1.8.2 Asymptotic Notations - Big Oh - Omega - Theta #2

Summary

The video explains key concepts of asymptotic notations—Big O, Omega, and Theta—used to describe algorithmic time complexity. It uses examples like polynomial functions and factorials to demonstrate how to determine upper and lower bounds for functions. Theta notation is highlighted as the most precise, while Big O and Omega provide broader estimates. The importance of selecting the nearest time complexity for meaningful analysis is emphasized, with analogies for clearer understanding. The video also prepares viewers for a subsequent discussion on the properties of these notations.

Takeaways

• 📐 The function 2n^2 + 3n + 4 is bounded by 9n^2 and n^2, indicating it is O(n^2) and Ω(n^2) but not Θ(n^2).
• 🔍 For the function n^2 log n + n, it is both O(n^2 log n) and Ω(n log n), making it Θ(n^2 log n).
• 📈 The function n! (n factorial) does not have a tight bound and is described as O(n^n) and Ω(1).
• 📊 log n! (log of n factorial) is bounded by n log n, making it O(n log n) and Ω(1), without a tight bound.
• 🌐 The importance of using the tightest possible bound is emphasized, with Θ notation being the most desirable for its precision.
• 📉 When a tight bound cannot be found, using upper bounds O and lower bounds Ω is recommended.
• 🔑 The script illustrates how to compare functions to determine their asymptotic relationships using inequalities.
• 📚 Understanding the concept of asymptotic notations is crucial for analyzing algorithm performance.
• 📝 The script provides practical examples to help viewers grasp the concept of asymptotic notations and their application.
• 🎯 The takeaway is that while Θ provides an exact measure, O and Ω offer insights into the upper and lower bounds of a function's growth.

Q & A

What does the symbol 'Big O' represent in the context of the transcript?

In the transcript, 'Big O' notation is used to describe the upper bound of a function, indicating that one function grows at most as fast as another function. It is denoted as f(n) = O(g(n)) where f(n) is less than or equal to c*g(n) for some constant c and for all n greater than or equal to some value.

Can you explain the example given for the function 2n² + 3n + 4 in terms of Big O notation?

The transcript explains that the function 2n² + 3n + 4 is less than 2n² + 3n² + 4n², which simplifies to 9n². Therefore, 2n² + 3n + 4 is O(n²), meaning it is bounded above by a constant times n² for sufficiently large n.

What is meant by 'Omega' in the transcript?

In the transcript, 'Omega' notation is used to describe the lower bound of a function, indicating that one function grows at least as fast as another function. It is denoted as f(n) = Ω(g(n)) where f(n) is greater than or equal to c*g(n) for some constant c and for all n greater than or equal to some value.

How is the function 2n² + 3n + 4 described using Omega notation according to the transcript?

The transcript states that 2n² + 3n + 4 is greater than or equal to 1*n², which means it is Ω(n²). This indicates that the function grows at least as fast as n².

What is the significance of 'Theta' notation as discussed in the transcript?

Theta notation in the transcript is used when a function is both Big O and Omega of another function, indicating that it grows at the same rate. It is denoted as f(n) = Θ(g(n)) where f(n) is bounded by c*g(n) and c'*g(n) for some constants c and c' and for all n greater than or equal to some value.

How does the transcript describe the function n² log n + n in terms of Big O and Theta notation?

The transcript explains that n² log n + n is less than or equal to 10n² log n and also greater than n^s log n for some s. This places it between n^s log n and n^s log n, making it both Big O of n² log n and Theta of n² log n.

What does the transcript suggest about the use of Big O, Omega, and Theta notations?

The transcript suggests that Theta notation is preferable when a tight bound can be found, as it provides an exact figure for the time complexity. However, when a tight bound cannot be found, Big O and Omega notations are used to describe the upper and lower bounds, respectively.

Why is it difficult to define a Theta notation for n factorial as per the transcript?

The transcript explains that n factorial does not have a tight bound or average bound that can be represented by Theta notation because it grows faster than any polynomial function. For small values of n, it may be closer to 1, but for larger values, it grows towards n^n, making it impossible to define a consistent Theta relationship.

How does the transcript describe the function log n factorial in terms of upper and lower bounds?

The transcript states that log n factorial is less than or equal to n log n, which means its upper bound is log n^n and the lower bound is 1. Since there is no tight bound for log n factorial, it cannot be represented by Theta notation.

What is the importance of choosing the 'right' value when using Big O notation as explained in the transcript?

The transcript emphasizes the importance of choosing the 'right' value when using Big O notation because it should be meaningful and useful. For example, if the function is n², then stating it as n² is the most useful and meaningful, whereas stating it as n log n or n might be correct but less useful.

What advice does the transcript give regarding the use of asymptotic notations?

The transcript advises that when using asymptotic notations, one should aim to use a tight bound if possible, which is represented by Theta notation. If a tight bound cannot be found, then Big O and Omega notations should be used, with an attempt to provide the nearest value for the upper or lower bound.

Outlines

00:00

📐 Understanding Asymptotic Notations

This paragraph introduces the concept of asymptotic notations used in computer science to describe the performance or complexity of algorithms. The script explains how to determine if a function is O(n²), Ω(n²), or Θ(n²) by comparing it to simpler expressions. It uses the example of the function 2n² + 3n + 4 and shows how it can be bounded by other quadratic expressions to classify its growth rate. The paragraph also discusses how to handle functions that do not fit neatly into these categories, such as n! (n factorial), and the importance of providing tight bounds when possible.

05:02

🔍 Exploring Upper and Lower Bounds

The second paragraph delves deeper into the practical application of upper and lower bounds, especially when a function cannot be described by a tight Theta notation. It uses the example of n factorial and log(n factorial) to illustrate how to find the bounds. The script emphasizes the importance of providing meaningful bounds that are neither too loose nor too tight, using the analogy of buying a mobile phone to explain the concept. It concludes by highlighting the preference for Theta notation when possible, as it provides an exact measure of time complexity, and the appropriate use of Big O and Big Omega when exact values are uncertain.

10:04

🎥 Next Video Teaser

The final paragraph serves as a teaser for the next video, which will cover the properties of asymptotic notations. It does not contain substantial content but indicates that further learning will be provided in the upcoming video, encouraging viewers to continue their education on the topic.

Keywords

💡Asymptotic Notation

Asymptotic Notation is a system used in computer science to describe the performance or complexity of an algorithm in the context of the size of the input data. It provides a way to classify algorithms according to how their running time or space requirements grow as the input size grows. In the video, asymptotic notation is central to understanding how functions relate to each other in terms of growth rates, with examples given such as comparing '2n² + 3n + 4' to 'n²'.

💡Big O Notation (O)

Big O Notation, often denoted as O, is used to describe the upper bound of an algorithm's growth rate. It represents the worst-case scenario in terms of time or space complexity. The video explains that '2n² + 3n + 4' is O(n²) because it grows at most as fast as '9n²' for sufficiently large n, illustrating the concept with a clear example.

💡Omega Notation (Ω)

Omega Notation, denoted as Ω, is used to describe the lower bound of an algorithm's growth rate. It signifies the best-case scenario for an algorithm's complexity. The script uses '2n² + 3n + 4' as an example to show that it is Ω(n²) since it grows at least as fast as 'n²'.

💡Theta Notation (Θ)

Theta Notation, represented by Θ, is used when both the upper and lower bounds of an algorithm's growth rate are known and are the same up to a constant factor. It signifies a tight bound on the growth rate. The video uses '2n² + 3n + 4' as an example, showing that it lies between 'n²' and '9n²', thus it is Θ(n²).

💡Growth Rate

Growth Rate refers to how quickly a function increases as its input size increases. In the context of the video, growth rate is crucial for understanding the relative performance of different algorithms. The script explains how different functions like 'n² log n' and 'n!' have different growth rates.

💡Upper Bound

An Upper Bound in the context of algorithm analysis is a function that an algorithm's running time or space requirement will not exceed. The video mentions 'n^n' as an upper bound for 'n factorial', indicating that the algorithm's complexity will not be worse than this.

💡Lower Bound

A Lower Bound is a function that an algorithm's running time or space requirement will not be less than. The script uses '1' as a lower bound for 'n factorial', meaning that the complexity cannot be better than constant time.

💡Tight Bound

A Tight Bound is an asymptotic notation that accurately represents an algorithm's growth rate without overestimating or underestimating. The video emphasizes the importance of finding a tight bound, like Θ(n²) for 'n²', as it provides the most accurate representation of an algorithm's complexity.

💡Factorial

Factorial, denoted as 'n!', is the product of all positive integers less than or equal to n. The video discusses 'n factorial' to illustrate a function for which it is difficult to find a tight bound, instead only providing an upper bound of 'n^n' and a lower bound of '1'.

💡Logarithm

Logarithm is the inverse operation to exponentiation. In the video, logarithms are used to analyze functions like 'log n factorial', where the script shows that the logarithm of the factorial of n is less than 'n log n', providing an example of how logarithms can be used to find bounds.

💡Time Complexity

Time Complexity is a measure of the amount of time an algorithm takes to run as a function of the length of the input. The video uses time complexity to explain how different algorithms scale with input size, using Big O, Omega, and Theta notations to describe this.

Highlights

Introduction to Big O notation and its application in comparing function growth rates.

Example of comparing 2n² + 3n + 4 to other quadratic functions to determine its growth rate.

Explanation that 2n² + 3n + 4 is O(n²) and Ω(n²), illustrating its upper and lower bounds.

Discussion on how to find the value from which a function starts behaving in a certain way.

The concept that 2n² + 3n + 4 is Θ(n²), showing it lies between 1n² and 9n².

Introduction to the function n² log n + n and its comparison to other functions to find its growth rate.

Explanation that n² log n + n is both O(n² log n) and Ω(n log n).

The importance of finding a tight bound (Θ) for a function's growth rate when possible.

Example of analyzing n factorial and its growth rate, showing it is not Θ(n^k) for any k.

Explanation of how to use upper and lower bounds when a tight bound cannot be found.

The concept of log n factorial and its growth rate analysis.

Discussion on the practical implications of using different notations like Big O, Ω, and Θ.

Advice on choosing the most meaningful and nearest values when describing function growth.

The significance of Θ notation in providing an exact figure for time complexity.

Explanation of when to use Ω notation and its implications on the lower bound of a function.

The application of asymptotic notations in real-world scenarios, using the mobile phone price example.

Encouragement to use Θ when possible for its precision, but acknowledgment that any notation can be used.

预告下一节视频内容：asymptotic notations的属性。

Transcripts

let just take few

examples F ofn if it is 2 n² + 3 n +

4 suppose this is a

function so 2 n² + 3 n + 4 is less than

2 n² + 3 n² + 4 n² so this 2 n² + 3 n +

4 is less than equal

to 9

n² for all n greater than equal to some

value you can find out from which value

it is starting I just put one

there so this is greater than equal to

this one so this is C and this is G of

n so it is f of n

is Big of G of n so G of n is what n²

F of n is Big of

n² now for the same function I can say 2

n² + 3 n + 4 is greater than = 1 into n²

so it is Omega of

n² so even I can write Omega of n² I

made it as 1 into n² now same thing 2 n²

+ 3 n + 4 4 is lying in between 1 n

n² and 9 n² so this is Theta of n² n² on

both the

sides next if F of n is n² log n + n

then n² log n + n is less than equal

to 10 n² log n i just wrote some value

greater value there and here it is less

than equal to 1 n² log n so put the side

I got n s log n and n sare log n and

this is a smaller this is greater than

this one and this is smaller than this

one so both the side I got n s log n and

n sare log n so this is big of n² log

n and Omega of n² log n

as well as this is Theta of n² L so all

three I have written them together here

now if You observe in

our this order we have not return n s

log n but if I write n Square log n

comes in between n² and

NQ n s log n is greater than n s but

less than n Cub so this is the

combination of n² and log n and for this

also you can and specify a class here in

between those two you can mention a

class next if a function is n factorial

then let us see n factorial is nothing

but n into n -1 into N - 2 into goes on

to 3 into 2 into 1 so if I write in

reverse then 1 into 2 into 3 goes on to

n

now this is greater than less than or

equal to what see as a practice what we

were doing everything we were making it

as n so here also I'll make it as n into

n into n into goes on to

n and this is less than equal to what so

as a practice we'll make everything as 1

into 1 into 1 into goes on to one so

this side I get 1 and this is n

factorial and this is n^ n

now for this

one for n factorial I don't have any

smaller value here and I have to take it

as one and if I take larger value it is

n power n upper bound is n^ n and both

the side I'm not getting same thing I'm

not getting same thing if I get same

thing then I can take it as Theta so

what I can write I can write B of n^n

and and Omega of

1 so the lower bond for n factor is one

upper Bond 4 n factorial is n^

n right so here we cannot find the tight

Bond or average bond or Theta for n

factorial so if you try to put n

factorial here class is not there for n

factorial but if you try to put it there

then for the smaller values of n it will

be nearer to this one and for the larger

values of n it will be increasing and

going up to n^ n so you cannot fix a

particular place for n factorial you

cannot say that n factorial is always

greater than n^ 10 and less than n power

11 you cannot say that you cannot find a

place for it you cannot find a place for

it

right so it's greater than equal to 1

but less than equal to n^n so upper

bound is n^ n and the lower bond is one

so this is the function for which you

cannot find the Theta so now upper bound

and lower Bond are useful as I already

told that when you cannot mention Theta

bond for any function then we go for

upper bound or lower bound so yes here

we have taken upper bound and lower Bond

of a function let's take log n factorial

so for log n factorial if I write log of

1 into 2 into 3 and so on to n then this

is less than or equal

to log of as a practice we'll make

everything as n so n into n into so on

to n and it is less than log 1 into 1

into so on to 1 so this side will be

zero but we write 1 and this is log of n

factorial and this is less than equal to

this is log of n to n and that will be

written as n log

n that part comes on this side and it is

n log n so now you can see that for n

log n factorial also upper bound is log

n^ n and the lower bond is one and there

is no tide bond for this function so the

factorial function we cannot Define the

tight Bond so we go for upper bound

as

analog and lower bound as one so there

is no tight bound so now you have

understood when to use a big old

notation when to use Omega and always

Theta is preferable if you are able to

find Theta for any function that's

better because that is a tight Bond and

if you're using upper bound big go also

try to use a tight Bond don't give a

very far away value a larger value and

if you're using low Bond don't give any

smaller value try to give a nearest

value for a function suppose your

function is n Square so try to give it n

Square only though writing n log n or n

or root n is also correct true but not

meaningful but not

useful if I ask you that I want to buy a

mobile phone for my requirement what

could be the better price so any price

for a mobile phone that's available in

the market so suppose you have some idea

about the mobile phone and you said that

you can buy a mobile phone around

20,000 so you are giving me nearest

figures if I say if you don't know the

nearest figures and if I ask you tell me

something minimum so you are saying that

18 19,000 or maximum you can say 21

24,000 like that so you are nearer only

but if you say minimum means 10 2,000

3,000 so your answer is correct I can

can get some mobile phone in 3000 also

smartphone also but that will not be a

suitable one for me but your answer is

right but it's is not useful and if I

ask maximum so you say 1 lakh 2 lakh so

even for 1 lakh 2 lakh mobile phones are

also available answer is correct but not

meaningful for me so same way when you

have any function n² then you can say n

sare that is the answer perfect answer

so giving Theta is the right one but if

you're writing Omega so I'm I'm not sure

whether you are giving me nearest or not

that's the problem with bigo it may be

nearest one it may be bigger than that

one also but Theta guarantees that you

have given the exact figure exact

notation exact time complexity when

you're using Omega then also it means

that you may be giving a lower value may

or may not be nearest one so that's why

big Omega used when you're not sure

about the exact one right that is the

meaning it KS so that is much much

suitable for uh n factorial and log n

factorial but you can use them for any

purpose it's not a rule that you must

not use you must use Theta only you can

use any notation so when you're using

Omega means maybe minimum this one and

maybe it is more than this the time may

be more than this also that's the

meaning Theta means exactly this much

time only so hope you have understood

this topic this is very very important

topic now in coming video that is next

video you can find the properties of

asymptotic notations you can watch that

video