1.8.2 Asymptotic Notations - Big Oh - Omega - Theta #2 - Phụ đề song ngữ
Let's take three examples.
F of n, if it is 2n squared plus 3n plus 4.
Suppose it is a function.
So 2n squared plus 3n plus 4 is less than 2n squared plus 3n squared plus 4n squared.
So the 2n squared plus 3n plus 4 is less than equal to 9n squared.
for all n greater than or equal to some value you can find out from which value is starting and just put one there.
So this is greater than or equal to this one.
So this is C and this is G of n.
So it is f of n is equal to f of g of n.
So g of n is what?
n squared.
f of n is big of n squared.
Now for the same function, I can say 2n squared plus 3n plus 4 is greater than or equal to 1 into n squared.
So it is omega of n squared.
So even I can write omega of n square, I made it as 1 into n square.
Now, same thing, 2n square plus 3n plus 4 is lying between 1n and n and nine n squared.
So this is theta of n squared, n squared on both sides.
Next, if the component is n squared log n plus n, then n squared log n squared
plus n is less than equal to 10 m squared log n.
I just wrote some value greater than there.
And here, it is less than equal to 1 m squared log n.
So, put the site a log n and n squared log n and this is a smaller, this is greater than this one and this is
smaller than this one.
So, put the site a log n squared log n squared So this is big O of n squared log n, and omega of n squared log n.
As well as this is theta of n squared log n.
So all three have written them together here.
Now this order, we have not written n squared log n.
But if I write n squared log n comes in between n squared and n cube,
n squared log n is greater than n squared but less than n cube.
So this is the combination of n squared and log n and for this also you can specify a class here in between those two,
you can mention a class.
Next if the function is n factorial, then let us see.
N factorial is nothing but n into n minus 1 into n minus 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 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, we are then equal to what?
So, as a practice we'll make everything as 1 into 1 into 1 into goes on to 1.
So, this side I get 1, and this is n factorial, and this is n bar.
Now, for this one, for n factorial, I don't have any smaller value here and I have to take
it as 1 and if I take larger value of this n power n, upper bound is n power n.
And on the side, I am not getting same thing, I am not getting same thing.
If I get same thing, then I can take it as 3 down.
So what I can write, I can write Big O n power n and omega power 1.
So the lower bound for n factor is 1, upper bound for n factor is n power n.
Alright, so here...
we cannot find a 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 power n.
So, you cannot fix a particular place for n factorial.
You cannot say that n factorial is always n over 10 and less than n over 11.
You cannot say that.
You cannot find a place for it.
You cannot find a place for it.
So, it's greater than or to 1, but less than or equal to n power n.
So, upper bound is n power n and a lower bound is 1.
So, this is the function for which you cannot find the theta.
So, now upper bound and lower bound are useful.
As you already told that when you cannot mention theta bound for any function, then we go for upper bound and lower bound.
So, yes, here we have to come upper bound.
function.
That's it 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 will 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 0 but we write 1 and this is log of n factorial and this is less than equal
to this is log of n raised to n.
and that will be written as n log n, the power comes on this side right is n log n.
So now you can see that for n log n factorial also upper bound is log n power n and the
lower bound is 1 and there is no tight bound for this function.
So the factorial function, we cannot define the tight bound.
So we go for upper bound as and log in and lower bound as 1.
So there is no type form.
So now we have understood when you use a representation when we use omega and all its theta is
preferable if you are able to find theta for any function that's better because that's a type form.
And if you are using upper bound we also try to use a type form.
Don't give away a value, a larger value.
And if you use a lower bound don't give any smaller value.
Try to give a nearest value for a function is n squared, so try to give it n squared only.
Door and login or n root n is also correct, true, but not meaningful but not useful.
It may 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.
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 speakers 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 near the rolling.
But if you say minimum means 23,000 so your answer is correct.
I can get some of my phone in 2000 also smart phone also but that will not be a suitable one for me but your answer is right but it's not useful.
So, you well, that do that.
So, even for one left will have mobile phones are also available, answer is correct, but not meaningful for me.
So, same way, when you have any function, n squared, then you can say n squared, that is the answer, perfect answer.
So, give it theta is the right one.
But if you are writing omega, so I am not sure.
whether you are giving me the nearest or not, that is the problem if we go.
It may be bigger than that one also,
but the key to guarantee is that you have given the exact figure, exact notation, exact time of the city.
When you are using omega,
then also it means that you may be giving a lower value, may or may not be in the nearest one.
So that is why, before omega are used when you are not sure about the exact one.
That is the meaning of the case.
So that is much suitable for n factorial and log n factorial, but you can use them for any purpose.
It is not a rule.
you must use the state of the way, you can use any notation.
So, when you are using open topics maybe minimum this one and maybe it is more than this.
the time may be more than this also.
that's the meaning.
Teeth Levins exactly this technique.
So if you have listened shopping this is a very important topic.
Now in the coming video of this next video you can find the properties of a synthetic notation.
You can move it
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