L-1.3: Asymptotic Notations | Big O | Big Omega | Theta Notations | Most Imp Topic Of Algorithm

Summary

This video from Gate Smashers delves into asymptotic notations, crucial for understanding algorithm time complexity. It covers Big O, Big Omega, Theta, and other notations, explaining how they help compare algorithms and determine the best one. The explanation uses examples like 2n^2 + n to illustrate upper and lower bounds, emphasizing the importance of Big O for maximum time complexity.

Takeaways

• 📚 Asymptotic notations are crucial for analyzing the time complexity of algorithms, especially for competitive exams and academic assessments.
• 🔢 The primary notations used are Big O, Big Omega, and Theta, each representing upper bound, lower bound, and average case time complexity, respectively.
• 📈 Big O notation (O(g(n))) indicates the upper bound or the maximum time an algorithm might take, often used to describe the worst-case scenario.
• 📉 Big Omega notation (Ω(g(n))) signifies the lower bound or the minimum time required, representing the best-case scenario.
• 🔄 Theta notation (Θ(g(n))) combines both upper and lower bounds, indicating the average time complexity of an algorithm.
• 📚 Understanding the concept of iteration and frequency is essential for analyzing time complexity without executing the algorithm.
• 🔎 The choice of constants in Big O notation is crucial; for example, choosing the closest upper bound for the quadratic term in an expression like 2n^2 + n.
• 🔍 In Big Omega notation, the focus is on the greatest lower bound, ensuring that the chosen term is the closest to the function's growth rate.
• 📘 The example of searching a book without indexing illustrates the concepts of best case (Big Omega), average case (Theta), and worst case (Big O) scenarios.
• 🔗 The script emphasizes that Big O notation is particularly important as it encompasses the maximum time, implicitly including best and average cases.
• 📌 The script also mentions lesser-known notations like little o (o(g(n))) and little omega (ω(g(n))), which exclude the equality condition compared to their Big counterparts.

Q & A

What is the main topic discussed in the video?

The main topic discussed in the video is asymptotic notations, which are mathematical ways of representing the time complexity of algorithms.

Why are asymptotic notations important in the context of algorithms?

Asymptotic notations are important because they allow us to analyze and compare the time complexity of different algorithms without actually executing them, which is crucial for understanding their performance in competitive exams or academic settings.

What are the different types of asymptotic notations mentioned in the video?

The video mentions several types of asymptotic notations including Big O, Big Omega, Theta, Little O, and Little Omega.

How is Big O notation used to represent the time complexity of an algorithm?

Big O notation is used to represent the upper bound of an algorithm's time complexity. It indicates the maximum time an algorithm will take to complete its task, often ignoring lower-order terms and constants.

What does it mean when an algorithm's time complexity is represented as O(n^2)?

When an algorithm's time complexity is represented as O(n^2), it means that the time taken by the algorithm grows quadratically with the input size n, making it a quadratic function.

How do you determine the dominant term in an algorithm's time complexity expression?

The dominant term in an algorithm's time complexity expression is determined by comparing the growth rates of different terms. Typically, higher-degree polynomial terms or exponential terms will dominate over lower-degree polynomials or linear terms as n increases.

What is the difference between Big O and Big Omega notations?

Big O notation represents the upper bound of an algorithm's time complexity, indicating the maximum time it will take. Big Omega notation, on the other hand, represents the lower bound, indicating the minimum time it will take.

How is Theta notation used to represent an algorithm's time complexity?

Theta notation is used to represent the average case time complexity of an algorithm. It indicates that the time complexity lies strictly between two bounds, typically expressed as Θ(g(n)), where g(n) is the function that describes the average case.

What is the significance of choosing the correct constant values in asymptotic notations?

Choosing the correct constant values in asymptotic notations is crucial for accurately representing the time complexity of an algorithm. It ensures that the notation accurately reflects the algorithm's performance, especially in terms of its upper or lower bounds.

How do Little O and Little Omega notations differ from their Big counterparts?

Little O notation (o) indicates that the time complexity of an algorithm grows strictly faster than the function g(n), without including the possibility of equality. Little Omega notation (ω) indicates that the time complexity grows strictly slower than the function g(n), also without the possibility of equality.

Outlines

00:00

📚 Introduction to Asymptotic Notations

The video begins with an introduction to asymptotic notations, emphasizing their importance in algorithm analysis. Asymptotic notations are mathematical representations of time complexity, crucial for comparing algorithms without executing them. The presenter explains that these notations help determine how many times a statement is executed (iteration frequency) and how many times a function calls itself. The focus is on three primary notations: Big O, Big Omega, and Theta, with Big O being the most common. The explanation includes a graphical representation of n (input values) on the x-axis and time on the y-axis, illustrating how these notations are used to compare algorithms.

05:04

🔍 Understanding Big O Notation

This paragraph delves deeper into Big O notation, which represents the upper bound or 'at most' time complexity of an algorithm. The presenter uses a function f(n) = 2n^2 + n as an example, explaining how to determine the dominant term (n^2 in this case) and how to choose a constant c that ensures f(n) ≤ c * g(n) for all n ≥ k. The emphasis is on selecting the least upper bound, illustrating that for large values of n, n^2 dominates n, making it the appropriate term for Big O representation. The explanation clarifies that the choice of c must be such that the inequality holds for all n greater than a certain threshold.

10:07

📈 Big Omega and Theta Notations

The third paragraph introduces Big Omega and Theta notations, focusing on their roles in algorithm analysis. Big Omega represents the lower bound or 'at least' time complexity, ensuring that the algorithm will take at least a certain amount of time. The presenter uses the same function f(n) = 2n^2 + n to demonstrate how to find the greatest lower bound, emphasizing the importance of choosing the closest lower value. Theta notation is then introduced as a representation of the average case, combining both upper and lower bounds. The presenter explains that Theta notation is used to describe the average time complexity, balancing the best and worst-case scenarios. The explanation also includes a practical example of searching through a book, illustrating the concepts of best, average, and worst cases.

Keywords

💡Asymptotic Notations

Asymptotic notations are mathematical representations used to describe the time complexity of algorithms. They provide a way to analyze the efficiency of algorithms without executing them. In the video, asymptotic notations are the central theme, as they are essential for comparing the performance of different algorithms, especially in competitive exams and academic settings.

💡Big O Notation

Big O notation is a commonly used asymptotic notation that represents the upper bound of an algorithm's time complexity. It indicates the maximum time an algorithm will take to complete its task. In the script, the concept is explained through the example of a function f(n) = 2n^2 + n, where it is shown that the time complexity can be represented as O(n^2), highlighting the quadratic nature of the function.

💡Time Complexity

Time complexity is a measure of the amount of time an algorithm takes to run as a function of its input size. It is a critical aspect in algorithm analysis and is discussed in the video in the context of asymptotic notations. The script uses the example of f(n) = 2n^2 + n to illustrate how time complexity can be determined and represented.

💡Input Values

Input values, denoted as 'n' in the script, are the variables that determine the size of the problem an algorithm is designed to solve. They are crucial in analyzing the time complexity of algorithms because the performance of an algorithm often depends on the size of the input. The script discusses how the number of input values affects the execution time of an algorithm.

💡Function f(n)

In the context of the video, 'function f(n)' refers to a specific algorithm or process whose time complexity is being analyzed. The script uses f(n) = 2n^2 + n as an example to demonstrate how to determine and represent the time complexity of an algorithm using asymptotic notations.

💡Iteration

Iteration refers to the number of times a statement is executed in an algorithm. It is a fundamental concept in algorithm analysis, as it helps in understanding how the execution time of an algorithm scales with the input size. The script mentions iteration in the context of determining the time complexity of an algorithm without its actual execution.

💡Big Omega Notation

Big Omega notation is another asymptotic notation that represents the lower bound of an algorithm's time complexity. It indicates the minimum time an algorithm will take to complete its task. The script explains this concept by stating that if f(n) = 2n^2 + n, then its time complexity can be represented as Ω(n^2), emphasizing the quadratic nature of the function.

💡Theta Notation

Theta notation is used to represent the average or exact time complexity of an algorithm. It combines both the upper and lower bounds, indicating that the algorithm's time complexity lies strictly between these bounds. The script uses theta notation to illustrate the balanced nature of an algorithm's performance, as seen in the example where f(n) = 2n^2 + n is shown to be Θ(n^2).

💡Little O Notation

Little O notation is a variant of Big O notation that excludes the possibility of equality. It indicates that the time complexity of an algorithm grows strictly faster than the function it is compared to. The script briefly mentions this notation to differentiate it from Big O notation, where equality is allowed.

💡Little Omega Notation

Little Omega notation is similar to Big Omega notation but does not include the possibility of equality. It signifies that the time complexity of an algorithm grows strictly slower than the function it is compared to. The script explains this by stating that if f(n) grows faster than g(n), then f(n) can be represented as o(g(n)).

💡Competitive Exams

Competitive exams are tests that assess a candidate's knowledge and skills in a competitive setting, often for admission to educational institutions or job opportunities. The script emphasizes the importance of understanding asymptotic notations for success in such exams, as they are crucial for analyzing algorithmic efficiency.

Highlights

Introduction to asymptotic notations as a crucial topic in algorithm analysis.

Asymptotic notation is used to represent time complexity in algorithm analysis.

Explains the concept of prior analysis without executing the algorithm.

Discusses the importance of iteration and frequency in determining time complexity.

Introduces Big O notation as the primary method for representing time complexity.

Describes Big O notation as an upper bound, indicating 'at most' time complexity.

Provides an example of representing a function f(n) in terms of Big O notation.

Explains the process of choosing the dominant term for Big O notation.

Discusses the concept of the least upper bound in Big O notation.

Introduces Big Omega notation as a representation of the lower bound or 'at least' time complexity.

Describes the process of choosing the greatest lower bound for Big Omega notation.

Introduces Theta notation as a representation of the average case time complexity.

Explains the concept of the greatest lower and least upper bounds in Theta notation.

Discusses the difference between Big O, Big Omega, and Theta notations.

Introduces little o notation, emphasizing its strict inequality without equality.

Introduces little omega notation, highlighting its strict inequality in the lower bound.

Emphasizes the importance of Big O representation in understanding time complexities.

Provides a practical example of time complexity in searching algorithms.

Transcripts

Hello friends, welcome to Gate Smashers

In this video we are going to discuss about asymptotic notations

And this is one of the most important topics of the algorithm

Because we discuss asymptotic notations in the syllabus

So guys in this video we are going to discuss all the important points from the basics

Which will be very beneficial for your competitive exams or for your college or university level exams

So guys like the video quickly, subscribe the channel if you haven't done it yet

And please press the bell button so that you get all the latest notifications

So let's start

What are asymptotic notations?

Asymptotic notation is the mathematical way of representing the time complexity

We have already seen that we talk about prior analysis

And in prior analysis means we do its analysis without execution of the algorithm

Now if we want to calculate time without execution

Then what we do is, how many times a statement is executed

That is called iteration, frequency

And we can say the number of times function is calling itself

So what we are doing here is, to represent the number of times

I need some notations so that I can properly represent

That this algorithm has this much time complexity

The other algorithm has this much time complexity

So that we can compare both of them and find out the better algorithm

So here in representation, first of all we use big O notation

Then big omega notation

Then we have theta notation

Apart from this, we have fourth, little o and little omega

So the main three are big O, big omega and theta notation

So first of all we start with big O notation

Because maximum time notations will be given in terms of big O

So what is big O notation here?

We have a graph, basically here is n, n is what? Input values

And what we have in y axis? Time

Now how to understand this?

We have a function f(n)

Now what is this function f(n)?

If I have any problem, any problem I have to solve

Let's say my time is or my function is f(n)

Now if I have to write fn in terms of let's say order of g(n)

What does this mean? That g(n) will always be greater than c.g(n) like this

Means f(n) is given to us

And if we have to represent this in terms of big O

So here g(n) should always be greater than equal to c.g(n)

And here c is a constant term which should always be greater than 0

And the value of n, this input value we are taking

This n should be greater than equal to k

And you can say k should be greater than equal to what?

It should be greater than equal to 0

We are talking about this k

So here f(n) is equal to order of g(n) means

f(n) is always less than equal to c.g(n)

So here if we discuss this with examples then you will be more clear

Let's say I have any problem, any algorithm

To solve it, the total time I took was 2n^2 plus n

We took it simple here, let's say f(n) is equal to 2n^2 plus n

Now I have to represent f(n) in terms of order

Means f(n) is equal to order of what?

Now here I have to put the filling the blank

So filling the blank means f(n) is equal to 2n^2 plus n

And it should always be what?

Less than equal to c.g(n)

And g(n) value, that n value you have to choose

Now obviously this function should be greater

to make it greater means that equal to upper bound

Bigger notation represents upper bound

That is also called at most

Means in doing any work, maximum time you will take

That is a big O notation

Now here upper bound means which is the bigger term from 2n^2 plus n

Now to find the bigger term obviously you have to do

Here you check as n^2, what is n^2? Quadratic function

And here if we talk n that is a linear equation

Now which will be dominating in quadratic and linear?

Obviously n^2 will be dominating

How? You put the value of n, let's say we put the value of n as 100

So if we put the value of n as 100, then it is 100

But how much will be n^2? 10,000

If you put the value of n as 1 crore, obviously n is 1 crore

And what is n^2? 1 crore power 2

So the value of n^2 is much bigger than n

If you want to represent it also, we represent it in this way

So n is what? Linear time

What about n^2? n^2 will obviously grow more than n

But not for the small values

Because if you put the value of n as 1 or 0

Then you are not getting much difference here

But for larger values, which is dominating? n^2

So here if I want to write upper bound

Means if I want to write a big value, then can I write n^2?

Can write

Can write n^3? Can write

Can write n4? n5? 2 power n? n power n?

All these are bigger than this term

But remember whenever we talk about upper bound

Always remember, least upper bound

Means we have to choose only the closest value

Upper value but very closest

So what is the closest here? n^2

So this point is representing here

Because 2n^2 plus n should always be less than equal to c.gn^2

Now what value do you want to take here?

You have to choose c value here

To take c value, obviously if you take c value as 2

If you take c value as 2, then it will become 2n^2

Means 2n^2 plus n should be less than equal to 2.n^2

But obviously this will not happen because here plus n is also there

So what do you take c value?

We take c value as 3

So if I take c value as 3

So if you solve this, then what will happen?

n is less than equal to 2n^2

If I divide both sides with n, then it will be 1 less than n

Which you can write n should be greater than equal to 1

Means for all the values of n

For all the values of inputs n greater than equal to 1

This condition will always hold

And the c value should be 3

So here I have c value 3

And for n greater than equal to 1

This condition will always hold

Means 2n^2 plus n can be written as order of n^2

But what should be the c value? 3

And for all the n value greater than equal to 1

This condition will hold

So this way we represent big O notation

So if my f(n) is increasing this way

So if I want to write big O of gn

So gn means you have to write upper bound

What does this representation mean?

That before this k value

It can be that this value is getting smaller

It is getting bigger

It is fluctuation

But after k value

Like if we put n value here

Now put n value 1

Put 2, put 3, put 4

So you check this value will always increase

You can check let's say

I put n value 5

So what will come here?

2 into 5 square plus 5

Is always less than equal to 3.5 square

So see this is 25 into 2, 50

That is 55

Less than equal to 25 into 3, 75

So you can also prove

That yes for all the values

From n is equal to 1

Whatever value you put for c3

It will always hold the condition of order of n^2

So this is how we represent the big O notation

Then we talk about big omega

What is the meaning of big omega here?

We talk about lower bound here

That is at least

Means how much time will it take to do any work?

So if we talk here

How to represent it?

So here the meaning of representation is

That F(n) is equal to

This is how we are representing

F(n) is equal to

Omega of g(n)

If I want to write this

Then what does it mean?

F(n) should always be greater than equal to c.g(n)

So see if F(n) is increasing in this way

Then c.g(n) should always be

Less than equal to the value of F(n)

So here if we have the same function

Let's say my F(n) is 2n square plus n

And if I want to represent in terms of g(n)

Now what should be the value of g(n)?

G(n) value should be less than this

Less than this means

You can take n square

You can take n alone

You can take log n

Log of log n

All these values are smaller than this term

But whenever we talk about lower bound

Always remember

We have to take the greatest lower bound

Means closest

Like smaller than 5

There is 4, 3, 2

But I don't want to take 1 or 0

I want to be closest to 5

The smallest 4

So I have to take the greatest lower bound

Like we are taking the upper bound

Least

So here we have the greatest lower bound

So here if I talk

You can take n square

You can write n alone

You can also take log n

But you have to write the nearest value

The nearest value is again n square only

Now if you are confused

How will n square come?

If you take c value as 3

So now we have proved that this term will increase

But remember

You have to carefully choose the value of c

Here you choose the value of c

2

Now if I choose the value of c

See 2n square plus n will be greater than equal to

2.n square

So see you can clearly see here

2n square

2n square is equal

And here plus n is also there

So this term will always be greater than this term

And if you want to prove it

Here you take n greater than equal to 2n square

So from 2n square to 2n square is minus

Means for all values of n greater than equal to 0

What does n mean here?

This key

Means you can put any value of n

But from 0 to if you will increase

0, 1, 2, 3, 4, 5 up to n

This condition will always hold

But c value should be what?

c value should be 2 only

c value can be 1 also

It will also hold for 1

But you have to choose the closest value here

So for c value 2

This term will always be less

If you want to check

So let's again put 5 value

It will come to be 55

And it will be what? 50

So this term will always be less than this

So this is how we represent the big omega

That is at least or greatest lower bound

Third comes theta notation

Theta notation means average

We represent it like this

That fn should be greater than c1.gn

And it should be less than c2.gn

See this

If we have a function fn

The value of fn is greater than c1.gn

But should be smaller than c2.gn

We have already calculated this

Here c1.gn means

If my fn is 2n2 plus n

So what will be the value of gn here?

n2

And here also what will be the value of gn?

n2

But c1 value

What you have to do with this value?

If you want to do less than

Then you have to choose c1 value 2

And here what is c2 value?

3

So that means 2n^2 plus n is always greater than 2n^2

And it will always be less than 3n^2

So we represent this in case of theta

Which is the average case time complexity

If you want to understand this in a simple way

Let's say I have a book

Now in this book if you are searching any topic

No indexing, no hashing, nothing

So what do you have?

Best case

Best case means omega

Means as soon as you read the first page

You got your topic

But what is the meaning of worst case?

What is the meaning of worst case?

At max

So what is the meaning of at max?

That you are searching one by one

But where did you get the desired topic?

On the last page

Means you are doing a linear search

Without any sorting and all

So normally if you are doing a linear search

Then you have to search all the n pages

And what is the meaning of average case?

Neither your topic is on the first page

Nor in the last page you have to move on an average half pages

So we generally represent big O

The reason for this is

That it will tell you the maximum time

So obviously best case and average case

Are included in this automatically

So this is how we represent the big O

Big omega and theta representation

If we talk about little o

Then there is a simple difference in little o

That in less than equal to

Here equal to does not come

In case of little o

It does not come equal to, rest of the funda is same

And if we talk about little omega

Then in little omega also

If we talk here

Then this greater than we had written

Let's say Fn should be greater than equal to c.gn

This is in the big omega

In little omega Fn should always be greater than c.gn

This is the only difference between these two

Now we will see all the time complexities

Of searching sorting algorithms

You will get representation in all

In terms of big O, best case, average case and worst case

But remember this is important

This big O representation is

Much more important than the other representation

Thank You.