explained in a detailed manner. He concluded that based on running time

selection sort is efficient [8]. The website provides the basic information about algorithms. It gives the

detailed information on different sorting and searching techniques. It provides the working procedure,

time complexity and pseudocode of searching and sorting algorithms [9]. Ramesh Chand Pandey explains

about the various sorting algorithms, their advantages and disadvantages. It compared sorting algorithms

based on time complexity, memory required and stability. It proposed a new sorting algorithm which

works on priority basis. The new algorithm gives the specific data first then general data if required [10].

They compared two-way linear search algorithm with linear search and provided the results with the help

of algorithm and comparison table. They concluded that in Linear Search it is not compulsory to arrange

an array in any order (Ascending or Descending) as in the case of binary search orderly [11].Manpreet

Singh Bajwa, Arun Prakash Agarwal and SumatiManchanda presented new algorithm for searching that is

Ternary search while comparing it with Binary search algorithm. The time complexity of ternary search is

reduced when compared to binary search. Hence, they concluded that it is efficient [12]. M. Goodrich and

R. Tamassia discussed that some sorting algorithms are stable by its nature such as insertion sort, merge

sort, bubble sort, while some sorting algorithms are not, such as quick sort, any given sorting algorithm

which is not stable can be modified to be stable [13]. Selection sorts the simplest of sorting techniques.

It's work very well for small files, also It's has a quite important application because each item is actually

moved at most once [14]. The third factor is memory space, algorithm that used recursive techniques need

more copies of sorting data that affect to memory space [15]. V.P.Kulalvaimozhi, M.Muthulakshmi,

R.Mariselvi, G.Santhana Devi, C.Rajalakshmi and C. Durai’s research dealt with most commonly used

internal sorting algorithms and their performance. It compared various sorting algorithms and found out

asymptotic complexity of each sorting algorithm. The study proposed a methodology for the users to

select an efficient sorting algorithm [19].

3. SEARCHING ALGORITHM

The searching technique can be classified into two categories:

Internal Searches: - If a file or table is kept in main memory this type of searching is called

internal search.

External Searches: - If a file or table is kept in auxiliary memory (hard disk, floppy, tape etc) this

type of searching is called External Search.

3.1. SEQUENTIAL SEARCH OR LINEAR SEARCH

The simplest form of searching technique is a sequence search or linear search. This search is

applicable to small size of arrays or linked list data structure. It finds out the searching element in

sequential manner on unordered list or ordered list. In this method searching process is started at the

beginning to end, it scans the elements one by one of the list from left to right until the searching

record/element is found. If we have taken an ordered list or table then the time given to fast searching and



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good efficiency as a comparison on unordered list.

Unordered Linear Search: Suppose that the given array was not sorted. This might correspond, for

example, to collection exams which have not yet been sorted alphabetically.

Ordered Linear Search: Now suppose that the given array is in a sorted order. In this case, we

need not necessarily search through the entire list to find a particular object or determine that it does not

exist in the collection

Worst Case: Θ(n)Θ(n), search key is not present or is the last element, Best Case: Θ(1)Θ(1) ,the first

element, No. of comparisons: Θ(n)Θ(n) in worst case and 11 in the best case.

Algorithm: LINEAR _SEARCH (A, N, M, P)

1. Initialize Q=0

2. For N=0 to length [A] //scan the element from start to end

3. If (MP = =A [N]) //check on searching element

4. Q=Q+1 // increment on location variable

5. Return A[N] //print on finding element

6. END If

7. END For LOOP

8. If Q=0 then return “did not find the element”

9. Exit

For Example: An unordered list 33 22 55 11 33 44 sort it using linear search

Unordered

Array A[1-6]

A[0] A[1] A[2] A[3] A[4] A[5] Output

33 22 55 11 33 44

Step:1 MP=33 33 22 55 11 33 44 Return A[0],Q=1

Step:2 MP=33 33 22 55 11 33 44

Step:3 MP=33 33 22 55 11 33 44

Step:4 MP=33 33 22 55 11 33 44

Step:5 MP=33 33 22 55 11 33 44 Return a[1],Q=2

Step:6 MP=33 33 22 55 11 33 44

Table 3.1: Linear Search

3.2. BINARY SEARCH

Another simplest form of searching technique and the best efficient method is a binary search. It

can be used if the table is stored in an array and mid size of array is taken. If we use linked list and

insert/delete on data in linked list during searching time then problem are generated. This algorithm is

also based on Divide-and-Conquer algorithm.



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Worst case/Average case: Θ(logn)Θ(logn), Best Case : Θ(1)Θ(1) ; when the key is the middle element ,

No. of comparisons : Θ(logn)Θ(logn) in the worst/average case & 11 in the best case.

Divide: The sorted list is divided into two parts.

Conquer: We first compare the search/input element with the mid element of the list. If they do not

match then we check search element and mid element if search element is less than the mid element then

go to first part/left part otherwise go to second/right part.

Combine: We apply the divide and conquer part whenever our search element is not found. When our

search element is found then list is automatically combined and returned.

Algorithm-

1. Assign n-1 to max and 0 to min.

2. Compute mid as the mean of max and min.

3. If array[mid] is equal to the search element, then stop and return mid.

4. If the mid is less than the search element then assign mid+1 to min.

5. If mid is greater than search element then assign mid-1 to max.

6. Go back to step 2.

For Example:

Unordered Array

A[1-6]

A[0] A[1] A[2] A[3] A[4] A[5]

33 22 55 11 33 44

Sort to Array A

and result Array

A is

11 22 33 33 44 55 Operation

Step: 1 Low=0, HIGHT=5, MID=0+5/2=2, A[MID]=33, SE=33

Check S==A[MID] 11 22 33 33 44 55 Return A[MID],Q=1

Step2: LOW=MID+1, 2+1=3, HIGHT=5, MID=3+5/2=4, A[MID]=44, SE=33

Check S==A[MID] 11 22 33 33 44 55

Step3: LOW=3, HIGHT=MID-1,4-1=3 MID=(3+3)/2=3, A[MID]=33, SE=33

Check S==A[MID] 11 22 33 33 44 55 ReturnA[MID],Q=2

Table 3.2: Binary Search



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4. SORTING ALGORITHM

The sorting technique is classified into types:

I. Internal Sort: In internal sorting, all the data to be sorted is stored in memory at all the times while

sorting is in progress.

There are 3 types of internal sorts.

i. SELECTION SORT: - Ex: Selection sort algorithm and Heap Sort algorithm.

ii. INSERTION SORT: - Ex: Insertion sort algorithm and Shell Sort algorithm.

iii. EXCHANGE SORT: - Ex: Bubble Sort Algorithm and Quick sort algorithm.

II. External Sort: In external sorting data is stored outside the memory (like on disk) and only loaded

into memory in small chunks. External sorting is usually applied in cases when data can't fit into memory

entirely. Different parts of data are sorted separately and merged together. For Example: - Merge Sort

4.1. BUBBLE SORT

The complexity of sorting algorithm is dependent upon the number of comparisons that are made.

Total comparisons in a Bubble sort is: n (n – 1) / 2 ≈ n 2 – n

Best case: O (n2), Average case: O (n2), Worst case: O (n2)

In the bubble sort method list is divided into two sub-lists sorted and unsorted. The smallest

element is bubbled from an unsorted sub-list. Compare adjacent elements and swap the smallest once with

the other item if they are in wrong order. This passing procedure is repeated until no swaps are required,

indicating that the list is sorted [15] [18]. The bubble sort was originally written to bubble up the highest

element in the given list. But there is no difference whether highest or lowest element is bubbled. This

method is easy to understand but it is time consuming. In this type there are two successive elements that

are compared and swapping is done. Thus, step-by-step the entire array elements are checked. When a list

of ‘n’ elements is given the bubble sort requires up to n-1 passes to sort the data.

Algorithm-

1. Compare each pair of adjacent elements from the beginning of an array and if they are in reverse

order then swap them.

2. If at least one swap is done, repeat step 1.

For Example:

5 3 8 4 6

Initial Unsorted array

5 3 8 4 6



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3 5 8 4 6

3 5 8 4 6

3 5 4 8 6

3 5 4 6 8

1. Step 1 : Compare 1st and 2nd (swap)

2. Step 2 : Compare 2nd and 3rd (do not swap)

3. Step 3 : Compare 3rd and 4th (swap)

4. Step 4 : Compare 4th and 5th (swap)

5. Step 5 : Repeat step1-5 until no more swaps needed

Table 4.1: Bubble Sort

4.2. INSERTION SORT

Insertion sort cannot exchange elements. In insertion sort the element is inserted at an appropriate

place which is similar to card insertion. Insertion sort provides several advantages Simple

implementation and, Efficient for small data sets [20]. Here the list is divided into two parts the sorted

and the unsorted sub-lists. In each pass, the first element of the unsorted sub list is picked up and moved

into the sorted sub list by inserting it in a suitable position. Suppose we have ‘n’ elements, we need n-1

passes to sort the elements in order. Insertion sort works in this way:

Algorithm-

INSERTION_SORT (B)

1. FOR j ← 2 TO length[B]

2. DO key ← B[j]

3. Put B[j] into the sorted sequence B[1 . . j − 1]

4. i ← j − 1

5. WHILE i > 0 and B[i] > key

6. DO B[i +1] ← B[i]

7. i ← i − 1

8. B[i + 1] ← key

For Example-

Unsorted list elements are:9,7,6,15,17,5,10,11 sort it using insertion sort.

9 7 6 15 17 5 10 11

9 7 6 15 17 5 10 11

7 9 6 15 17 5 10 11

6 7 9 15 17 5 10 11

6 7 9 15 17 5 10 11

6 7 9 15 17 5 10 11

5 6 7 9 15 17 10 11

5 6 7 9 10 15 17 11

5 6 7 9 10 11 15 17



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Table 4.2: Insertion Sort

4.3. SHELL SORT

The shell sort, sometimes is called the “diminishing increment sort,” it improves the insertion sort

by breaking the original list into many number of smaller sublists, each of which is sorted using an

insertion sort. The unique way to choose these sublists is the key to the shell sort. Instead of breaking the

list into sublists of adjacent items, the shell sort uses an increment 'i', sometimes it is called the gap, it

used to create a sublist by choosing all the items which are 'i' items apart.

Algorithm-

1. Initialize the value of p

2. Divide the given list into smaller sub-list of equal intervals of i

3. Sort the sub-lists using the insertion sort

4. Repeat until complete list is sorted

For Example- Given 17, 26,20,44,55,31,54,77,93 is an unsorted list. Sort using shell sort.

17 26 20 44 55 31 54 77 93

1 shift for 20

17 20 26 44 55 31 54 77 93

2 shifts for 31

17 20 26 31 44 55 54 77 93

1 shift for 54

17 20 26 31 44 54 55 77 93

Sorted list

Figure4.3: Shell Sort

4.4. SELECTION SORT

This is a sorting algorithm, specifically it is an in-place comparison sort. It has O (n2) time

complexity, making it inefficient on large lists. Selection sort has one advantage over other sort

techniques[16][17] It is simple and has high performance on complicated algorithms during some

situations specially when the secondary memory is limited. Assume that they have taken a list of n

elements. By applying the selection sort, the first element is compared with all the remaining (n-1)

elements. The smallest element of the list is placed at the first location. Again, the second element is

compared with the remaining (n-1) elements. At the time of comparison, the smaller element is swapped

with the larger element. Similarly, entire array is checked for the smallest element and then swapping is



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done according to it. Here they need n-1 passes or iterations to complete the rearranging of data.

Algorithm-

1. Set MIN to the location 0

2. Search for the least element in the considered or given list.

3. Swap it with the value at the location MIN.

4. Increment MIN so that it points to the next element in the list.

5. Repeat the same until list is sorted.

Example-26, 54, 93,17,77,31,44,55,20 is an unsorted list, sort it using selection sort

26 54 93 17 77 31 44 55 20

93 is largest

26 54 20 17 77 31 44 55 93

77 is largest

26 54 20 17 55 31 44 77 93

55 is largest

26 54 20 17 44 31 55 77 93

54 is largest

26 31 20 17 44 54 55 77 93

44 is largest and stays in its place

26 31 20 17 44 54 55 77 93

31 is largest

26 17 20 31 44 54 55 77 93

26 is largest

20 17 26 31 44 54 55 77 93

20 is largest

17 20 26 31 44 54 55 77 93

The list is sorted



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Table 4.5: Selection

4.6. MERGE SORT

The basic concept of merge sort is to divide the list into two smaller sublists of approximately

equal size. Repeat this procedure till only one element is left in the sublist. After this, various sorted

sublists are merged together to form the sorted parent list. This process goes on recursively until the

original sorted list is obtained.

Algorithm -

Merge sort is based on the divide-and-conquer algorithm. Its worst-case run time has a lower order of

growth than insertion sort. Since we are dealing with subproblems, they state that each sub problem are

sorted as a sub-array A[q .. r]. Initially, q = 1 and r = n, but these values change as they recurse through

sub-problems. To sort A[q .. r]:

1. Divide Step: If a given array A has zero or one element, simply return as it is already sorted.

Otherwise, split A[q .. r] into two sub-arrays A[q .. s] and A[s + 1 .. r], each containing

approximately half of the elements of A[q .. r]. That is, s is the halfway point of A[q .. r].

2. Conquer Step: Conquer by recursively sorting the two sub-arrays A [q ... s] and A[s + 1 ... r].

3. Combine Step: Combine the elements back to A[q .. r] by merging the two sorted sub-arrays A[q

.. s] and A[s + 1 .. r] into a sorted sequence. To accomplish this step, they will define a procedure

MERGE (A, q, s, r). The recursion stops when the sub-array has just one element, so that it is

trivially sorted.

To sort the entire sequence A[1 .. n], make the initial call to the procedure as MERGE-SORT (A,

1, n).

MERGE-SORT (A, q, r)

1. IF q < r // Check for base case

2. THEN s= FLOOR[(q + r)/2] // Divide step

3. MERGE (A, q, s) // Conquer step.

4. MERGE (A, s + 1, r) // Conquer step.

5. MERGE (A, q, s, r) // Conquer step.

For Example-



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38, 27, 43,3,9,82,10 is an unsorted, sort the list using merge sort.

38 27 43 3 9 82 10

Figure 4.6: Merge Sort

5. CONCLUSION

In this survey paper various searching and sorting techniques have been discussed. Authors have

seen sorting techniques based on time and space complexity. Various examples have been used to explain

the searching and sorting techniques. They have seen how a binary search is used in a sorted list. They

have also seen how quick sort and insertion sort are productive for large items and small items

respectively. They have also compared the searching algorithm on the basis of various factors like

complexity, searching type, etc.

38 27 43 3 9 82 10

10 38 27 43 3 9 82

38 27 3 9 43 10 82

9 82 10 27 38 3 43

3 27 38 43 9 10 82

3 9 10 27 38 43 82



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6. REFRENCES

[1 ] Thomas Niemann, “Sorting and Searching Algorithms.

[2 ] Brad Miller and David Ranum, Problem Solving with Algorithms and Data Structures,

2013.

[3 ] Debadrita Roy and Arnab Kundu, A Comparative Analysis of Three Different Types of

Searching Algorithms in Data Structure, International Journal of Advanced Research in

Computer and Communication Engineering, 2014.

[4 ] Ajay Kumar, Bharat Kumar, Chirag Dawar and Dinesh Bajaj, Comparison Among

Different Sorting Techniques, International Journal for Research In Applied Science And

Engineering Technology (IJRASET), 2014

[5 ] Amy CsizmarDalal, Searching and Sorting Algorithms, 2004.

[6 ] “Analysis of Binary Search algorithm And Selection Sort algorithm”,

http://www.cartagena99.com/recursos/alumnos/apuntes/Binarysearch_sorting.pdf

[7 ] P.Adhikari, Review on Sorting Algorithms, "A comparative study on two sorting

algorithm", Mississppi state university, 2007.

[8 ] FahriyeGemciFurat, A Comparative Study of Selection Sort and Insertion Sort

Algorithms, International Research Journal of Engineering and Technology, 2016.

[9 ] http://www.tutorialspoint.com/data_structures_algorithms/data_structures_algorithms_tut

orial.pdf

[10 ] Ramesh Chand Pandey, Study and Comparison of various Sorting Algorithms, 2008.

[11 ] Nitin Arora, GarimaBhasin and Neha Sharma,”Two way Linear Search Algorithm”,

International Journal of Computer Applications, 2014.

[12 ] Manpreet Singh Bajwa, Arun Prakash Agarwal and SumatiManchanda, “Ternary Search

Algorithm: Improvement of Binary Search”, International Conference on Computing for

Sustainable Global Development, 2015.

[13 ] M. Goodrich and R. Tamassia, Data Structures and Algorithms in Java,John wiley & sons

4th edition, 2010,pp.241-243.

[14 ] R. Sedgewick and K. Wayne, Algorithms,Pearson Education, 4th Edition, 2011,pp.248-

249.

[15 ] D.Knuth, The Art of Computer Programming,Addison-Wesley,Third Edition 1997,pp.

106– 110 .

[16 ] R. Sedgewick, Algorithms in C++, Addison–Wesley Longman,1998,pp 273–274.

[17 ] Levitin, Introduction to the Design & Analysis of Algorithms, Addison–Wesley

Longman, 2007, pp 98–100.

[18 ] http://www.techopedia.com/definition/3757/bubble-sort.

[19 ] V.P.Kulalvaimozhi, M.Muthulakshmi, R.Mariselvi, G.Santhana Devi, C.Rajalakshmi and

C. Durai, Performance Analysis of Sorting Algorithm, International Journal of Computer

Science and Mobile Computing, 2015.

[20 ] http://corewar.co.uk/assembly/insertion.htm.



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ANALYSIS ON SEARCHING AND SORTING ALGORITHMS · A number of searching algorithms have been developed like that sequential search, binary search and hashing etc. Every searching algorithm

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