I was really struggling with some very dry literature on amortised complexity for my data science degree and this has helped a huge amount. Great work :)
I sincerely appreciate this content! I am taking an online class that is pure textbook reading and it's murderous. This video seriously saved me from hours of pain and suffering.
i had been trying to find good material for this topic and reading for different sources but couldn’t understand anything and then I came across your video and now have no doubts. thanks bro
The best explanation of these 3 methods on the Internet! I watched a Corsera video but I did not get the point. Your video is more intuitive and clear! Thank you very much!
Your example is surprisingly similar to the question I got in my assignment, just worded differently, which helped me to look at it from dynamic array perspective. Thank you a lot!
Wholesome and amazing explanation, thank you ! I finally understand the bank account method which my professor and the MIT 6.046 video failed to make me understand. keep up with the amazing content !
Because of Corona my Algorithm & Data-Structure lecture was severely downgraded, and a lot of subjects were not covered... Thanks to you, i can still get some of that knowledge.
Wow. Wow wow wow wow wow wow wow wow wow wow...... I am speechless. Just speechless. You are a genius. Best video on this topic. I want to give this video 1 million likes.
Just subscribed because of this video...it's 100x better than my professor's method of explaining this concept. However, could you please explain how amortized analysis works for deleting elements in a dynamic array?
Hello and thank you for your comment! I'm afraid it's not something I've given much thought to so I don't have an answer for you. However, I found something online that may be of use: www.cs.cmu.edu/afs/cs/academic/class/15451-s15/LectureNotes/lecture06/growing-shrinking-table.txt
This helped a lot. Thanks! Initially I didn't want to click on this video because the thumbnail looked bad though. So if you made it better I think it would get more views.
Hello and thank you very much for your comment! I don't usually give a lot of thought to thumbnails, so this is valuable feedback - Could you elaborate more on what was off-putting about it? I'll see what I can do to make it better!
i find it amazing when you can explain to me better compared to my lecturer in a 2 hour lecture. and heck most of the example came from your video 🤣 #keepUpTheGoodWork!
Hello and thank you very much for your comment! The example given is the most classic example, that's why you'll see it common to a lot of explanations on this topic!
You're welcome! Very happy to be of help, and thank you for the feedback! Pacing is something I'm always very unsure about, so I'm glad to know ti worked well for you!
Thank you! Question about accounting method - how do you decide how much to charge? I'm sure you're not randomly guessing or pulling numbers out of thin air. Is there a method to deciding costs?
Hello and thank you for your comment! The way I see it, some amount of trial and error is unavoidable. A good gauge would be to consider how the expensive operations can be "divided into" a number of "cheaper" operations. Set up some numbers from there and try. If the number keeps increasing, consider assigning less to each operation. If the value dips below zero, then the cost must be increased. For deriving potential functions, it may be advantageous to first consider the aggregate method, which allows you to consider step-by-step costs which you can then "generalize".
It was a really great video and has given a great understanding for amortized analysis.... I just had one doubt, that why do we have 1 in (1+i) because 'i' is used for moving 'i' elements but likewise we are also inserting 'i' elements right?.. So shouldn't it be (i+i) ?
Hello and thank you for your comment! Don't forget, (1+i) is discussed at a time when we're talking about what's going on *step by step* , in other words, on a *per-operation basis* . At any point of time, we are inserting 1 item only. However, there are some cases in which inserting that 1 item causes a chain reaction in which _i_ additional things happen. In that case, the amount of work done *for that one step* is (1+i).
Hello and thank you for your comment! Unfortunately there is no "one rule" or method for doing that. You'll have to just get your head into the right space and reason it out. There's a good writeup here to start building that intuition: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
Thank you for the amazing explanation. Just a doubt In Aggregate analysis isn't the cost of an expensive operation is 4 when 3 items instead of 4 items. This is due to the new array creation point is when we try to insert 3 item to the dynamic array. Correct me if I am wrong.
Hello and thank you for your comment! It depends on when you want to expand the array. In the example shown, we expand the array when it becomes full as the result of an insertion operation. You can of course choose to do this earlier.
Hello, great video! Just one thing. At minute 7:16, why did we inflate the numbers and forced an expansion? Could have we just done it with out the inflation? Thanks.
Hello and thank you for your comment! The inflation starts from 6:30 actually. The reason why we do that is to create a series of additions 1+2+4+8+... instead of having it start as 4. This helps us see the pattern more clearly later on, where the total cost is almost 2x the number of operations (7:23).
@@NERDfirst yes sorry I put the time wrong. Okay so this just for demonstration purposes? If we leave it as it is, can we prove that its the double without the inflation? Thanks
Hello again! Personally I feel it is more convincing for a proof to start with an array of size one (ie. The "inflated" version as I have shown). Otherwise, you'll have to explain your arbitrary choice of starting array size, and you could be questioned about whether your proof holds true for smaller array sizes. The clearest proof shows that the _worst case time_ (ie. Even if we start with an array of size one), still gives us under 2x number of operations.
Hello and thank you for your comment! I don't think you'll really ever need to blindly guess in the dark. If it helps, start with the aggregate method first to see a pattern. Of course, there's no single method that will always give you a perfect answer from the get go, so some experience and pattern recognition skills are required, but there are some frameworks for reasoning about this. Here's a pretty good writeup: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
fantastic video: just a tiny comment. is the cost correct in the table with the aggregate stuff. I would expect the cost to be: 1,2,3,1,5,1,1,1,9,1,1,1,1,1,1,17……because we copy when we want to insert to a full table that has no more space, not when inserting to the last slot in the array.
Hello and thank you for your comment! An interesting thought, but it works out all the same, because the work kicks in at comparable points in time - The total amount of work done is exactly equal. I would say both strategies are equivalent, though round powers of two are nicer to look at =)
could you explain for me what is the mean of cost you use in aggregate method ? is it related to big o notation ? Again, many thanks for your videos.i really appreciate it
Hello and thank you for your comment! Cost "leads" us towards coming up with a conclusion for Big O time complexity, though when we're discussing "cost", we're not quite there yet. Think of cost as a count of the number of steps, as in 1 unit of cost is equivalent to one step.
With the accountant's method... I don't get why you assigned $3 per operation.... I mean what's to stop me from assigning $10, or $20 per operation, or even infinity and then I could use that same argument to argue that any operation amortized costs O(1) only! Where did I get these $3 from? I can't seem to grasp that... Can anybody explain that to me please?
Hello and thank you for your comment! The costs assigned are arbitrary, but there is a strategy to picking them. The general idea is that we want to _count_ the number of fundamental units of operation (which we usually assign as $1), and see how many of them take place on average "per step" over a data structure of size n. Of course you could give it $10 or $20, but: 1) In general, we are talking about worst case time complexity, so we want a tight bound. It's the same way saying bubble sort has a worst case time complexity of O(2^n) isn't _technically_ wrong (Big O notations only specify an upper bound), but saying this becomes meaningless because _tight_ upper bounds are more useful. 2) If the algorithm can't be amortized to O(1), using a large constant cost won't help you - You will get to a point where the $10 or even $100 or $1000 you assign won't be enough. Of course, infinity is another question altogether. If you use infinity the whole exercise becomes pointless because you'll never run out, so you can't do that.
Hello and thank you for your comment! Intuitively I'd say _creating_ the array is O(1) since that just involves setting aside a contiguous block of RAM (assuming availability). It's the _shifting_ of items from the old array into the new that takes O(n).
Hello and thank you for your comment! To greatly oversimplify, the Big O Notation just indicates the fastest growing component. So usually when we see c*n + a (where c and a are constant numbers), we'll just represent it as O(n). For more on this topic, you may want to check out my playlist on asymptotic notations: th-cam.com/play/PLJse9iV6Reqh5B_w9koGyT7nlYm92iITk.html
Hello and thank you for your comment! This is generally a mix of my own experience plus formal university education, so no exact sources here. If you're looking for more reference material, you can refer to the Reference section of the Amortized Analysis Wikipedia page: en.wikipedia.org/wiki/Amortized_Analysis
Hi, Thanks ever so much! this is really useful.. I'm actually interested in the potential function method and need more info about the way or in which sense one can choose the potential function for certain problem. So, could you please give me a hint or any reference for it? Appreciate your help!
Hello and thank you for your comment! Unfortunately I'm no good at the specifics of it either! Here's a stack exchange answer that can get you started on the basic intuition: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
More helpful than a 3 hour lecture. Thank you so much! Subscribed.
Hello and thank you very much for your comment and support! Very happy to be of help :)
You just managed to explain in 20 minutes, what my professor could not in 2 hours. Best video I've seen on the topic.
Hello and thank you very much for your comment! Glad to be of help =)
I was really struggling with some very dry literature on amortised complexity for my data science degree and this has helped a huge amount. Great work :)
Hello and thank you very much for your comment! Very happy to be of help =)
I sincerely appreciate this content! I am taking an online class that is pure textbook reading and it's murderous. This video seriously saved me from hours of pain and suffering.
Hello and thank you very much for your comment and support! Very happy to be of help, all the best for your class =)
This was such an excellent video. Thank you
You're welcome! Very happy to be of help =)
i had been trying to find good material for this topic and reading for different sources but couldn’t understand anything and then I came across your video and now have no doubts. thanks bro
Hello and thank you very much for your comment! Very happy to be of help =)
Dude, thanks a lot !
Your explanations are a zillion time more clearer than the best University professor :)
+Daniel Madar Hello and thank you very much for your comment! That's great to hear, that means I've achieved what I've set out to do :)
OMG this' the best amortized cost analysis lecture I've ever seen
Hello and thank you very much for your comment! Glad you liked the video!
The best explanation of these 3 methods on the Internet! I watched a Corsera video but I did not get the point. Your video is more intuitive and clear! Thank you very much!
You're welcome! Very happy to be of help =)
I'm studying for finals, and this was SUPER helpful. Thanks!
You're welcome! Glad to be of help and all the best for your finals =)
Im so grateful for this video! Thanks for explaining that topic so well. It was kompact, but every important detail was mentioned. Chapeau!
You're welcome! Very happy to be of help and I'm glad you found it useful =)
Cool explanation on the intuition of amortized analysis :D the rough idea does helps a lot in the understanding of the topic. Thank you very much
You're welcome! Very happy to be of help :) What remains is to apply!
Best explained video ,I searched for shorter videos on this concept but they didn't explain in easy way .I wish I had watch this before❤
Hello and thank you for your comment! Very happy to be of help =)
Best video on youtube that explains amortization analysis so far.
Hello and thank you very much for your comment! Glad you liked the video =)
Your example is surprisingly similar to the question I got in my assignment, just worded differently, which helped me to look at it from dynamic array perspective. Thank you a lot!
Hello and thank you for your comment! Array expansion is a classic example for amortized analysis, so I'm not surprised! Glad to be of help =)
You look so young, and yet you are better at teaching than most professors. THANKS!!!!!!!!
Hello and thank you very much for your comment! Really appreciate the kind words! Very happy to be of help =)
Wholesome and amazing explanation, thank you ! I finally understand the bank account method which my professor and the MIT 6.046 video failed to make me understand. keep up with the amazing content !
Hello and thank you very much for your comment! Very happy to be of help =)
Because of Corona my Algorithm & Data-Structure lecture was severely downgraded, and a lot of subjects were not covered... Thanks to you, i can still get some of that knowledge.
Hello and thank you very much for your comment! Glad to be able to fill in that role for you =)
I genuinely cannot thank you enough for this! Thanks a lot man, thanks a lot :)
You're welcome! Very happy to be of help =)
You're better than my professor !!! thank you so much
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This was the far best video on Amortized Analysis. Thanks a ton❤️
You're welcome! Very happy to be of help =)
Very clear and concise. Thank you!
You're welcome! Very happy to be of help =)
Awesome video on amortized analysis! Thank you!
Welcome! Very happy to be of help =)
Just amazing. What a great explanation! You just earned a subscriber.
Hello and thank you for your comment and support! Glad you liked the video =)
This saved me a lot of time. Appreciated!
Hello and thank you for your comment! Happy to be of help =)
thank you! You teach wayyyyy better than my professors do!
You're welcome! Very happy to be of help =)
@@NERDfirst you are so sweet!!!!
That was very clear. Thank you.
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Nicely presented! Very clear and well thought-out.
Hello and thank you very much for your comment! Glad you liked the video =)
Wow.
Wow wow wow wow wow wow wow wow wow wow......
I am speechless. Just speechless.
You are a genius. Best video on this topic.
I want to give this video 1 million likes.
Hello and thank you very much for your comment! Very happy to be of help =)
Great explanation! Thank you
You're welcome! Glad to be of help =)
Just subscribed because of this video...it's 100x better than my professor's method of explaining this concept. However, could you please explain how amortized analysis works for deleting elements in a dynamic array?
Hello and thank you for your comment! I'm afraid it's not something I've given much thought to so I don't have an answer for you.
However, I found something online that may be of use: www.cs.cmu.edu/afs/cs/academic/class/15451-s15/LectureNotes/lecture06/growing-shrinking-table.txt
Wow this is great. Thank you so much!
You're welcome! Very happy to be of help =)
Thank you so much! Extremely helpful video!
You're welcome! Very happy to be of help =)
Wow, very well explained. Please, keep them coming!
Hello and thank you for your comment! New videos every week =)
This helped a lot. Thanks! Initially I didn't want to click on this video because the thumbnail looked bad though. So if you made it better I think it would get more views.
Hello and thank you very much for your comment! I don't usually give a lot of thought to thumbnails, so this is valuable feedback - Could you elaborate more on what was off-putting about it? I'll see what I can do to make it better!
This is the best video about Amortized Analysis! Thank you
Hello and thank you very much for your comment! Very happy to be of help =)
i find it amazing when you can explain to me better compared to my lecturer in a 2 hour lecture. and heck most of the example came from your video 🤣 #keepUpTheGoodWork!
Hello and thank you very much for your comment! The example given is the most classic example, that's why you'll see it common to a lot of explanations on this topic!
Helped me a lot for my upcoming test! Thanks a lot!
You're welcome! All the best for your test =)
Thanks for your explanations about amortized analysis. It was helpful for me :)
You're welcome! Glad to be of help =)
Thank you for the video!
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This video is so awesome... save me a bunch of time to study
Hello and thank you very much for your comment! Glad you liked the video =)
This was really helpful, thank you
You're welcome! Very happy to be of help =)
thank you for making this video. the pace of the video was perfect.
You're welcome! Very happy to be of help, and thank you for the feedback! Pacing is something I'm always very unsure about, so I'm glad to know ti worked well for you!
Perfect explanation. Great Job
Hello and thank you very much for your comment! Glad you liked the video =)
Fantastic!!! Thank you so much
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Thank you so much. I finally know the difference between accounting and potential in real examples and know how to calculate them now (cry)
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Explained it so well. Kudos to you and subscribed:)
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great explanation! good job sir. very helpful
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Thank you so much. this is super helpful.
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Thank you for this. So helpful!
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I like your shadow on the wall
Shadow likes you too!
Amazing video, thank you
You're welcome! Happy to be of help =)
Thank you! Question about accounting method - how do you decide how much to charge? I'm sure you're not randomly guessing or pulling numbers out of thin air. Is there a method to deciding costs?
Same question regarding how you came up with the potential function
Hello and thank you for your comment! The way I see it, some amount of trial and error is unavoidable. A good gauge would be to consider how the expensive operations can be "divided into" a number of "cheaper" operations. Set up some numbers from there and try. If the number keeps increasing, consider assigning less to each operation. If the value dips below zero, then the cost must be increased.
For deriving potential functions, it may be advantageous to first consider the aggregate method, which allows you to consider step-by-step costs which you can then "generalize".
Sir,you explanation is tooooooo good🙂🙂🙂
Hello and thank you for your comment! Glad you liked the video =)
Why anyone would give this a thumbs-down is beyond me.
Hello and thank you for your comment! I wouldn't worry about that =) What's important is that people have found this useful!
It was a really great video and has given a great understanding for amortized analysis.... I just had one doubt, that why do we have 1 in (1+i) because 'i' is used for moving 'i' elements but likewise we are also inserting 'i' elements right?.. So shouldn't it be (i+i) ?
Hello and thank you for your comment! Don't forget, (1+i) is discussed at a time when we're talking about what's going on *step by step* , in other words, on a *per-operation basis* .
At any point of time, we are inserting 1 item only. However, there are some cases in which inserting that 1 item causes a chain reaction in which _i_ additional things happen. In that case, the amount of work done *for that one step* is (1+i).
Great video! May I ask you what is a good way to come out with the potential function?
Hello and thank you for your comment! Unfortunately there is no "one rule" or method for doing that. You'll have to just get your head into the right space and reason it out. There's a good writeup here to start building that intuition: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
Nicely covered
Hello and thank you for your comment! Glad you liked the video =)
Great explanation.. thanks bro
You're welcome! Happy to be of help =)
finally got that stuff :D my teacher failed explaining it for over a week now xD thx a lot m8!
You're welcome! Very happy to be of help =)
Thank You buddy you explained it very well!!
You're welcome! Very happy to be of help =)
Very helpful video. Thanks :)
You're welcome! Very happy to be of help =)
Great content!!
Thank you very much! Glad you liked the video =)
Thank you for the amazing explanation. Just a doubt In Aggregate analysis isn't the cost of an expensive operation is 4 when 3 items instead of 4 items. This is due to the new array creation point is when we try to insert 3 item to the dynamic array. Correct me if I am wrong.
Hello and thank you for your comment! It depends on when you want to expand the array. In the example shown, we expand the array when it becomes full as the result of an insertion operation. You can of course choose to do this earlier.
@@NERDfirst Thank you.
Hello, great video!
Just one thing. At minute 7:16, why did we inflate the numbers and forced an expansion? Could have we just done it with out the inflation?
Thanks.
Hello and thank you for your comment! The inflation starts from 6:30 actually. The reason why we do that is to create a series of additions 1+2+4+8+... instead of having it start as 4. This helps us see the pattern more clearly later on, where the total cost is almost 2x the number of operations (7:23).
@@NERDfirst yes sorry I put the time wrong. Okay so this just for demonstration purposes? If we leave it as it is, can we prove that its the double without the inflation? Thanks
Hello again! Personally I feel it is more convincing for a proof to start with an array of size one (ie. The "inflated" version as I have shown). Otherwise, you'll have to explain your arbitrary choice of starting array size, and you could be questioned about whether your proof holds true for smaller array sizes. The clearest proof shows that the _worst case time_ (ie. Even if we start with an array of size one), still gives us under 2x number of operations.
Quick n neat, do you have to guess the cost for accounting and function for the potential method?
Hello and thank you for your comment! I don't think you'll really ever need to blindly guess in the dark. If it helps, start with the aggregate method first to see a pattern.
Of course, there's no single method that will always give you a perfect answer from the get go, so some experience and pattern recognition skills are required, but there are some frameworks for reasoning about this. Here's a pretty good writeup: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
@@NERDfirst thank thee :)
omg this deserves a subscribe thanks bro
You're welcome! Glad to be of help =)
Bro you are a legend. 😭
Hello and thank you for your comment! Well, I'm glad you liked the video =)
awesome video man ...very well explained each and every step .
Hello and thank you very much for your comment! Glad you liked the video!
The explanation was good...It really helped me
That's great to hear! Very happy to be of help :)
Thank you brother
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thanks, it was very helpful :)
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Wow better than teachers there in university. I wonder about people who disliked 🤔🤔
Hello and thank you very much for your comment! I wouldn't worry about dislikes, what's important is that the people who need my help get it :)
sir, thank you very much. this is a great explanation and great intuition content
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Thank you sir.
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Very clear! Recoommend
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Great content. Thanks a lot. It helped me big time
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fantastic video: just a tiny comment. is the cost correct in the table with the aggregate stuff. I would expect the cost to be: 1,2,3,1,5,1,1,1,9,1,1,1,1,1,1,17……because we copy when we want to insert to a full table that has no more space, not when inserting to the last slot in the array.
Hello and thank you for your comment! An interesting thought, but it works out all the same, because the work kicks in at comparable points in time - The total amount of work done is exactly equal. I would say both strategies are equivalent, though round powers of two are nicer to look at =)
Simply Superb !
Thank you very much! Glad to be of help :)
Great video
Hello and thank you for your comment! Glad to be of help =)
great video!
Hello and thank you for your comment! Glad you liked the video =)
good work
Hello and thank you for your comment! Glad to be of help =)
could you explain for me what is the mean of cost you use in aggregate method ? is it related to big o notation ?
Again, many thanks for your videos.i really appreciate it
Hello and thank you for your comment! Cost "leads" us towards coming up with a conclusion for Big O time complexity, though when we're discussing "cost", we're not quite there yet. Think of cost as a count of the number of steps, as in 1 unit of cost is equivalent to one step.
With the accountant's method... I don't get why you assigned $3 per operation.... I mean what's to stop me from assigning $10, or $20 per operation, or even infinity and then I could use that same argument to argue that any operation amortized costs O(1) only!
Where did I get these $3 from? I can't seem to grasp that... Can anybody explain that to me please?
Hello and thank you for your comment! The costs assigned are arbitrary, but there is a strategy to picking them. The general idea is that we want to _count_ the number of fundamental units of operation (which we usually assign as $1), and see how many of them take place on average "per step" over a data structure of size n.
Of course you could give it $10 or $20, but:
1) In general, we are talking about worst case time complexity, so we want a tight bound. It's the same way saying bubble sort has a worst case time complexity of O(2^n) isn't _technically_ wrong (Big O notations only specify an upper bound), but saying this becomes meaningless because _tight_ upper bounds are more useful.
2) If the algorithm can't be amortized to O(1), using a large constant cost won't help you - You will get to a point where the $10 or even $100 or $1000 you assign won't be enough. Of course, infinity is another question altogether. If you use infinity the whole exercise becomes pointless because you'll never run out, so you can't do that.
thanks man, very good explanation
You're welcome! Glad to be of help =)
During the accounting method, what is the cost for creating the new, double in size, array?
Hello and thank you for your comment! Intuitively I'd say _creating_ the array is O(1) since that just involves setting aside a contiguous block of RAM (assuming availability). It's the _shifting_ of items from the old array into the new that takes O(n).
Great explanation! You are made to teach
Hello and thank you for your comment! Glad you found my work useful =)
Nice ! Why the name 0612 ? Is there any conversions or some cipher text ?
Hello and thank you for your comment! Not at all, that just comes from my birthday.
Awesome video, wish my boring professor just showed us your video in class instead of wasting our time for 1.5hrs lol
Hello and thank you for your comment! Heh, it's not an easy topic to deliver on the go, so I can understand. Either way, glad to be of help =)
GOOD JOB!
Thank you very much! Glad you liked the video =)
Great Video! :)
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good stuff thanks
You're welcome! Happy to be of help =)
Even if the number of cheap operations are NEARLY (and not equal to n) n times, is it correct to say it's total cost is of order O(n)?
Hello and thank you for your comment! To greatly oversimplify, the Big O Notation just indicates the fastest growing component. So usually when we see c*n + a (where c and a are constant numbers), we'll just represent it as O(n).
For more on this topic, you may want to check out my playlist on asymptotic notations: th-cam.com/play/PLJse9iV6Reqh5B_w9koGyT7nlYm92iITk.html
can you please share any resource you learned this from?
Hello and thank you for your comment! This is generally a mix of my own experience plus formal university education, so no exact sources here. If you're looking for more reference material, you can refer to the Reference section of the Amortized Analysis Wikipedia page: en.wikipedia.org/wiki/Amortized_Analysis
@@NERDfirst thank you!!
Great video ever!
Hello and thank you for your comment! Glad you liked the video =)
Hey man. Thank you. It helped a lot.
You're welcome! Very happy to be of help =)
Perfect, thanks
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thanks very clear
You're welcome! Glad to be of help :)
Hi,
Thanks ever so much! this is really useful.. I'm actually interested in the potential function method and need more info about the way or in which sense one can choose the potential function for certain problem. So, could you please give me a hint or any reference for it?
Appreciate your help!
Hello and thank you for your comment! Unfortunately I'm no good at the specifics of it either!
Here's a stack exchange answer that can get you started on the basic intuition: cs.stackexchange.com/questions/30543/what-is-the-intuition-behind-the-potential-function-in-amortized-analysis-of-som
It is cool bro. Thank you
You're welcome! Glad to be of help =)