1.8.2 Asymptotic Notations - Big Oh - Omega - Theta #2
Summary
TLDRThe 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
📐 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.
🔍 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.
🎥 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.
Mindmap
Keywords
💡Asymptotic Notation
💡Big O Notation (O)
💡Omega Notation (Ω)
💡Theta Notation (Θ)
💡Growth Rate
💡Upper Bound
💡Lower Bound
💡Tight Bound
💡Factorial
💡Logarithm
💡Time Complexity
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
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