Big Oh(O) vs Big Omega(Ω) vs Big Theta(θ) notations | Asymptotic Analysis of Algorithms with Example

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--------------------------------------------------------------------------------------------- In this tutorial we will understand the 3 different Asymptotic Time Complexity analysis of Algorithms namely -
Big Oh(O)
Big Omega(Ω)
Big Theta(θ)
We will understand each Complexity by taking its mathematical definition as well as example with graph.
Lastly we will understand its practical usage & understand why we really need 3 different time complexity measures.
Big O notation -
Big O notation specifically describes worst case scenario.
It represents the upper bound running time complexity of an algorithm.
Mathematically -
Let f and g be functions of n - where n is natural no denoting size or steps of the algorithm then -
f(n) = O(g(n))
IFF
f(n) less than or = c.g(n)
where n greater than = n0, c greater than 0, n0 greater than = 1
Big Omega notation -
Big Omega notation specifically describes best case scenario.
It represents the lower bound running time complexity of an algorithm.
Basically it tells you what is the fastest time/behavior in which the algorithm can run.
f(n) = Ω(g(n))
IFF
f(n) greater than or = c.g(n)
where n greater than = n0, c greater than 0, n0 greater than = 1
Big Theta (θ) notation -
Big Omega notation specifically describes average case scenario.
It represents the most realistic time complexity of an algorithm.
f(n) = θ(g(n))
IFF
c1.g(n) less than or = f(n) less than or = c2.g(n)
where n greater than = n0, c1,c2 greater than 0, n greater than = n0, n0 greater = 1
Big Ω - Best Case
Big O - Worst Case
Big θ - Average Case
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How can we imagine the value of g(n) to be n,n2 like that

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7mins into the video, I understood Big Oh better. Well played.

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the most thoroughly and easily explained tutorial I have ever seen. Thank you a bunch!

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You are 3x better at explaining this than my college professor at ASU. You should be making the absurd tuition money she does

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I am also a student of discrete mathematics at ASU who is finally getting a clear explanation. Thank You!

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[@nikolajnguyen4273]

Whenever you choose a constant value c = ___ and a n0 value as n0 = ____, is it random that you choose the constant you chose? Is there a systematic way to do this, or would you just keep going with different n-values?

[@atlanta2203]

Thank you so much for this! Honestly saving my exams by explaining it so clearly I finally understand :')

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Thank you some much for this video!! Thanks to you in 30 min I understood perfectly what my professor didnt explain properly in 10 hours :))

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Glad it helped!

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I love it - my professor should learn from you

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Glad you like them!

[@ProBuilder-ck2vk]

I don't understand where these constant values are taken from. How do determine if c should be 1 or 2 or whatever? Is it just pick a random number, or are there some logic behind it?

[@punkgrl325]

You just drop the constants because what matters is the type of operation happening, not how many times it happens. So, O(2n), O(3n), O(4n), etc. can have their consants dropped to O(n), because they're all the same type of operation regardless (linear).

[@Albert-of4pg]

hi, just a little suggestion: it's better to say f(n) is O(g(n)) or f(n) belongs to O(g(n)) instead of saying f(n) = O(g(n))

[@jay-rathod-01]

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[@Insan123_]

your explanation is the best!!! Thank you a lot!

[@prempresents8776]

In big o notation what is c constant, like u took c as 5 in example so how we have take and what's it's role I'm not understanding 😶

[@alexandraherr9530]

This is a life saver man, thank you!!

[@Sunny-qe5el]

Quite exemplary and to the point. Thanks for your work.

[@konodioda1268]

Thanks, this will surely help me out in my midterm

[@gustavrisager8939]

Within the first 100 seconds this video explained Big-O better than my $200 textbook and my professor… combined.

[@SimpleSnippets]

Haha thank you for this feedback. Would be great if you can transfer that 200 dollars to me 🤣 Just kidding. Don't need donations. I'm happy that this video helped you 😊

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[@certifiedsmartass4122]

I have a doubt. If we need to find the closest fit to the best case time like you said, then shouldn't Big-Omega(n) have the constant as 2 instead of 1?? 2n < 2n+3 always Instead of 1n as 2n is a closer fit. Please tell me if I'm wrong with reason

[@sangodan3031]

With big omega you don't actually need to write what constant you use, whether it's 2n or 2000n it's still just O(n), you just have to find any constant to satisfy g(n) being bigger after n0 and you're set

[@certifiedsmartass4122]

@@sangodan3031 You're right mate. Thanks.

[@JV-jc7ci]

I can tell this is a good teaching video. Just wished the accent was a little less heavy for those who are not Indian. It's very difficult to understand and it's frustrating because I know how intelligent Indians are but can't seem to find videos where they don't speak with such heavy accents. Oh and I'm also Asian btw before anyone wants to pull the race card.

[@poisonedrice]

Thank you, excellent explanation

[@flipper71100]

I don't understand one thing in this equation 2n+3

[@ganashree8342]

omg d same doubt flashed to me as soon as he explained it.can someone please explain this.

[@Oscar-we5ke]

@@ganashree8342 Yeah, I understand what Big O and the others; for easy f(n), it is easy, but the problem for me comes when the f(n) is more complex. I have a lot of issues finding c1, c2, and n0.

[@moshibudimathabatha2611]

Well explained

[@availkrishmytube]

This covers theory quite well unlike other videos

[@mellonviskaino9537]

u r amazing. Thank u soooo much

[@geschichte4u251]

Thank you, men. Really helped me

[@pankajmhaske09]

Nice explanation, but cases(best, worst and average) and asymptotic notations are two independent terms, like best case of linear search also can be mentioned as O(1).

[@amrutachavan7686]

I like your ds lecture now I will complete your ds course thank you I am last year student

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@@SimpleSnippets you upload your videos on edyoda learning

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@@amrutachavan7686 yes I have uploaded some video on edyoda platform 😊

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Thank you Vaishnavi 😊

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Excellent explanation.

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[@deepakmaidasani1512]

As always amazing video and very nice explanation. Thank you so much!

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Am even using the tutorial to prepare for an exam this morning and is so helpful

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Bhaiya from where can i solve DSA questions...? Coz in geekforgeek , interviewbit they have only solution but not explaination (video explaination)

[@georgey4151]

THANK YOU VERY MUCH SIR

[@cybermember2939]

Sir is f(n) is different algorithm for same problem because u took differ equation for f(n) and g(n), is f(n) is like refrence and we r comparing with g(n) to find best best ,worst and average case?

[@Kucchuu]

Do not get confuse. Lets clear if f(n) = n^3+n^2+1 then g(n) is some derived portion of f(n) which is impacting your algorithm. Therefore here, g(n) can be n^3 i.e. g(n) = n^3 or g(n) = n^3+n or g(n)=n^3+5 etc. Both f(n) and g(n) belongs to same algorithm.

[@black_eye7105]

@@Kucchuu I had also the same problem but i can't understand where the g(n) comes from can you explane. you saying derived portion what is derived portion

[@tangent905]

thanks a lot for such a amazing explanation :)

[@cheems3202]

You are a life saver bro

[@abhinavl999]

superb explanation

[@Oscar-we5ke]

Could anyone help me with this one? I understand Big O and the others, but my problem is finding c1,c2, and n0 for complex functions. Example: n^3/1000 - 100n^2 - 100n + 3. I need to express that one in order of theta notation.

[@isfoo]

Equating O(.) notation with worst-case, Ω(.) notation with best-case and Θ(.) with average-case is incorrect. The O/Ω/Θ notations and "caseness" (worst/best/average) are independent concepts. It's a common misconception and I see nobody has pointed it out yet in the comments so I will explain why it's wrong. Let's start with that your mathematical definitions of the O/Ω/Θ notations are generally correct. Maybe would only highlight the fact that this notations are not exclusive to computer science or algorithms, but just describe the upper/lower/tight asymptotic bounds on the growth rate of a given function. Ok so the first minor inaccuracy is that when in 13:51 you have found out that f(n) is O(n) and f(n) is O(n^2) you've said that "when we try to find the O(.) notation we have to find the closest one which matches the f(n)". Well, no we don't have to. We have shown that indeed both O(n) and O(n^2) satisfy the mathematical definition and thus both are true. The reason we prefer the O(n) to O(n^2) is just because it gives as more information (it's a tighter bound). Now the big problem. At 24:10 you decided to analyse the time complexity of the linear search algorithm. So now it's true that it's Ω(1) and it's O(n), however it's NOT Θ(n). There is actually no function g(n) such that the time complexity is Θ(g(n)). That is because indeed Ω(1) is the tightest lower bound (for example it's not Ω(log(n))) and O(n) is the tightest upper bound (for example it's not O(log(n))). So you can see there is no g(n) which satisfies the condition c1*g(n) Θ(1) In the average case we have: Ω(n) and O(n) => Θ(n) // here we could say it's n/2, but we omit the constants So it's worst case Θ(n), best case Θ(1) and average case Θ(n). See that I used Θ(.) notation for each worst/best/average case. And the benefit of using Θ(.) for all cases is that it shows the tight bound. That is for example when we say it's worst case Θ(n) it means that it is not worst case Θ(1) and it is not worse case Θ(n^2). When we would use O(.) notation to describe worse case we can indeed say that it's O(n), but it's also true that it's O(n^2). So using Θ(.) gives us more information (it "forces" us to give the tight bound). This means that we should generally use Θ(.) notation as it gives us the most information. The problem however is that if we want to look at the general case of the algorithm the Θ(.) simply might not exist. So in that circumstance the best we can do is say that in general case this algorithm is O(n) and Ω(1). The only algorithms for which we can describe the general case complexity using Θ(.) notation are once for which worst case Θ(.) is the same as best case Θ(.). For example the problem of finding minimum value in n element array is worst case Θ(n), best case Θ(n) and average case Θ(n). So we can say that this algorithm has (in general) Θ(n) time complexity.

[@funnyshiittt]

It was amazing. Thank you.

[@ahmadxgame8885]

at 6:10 why did consider that c =5 but when n was powered by 2 we consider c =1 at 11:20 ?

[@The_Programming-Teacher]

Thank you very much. You are a hero!

[@muktagavli1106]

Very good lecture

[@DavidParathyras]

Your explanation is excelent!

[@MD-zw5nl]

Finally understood it. Thank you so much.

[@availkrishmytube]

Is there a reason why you chose 2n+3 for f(n)

[@fang8660]

Incredibly helpful video ~ thank you

[@freezinfire]

Thank you very much.

[@jamesstark4136]

Thank you! Note: small error on Big theta slide, description says "Big Omega"

[@som_girl6702]

You rock! Thank you for sharing your knowledge

[@santoshpalli2109]

Nice explanation Sir

[@diegoferreirarapaci6856]

6:10 in this example what if we consider c=2 ad n=2? We need to desconsider the number without an n quocient for it to work?

[@user-dl7ui4ii6r]

thanks a lot, you are the best 😍

[@nitismita1035]

I have a doubt, It's that can we get various pairs of c and n which satisfy the f(n)=o(g(n)). i. e for f(n)

[@liquidred257]

So when I saw the example I paused the video and got Big O=n Big Ω=1 Big θ= (n+1)/2 (because the average of n and 1 is n+1/2) I get why we get rid of the /2 for big θ, as it becomes negligible, so could the same be said of the +1?

[@codingwithanonymous890]

fantastic..pls upload more videos for clearing concept

[@cf0e6d7b83]

Thanks for explanation, nice video !

[@isaacwamala6836]

But please that means all notations just need testing through inputs in order to fulfill their conditions??

[@naweddiwan]

when f(n) = 2n + 3 Big Omega is Ω(n) Big Theta is θ(n) But for linear seach algorithm f(n) would also be like f(n) = a*n + b; where a and b are some constants Then why Big Omega is Ω(1) in this case?

[@bhargavnagacharan1899]

Best explanation ever ❤️❤️❤️

[@oliviazhai1831]

Thanks so much man

[@cinders-and-smoke]

Best video on notations 💪

[@sudakishorekumar]

Thanks man for making such an awesome content

[@giggleglyphs]

thanks for this video, even thanks for this playlist dude....:)

[@sunraiii]

Decent tutorial! Thank you

[@SimpleSnippets]

Glad it was helpful!

[@Spider-mf6be]

sir, in the end of the video, you give an example , O(1),O(n),O(n/2)or O(n)....we understand it , but sir when O(logn),O(nlog(n))...same thing happed in same process.... but any example for O(logn),O(nlog(n))?!🤔 tnq.... sir for this type of OSM!😍😍😍 video... as always osm explanation . hope you replay.❤

[@SimpleSnippets]

These time complexities can be seen in recursive kind of algorithms like mergesort :) Also thank you very much for the compliments. ✌

[@thehindu9972]

Hello for big omega Why dont we choose g(n) as 1 as 1 is always less than any f(n) so time complexity would be omega(1) for any program

[@siddharthkumaryadav575]

Maybe bcuz it is not closest to f(n). if linear g(n) = n is closer to f(n) then we will choose it instead of 1. could be wrong but i think it's the answer.

[@samarthyapatel2157]

Keep up with the good work, thanks.

[@samidelhi6150]

Hi simple , great explanation , would you kindly provide an example out of say ML algos where it is better to use say Big theta relative say to big O and big Omega ? Thanks

[@kodandaraochellapilli6212]

Do you have implementation for these concepts. Thank you for your help. It is very clear and simple. It is way better than my university teachings.

[@ruskindrag9649]

Why do we take the closest to f(n) fn as the best case and the worst case scenario ,it has to be the farthest one right?So that for the best case if you take Omega(1) that will be the fastest taking less time compared to Omega(n).

[@izharkhankhattak]

Excellent job, man!

[@albertd.bangura3794]

You are great!

[@Abinash0323]

Excellent video

[@SimpleSnippets]

Thank you very much!

[@dalisalvador9167]

Thanks bro

[@AnujKumar-ev4fm]

really good explanation! sir

[@SimpleSnippets]

Glad you liked it! Please support me by sharing the videos and our channel with your friends too. Thats the biggest help and support you can provide 😇

[@siddharthkumaryadav575]

At 21:32, we say that mathematically saying, Big O, Big Omega, Big Theta could be equal, so does that mean there exist a algo that has same fast case, worst case and realistic case? Anyways, your videos are amazing!!

[@user-zx2et9lf8y]

Selection sort !

[@ferhadmehdizade4772]

Thanks, it helped a lot👍

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great video!!!

[@RAKSHITHPGBBTCSBTechCSE]

why they is a curve in the f(n) line

[@omarrefaye3105]

Very well done

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Thank you very much!

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Amazing

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thnx for the help brother

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Thank you ily

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You're a god

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Thank yooou!