Time Complexity Analysis in Data Structure.docx

Time Complexity Analysis in Data
Structure
Overview
Whenever we are thinking of a solution to a problem, there can be N number of solutions. For
instance, any problem can be solved in multiple different ways. But the Question that arises
here is, how can we recognize the most efficient solution if we have a set of different
solutions?
To answer this question let us understand What is Time Complexity? Through this article we
shall also cover the different notations to calculate time complexity.
Let us consider a simple example to understand what we we are trying to figure out. Let us
say we want to find square of a number. Now to check that let us suppose we have two
approaches. The first approach towards this problem would be to simply use the
mathematical operator * to find the square.
On the other hand, the second approach towards finding the solution would be to run a loop
of 'n' times which starts with the number 'n' and then we gradually keep adding 'n' to it
everytime. Now it is clearly understood that if we execute both the approach, both will take
different time to execute and give output. And in this case, the first approach will take less
time and hence is said to be the efficient way.
Scope of the Article
1. Through this article we will cover the Basics of finding the efficient solution which is where
the Time Complexity comes into picture.
2. We will deep dive into What is Time Complexity? The asymptotic notations associated with
it? We will learn this by in detail explanation of Big oh, Omega and Theta.
3. We also will be covering different types of time complexity notations and how can we
evaluate the time complexity algorithm.
4. This article also covers the Time complexity of sorting algorithms and searching algorithms.
5. Also, moving further in the module we shall also find few examples to get the whole
understanding of how the actually can be utilised.
6. This article is not covering any Data Structures and Algorithms basics and assumes the users
of the manual to be well aware with the same.
7. Also, The space complexity part is not the part of this module.
Takeaways

Time complexity can be defined as the amount of time taken by an algorithm to execute each
statement of code of an algorithm till its completion with respect to the function of the length
of the input
What is Time Complexity?
To answer this question first we must keep in mind that for number of problems that must be
solved using an algorithm, there can be infinte number of solutions. And it then becomes
important to do the analysis of algorithms to find the most efficient solution for solving that
problem.
Now to know which solution is the most efficient one, the concept of space and time
complexity of algorithms comes into picture.
The Space and Time complexity can be defined as a measurement scale for algorithms where
we compare the algorithms on the basis of their Space (i.e. the amount of memory it utilises )
and the Time complexity (i.e. the number of operations it runs to find the solution).
There can more than one way to solve the problem in programming, but knowing how which
of the algorithm works efficiently and hence, can add value to the way we do programming .
So, in order to dive deep to find the effectiveness of the program, we must know how to
evaluate them using the Space and Time complexity. This use of Space and Time complexity
can help us make the program behave according to the required optimal conditions, which in
some way makes us an efficient programmers too. As with this module we are focussing
more on the Time Complexity domain let us take a look at the definition of the same.
Time Complexity Definition: The Time complexity can be defined as the amount of time
taken by an algorithm to execute each statement of code of an algorithm till its completion
with respect to the function of the length of the input.
The Time complexity of algorithms is most commonly expressed using the big O notation.
It's an asymptotic notation to represent the time complexity. We will study about it in detail
in the next tutorial.
The Time Complexity can also be elaborated by estimating the counts of the number of
elementary steps that are performed by any algorithm to finish the execution. As we saw in
above example where we shared the two approaches to find a square of a number, then with
the first approach the time taken will remain constant while for the second it varies with
respect to the 'n' that runs on the loop to give an output.
It should also be noted that as the algorithm's performance varies with respect to the multiple
types of input data, so we usually we also keep a check for the worst-case scenario for an
algorithm so that we are well aware of the maximum time taken for any input size.
Summing up above points, we can concude that the time complexity is explained as the
number of operations an algorithm performs in order to complete a task with respect to the
size of the input. And to find out which algorithm must be considered as the most efficient
one, we pick the one which takes the smallest number of operations in terms of the time
complexity.

Asymptotic Notations
Now we shall we understanding the Notations of Time Complexity. There are major 3
notations which we will hear about.
 Big O(expression): The Big-O notation is used to define if the set of functions is
going to grow slower than or at the same rate with respect to the expression. This is
how we define the worst case of an algorithm's time complexity. This also elaborates
on the the maximum amount of time required by an algorithm considering all input
values.
 Omega(expression) The Omega notation is used to define if the set of functions is
going to grow faster than or at the same rate with respect to the expression. This is
how we define the best case of an algorithm's time complexity. This also elaborates
on the the minimum amount of time required by an algorithm considering all input
values.
 Theta(expression) The Theta notation is used to define consist if the set of functions
is going to lie in both O(expression) and Omega(expression). This is how we define
the average case of an algorithm's time complexity. This also elaborates average
bound of an algorithm.
NOTE:
We can understand that when we do something with every item in one dimension its
considered linear, in two dimensions its considered quadratic, and keep dividing the sample
space in half is considered logarithmic.
Time Complexity of Sorting Algorithms
The Time complexity of sorting algorithms can be elaborated as the best posiible way to pick
out the best sorting technique in a situation. Understanding the time complexities with respect
to the sorting algorithms can help us in picking up the best sorting technique, listed below are
few of them:
1. The Time Complexity of Insertion Sort: The time complexity of Insertion
Sort is Ω(n) in its best case possible and O(n^2) in its worst case possible. It has been
observed that for very small 'n',the Insertion Sort is faster than more efficient
algorithms such as Quick sort or Merge Sort.
2. The Time Complexity of Merge Sort: The time complexity of Merge
Sort is Ω(nlogn) in its best case possible and O(nlogn) in its worst case possible. This
can be explained as in merge sort implements the same number of sorting steps for all
kinds of cases. Out of all the sorting technique, the merge sort has the most stable
time complexity for all kinds of cases because the steps which are executed for all
kinds of cases are same.
3. The Time Complexity of Quick Sort: The time complexity of Quick
Sort is Ω(nlogn) in its best case possible and O(n^2) in its worst case possible. As we
know QuickSort is a Divide and Conquer algorithm where the element is picked as a
pivot and partitions are made on the given array around the picked pivot. This makes
it the fastest of the sorting algorithms as due to its performance in best case and alos
in average cases are said to of O(nlogn).

4. The Time Complexity of Bubble Sort: The time complexity of Bubble
Sort is Ω(n) in its best case possible and O(n^2) in its worst case possible. As is
widely known that the The Time Complexity of Bubble Sort is a reliable sorting
algorithm as runs through the list repeatedly, compares adjacent elements, and swaps
them if they are out of order. This process is repeated until the list is sorted and hence,
this is also considered the simplest sorting algorithm.
Time Complexity of Searching Algorithms
The Time complexity of searching algorithms can be elaborated as checking for an element
or retrieving an element from any data structure where it was once stored. Understanding the
time complexities with respect to the searching algorithms can help us in picking up the best
searching technique and analyse which of them is faster, listed below are few of them:
1. The Time Complexity of Linear Search: The Time Complexity of Linear
Search has the best case defined by Ω(1) and the worst case defined by O(n). Here we
can see that the Linear Search follows the sequential access.
2. The Time Complexity of Binary Search: The Time Complexity of Binary
Search has the best case defined by Ω(1) and the worst case defined by O(log n).
Binary Search is the faster of the two searching algorithms. However, for smaller
arrays, linear search does a better job.
Example to demonstrate the Time complexity of searching algorithms:
let us dive deep into this following example to understand more on the Time complexity of
searching algorithms. Here we shall be comparing two different algorithms which were used
to solve a problem. Now the problem that we are talking here is searching.
Let us assume here that we have a sorted array in ascending order and we need to search for a
particular element in this array. Now to solve this problem we have two algorithms:
1. Linear Search.
2. Binary Search.
Consider an array which contains five elements, and we have to find the number 24 in the
array.
const array = [3, 7, 11, 21, 24];
const search_value = 24;
Now when we search via the Linear search algorithm, it will compare each element of the
array to the specific search_value. Now when the search_value is found, it shall return true.
Seems simple right!
Now let’s analyse the count of the number of operations that it needs to performs to return its
output. We find the answer to this as 5 as it compares every element of the array one by one.
We can now say that by linear search it uses five operations (the maximum number of
operations) to find the given element. This is also the worst case of an algorithm in the case
of Linear search.

Summing up, The Linear search takes 'n' (where n is the size of the array) number of
operations in its worst case.
Now Let’s examine the Binary search algorithm by diving deep into a simple example.
const array = [3, 7, 11, 21, 24, 36, 47, 61, 99];
const search_value = 21;
Now for the given array , if we were to find the search_value then by the binary search
algorithm we first check the middle of the array. Now as middle of array is 24, we then
check 24>21 so we start searching left, then by searching elements to the left we see that we
will find the number say 7, then as 7<21 then we start searching to the desired element and
once we see 21, then we compare 21=21.
We see that when we count number of operations binary search took to find the desired
element was four operations. This justify that there is a logarithmic approach for the binary
search.
number of operations = log(10) = 4(approx)
for base 2
Standard representation when we search through binary search algorithm can be represented
as ,
For an array of size n, the number of operations performed by the Binary Search is: log(n)
How to Compare Algorithms?
When we want to find out what are the parameters to compare algorithms in the time
complexity domain, listed below are the three points which must be evaluated to compare the
algorithms:
1. The Execution Time: The execution time can be one of the measure but majorly this also
gets manipulated to the particular computer we are using so this measure can be one of the
aspect but not the only aspect to define and compare algorithms.
2. The number of statements executed: The number of statements executed can too vary
with respect to which expertise the programmer is having as well as the number of statements
varies with the programming language. Again, this measure can be one of the aspect but not
the only aspect to define and compare algorithms.
3. The Ideal solution: When we want to the express the running time of a given algorithm
with respect to the function of the input size lets say ,'f(n)'. Now to compare these different
functions corresponding to running times we see that this type of comparison are independent
of machine time, programming style, etc.

Conclusion
1. The Time complexity can be defined as the amount of time taken by an algorithm to
execute each statement of code of an algorithm till its completion with respect to the
function of the length of the input.
2. The three important asymptotic notations of Time complexity are:
O: This notations defines if functions grow slower than or at the same rate with
respect to the expression.
Ω: This notations defines if functions grow faster than or at the same rate with respect
to the expression.
Θ: This notations defines if functions' growth lie in both O and Ω.
3. The different types of Time complexity of sorting algorithms are:
o The Time Complexity of Insertion Sort : The time complexity of Insertion
Sort is Ω(n) in its best case possible and O(n^2) in its worst case possible.
o The Time Complexity of Merge Sort : The time complexity of Merge Sort
is Ω(nlogn) in its best case possible and O(nlogn) in its worst case possible.
o The Time Complexity of Quick Sort: The time complexity of Quick Sort
is Ω(nlogn) in its best case possible and O(n^2) in its worst case possible.
o The Time Complexity of Bubble Sort : The time complexity of Bubble Sort
is Ω(n) in its best case possible and O(n^2) in its worst case possible.
4. The Time complexity of searching algorithms can be elaborated as checking for an
element or retrieving an element from any data structure where it was once stored.
5. The different types of Time complexity of searching algorithms are:
a. The Time Complexity of Linear Search: The Time Complexity of Linear Search
has the best case defined by Ω(1) and the worst case defined by O(n).
b. The Time Complexity of Binary Search: The Time Complexity of Binary Search
has the best case defined by Ω(1) and the worst case defined by O(log n).

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