L-1.3: Asymptotic Notations | Big O | Big Omega | Theta Notations | Most Imp Topic Of Algorithm
Summary
TLDRThis 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
📚 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.
🔍 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.
📈 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.
Mindmap
Keywords
💡Asymptotic Notations
💡Big O Notation
💡Time Complexity
💡Input Values
💡Function f(n)
💡Iteration
💡Big Omega Notation
💡Theta Notation
💡Little O Notation
💡Little Omega Notation
💡Competitive Exams
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.
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