Lecture 19: Dynamic Programming I: Fibonacci, Shortest Paths

MIT 6.006 Introduction to Algorithms, Fall 2011
View the complete course: ocw.mit.edu/6-006F11
Instructor: Erik Demaine
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@JanacMeena]

The instructor, Erik Demaine, is the youngest professor of all time at MIT and a child prodigy. He has published several articles in scientifical journals, and and now has helped several students publish their own articles. He is one of the modern day geniuses of our time. We are extremely lucky to have a free video of his lecturing.

[@lukeTHEDUUKE]

he's no genius

[@alesc3252]

@@lukeTHEDUUKE he was a child prodigy when he was 7, he finished college at 14 and got his phd when he was 20. He has won plenty of awards and is the youngest teacher at mit for a reason. Most people think that if someone speaks a lot and you dont undestand him then he must be smart, but actually intelligent people, like mr. Erik demaine, can take complicated concepts and explain them in a way that a lot of people like ourselves can undestand them

[@sammyholdem2492]

So this guy is like dumb smart? A 🐐 of numbers? I'm a dummy and have no clue what is going on in this video but guys and girls that know this stuff it blows my mind. All I'm wondering is does this really figure things out or is this just made up by some super smart people that make these things up for fun

[@lx4302]

​ @sammy holdem it's not about being smart, people who have been doing one thing all their life can be extremely good at it.

[@manmeetworld]

@@sammyholdem2492pretty sure it's for faster run times i.e., you're in the program and it outputs your results faster. Or in some cases you may never get the result, like the work on the processor to store all the bits gets used up to the point of failure or low mem. Because I think, what's not explained real well right off the bat is that this program is constantly looping to get to the next fib.

[@sohntv8105]

Watching this 6 years after graduation. The irony of skipping class...

[@ShubhamSinghYoutube]

Never too late to start.

[@Stl71]

I was lucky enough to find out that there are a lot of videos in yt that could help me during my CS programm...And they really did help me.

[@yanfranca8382]

how about 11 years? kkk

[@AbhishekEkaanth]

I just googled him to know that he finished his bachelor's degree at 14 years of age at Dalhousie University in Canada. damnn!!!!

[@osmedia7239]

geez. Smart guy.

[@danielalemu3029]

When you have professors father at young age who inspires you and teach you, that helps a lot. His father is also professor at MIT.

[@osmedia7239]

@@danielalemu3029 true

[@winrx]

Daniel Alemu Having the right genes may also help I think......

[@ikrakkentm9588]

@@winrx no a lot of people are actually as intelligent and perhaps even more than him. Thing is not everyone had the spark by their enviroment from a young age to get interested in specialising in a specific area. Also in other countries there is no alternative to finish high school earlier and apply to university. Also SAT is easy i can imagine back then it was way easier. Im not saying he isnt smart. Hes definetely above average for sure.

[@santosht6644]

stop scrolling down listen to this guy

[@utkarshsrivastava]

Why not both??

[@dartme18]

/me scrolls back up

[@calvinsylveste8474]

I can hear him while scrolling down because i hear with my ears and not my eyes.

[@adott55]

Ok

[@isoccerpluse]

Why?

[@nbro5529]

Introduction to Dynamic Programming: 00:21 Fibonacci Numbers: 6:01 Fibonacci Numbers with Memoisation: 11:11 Fibonacci Numbers using bottom-up approach: min 23:25 Shortest paths: 31:05

[@aashudwivedi]

This is so helpful. Thanks a Ton :)

[@pikiwiki]

thank you

[@dcmarvel3031]

The world needs you

[@dscheme4427]

Nice breakdown -- much appreciated

[@CodersField]

life sever

[@thefreakingmindistaken]

I watched this guys couple of old lectures quite sometime back and he made me fall in love with dynamic programming. Love this guy.

[@sandeepmishra3972]

He is really putting an effort for students....these kind of teachers r very rare to find

[@markpurslow7446]

I agree totally

[@singhishandeep]

rare

[@ScapeGoat2099]

true

[@Tavorath]

Thank you for caring about the subtitles quality, they made it possible to follow up the whole class.

[@Jason_Kang]

I love this guy. I can tell he enjoys teaching.. best trait a lecturer can have :)

[@penips]

+Jason Kang nice

[@fruitjam6529]

+Jason Kang Indeed.

[@mriegger]

+Jason Kang I know, I was struggling with Dynamic Programming until I saw this series. He's awesome.

[@TheMasonX23]

Agreed! Erik is an amazing teacher, I love watching his lectures. He manages to make very difficult subjects relatively easy to understand and is very thorough and precise when defining things. And as you pointed out, his obvious love of teaching and the subject matter inspires an eagerness to learn.

[@adityakulai6419]

Yes true!

[@riverofcustard5027]

Not only is he knowledgeable but he’s also one of those teachers that makes you really get interested in the lesson

[@Goateduzi]

So apparent why MIT students are industry leaders.... their teachers are amazing. My data structures and algorithms professor was trash at my university, and it is a well respected school.

[@cmubill]

Which university you went to?

[@M4D4F4K4.]

@@cmubill prolly outside 100th rank unis

[@pythonprofreak7522]

I just want to thank you for this video and other related videos from this channel. While working fulltime and schooling at night it is hard sometimes to follow through my classes. However, I managed to earn a MS Computer Science concentration on Network and Security with a big thanks to these videos. Thank you!!

[@pman-codes]

Thank you guys. I am doing my masters though I couldn't get in MIT, you give me exposure to top class teaching materials.

[@kkk3252]

Professor Demaine, I really enjoy your confident lectures.

[@keyyyla]

Absolutely fascinating lecture. This guy received his master of science in mathematics at the age of 16.

[@NicolasPare]

Really? That's impressive.

[@NazriB]

Lies again? DMP DUO

[@brucesbanner5057]

Another post on this video says 14 yrs of age which one is it 🤔

[@Sarajboos12]

@@brucesbanner5057 he was bachelor

[@rschmidtschmidt6810]

@@brucesbanner5057 his Bachelors at 14 and masters at 16. It matches up.

[@aayushshah7193]

Best Algorithm lecture ever seen...Thank You MIT for this Lecture and Eric is one the best lecturer with great teaching skills.

[@CodeJeffo]

I love how Erik is so humble and approachable. It's so great that MIT recognized him so early as a faculty member. This is so important for students and progress in education and research. MIT is really showing the way for other universities how the modern education is meant to be. Erik stay cool and keep improving this wonderful series of lectures. All the best!

[@lapipesmoker3751]

"This is kind of obvious" Me: Leaving the lecture hall with my head down.

[@shirish3008]

Mannn, Sir Erik is so passionate professor I have ever seen. I just love the way he explains🙏🙏🙏

[@rmpx2gh532]

I love how much passion and energy the Prof got

[@sergeykholkhunov1888]

00:58 dynamic programming (DP) 05:06 Fibonacci numbers 11:04 memoized DP algorithm 23:14 bottom-up DP algorithm 31:00 shortest paths 41:14 example

[@fastacelzapacescu5445]

Thank you!

[@jose000]

Hh

[@sohambasu6375]

thanks brada

[@fckdahlloff]

I'm from Perú and following a Software Engineering, although we sometimes may get an overall view of these kind of topics, we tend to stay in the easy zone and never go any further from that. Having access to these many MIT real lectures just makes me so happy because, even if you don't enjoy the best education, you stil have the opportunity to learn more from reliable sources. And for me, that's what the MIT, give opportunities and hope. Thank you so much!

[@axelmori9284]

Another Peruvian here!

[@chayannesamaaa7867]

Now we are three. :)

[@patrickmcpartland1398]

Even at MIT this is a graduate level course, you're not taking this in a regular undergrad haha

[@kartiksangwan3302]

@@patrickmcpartland1398 What's the difference? 🤔

[@justinliburd8996]

@@kartiksangwan3302 How much debt your willing to go into.

[@user-lp4kg7iv5i]

Amazing lecture from Erik Demaine. It's impressive how easy he explains a topic like dynamic programming and how he can have the attention of the students in the hall. Thank you very much to Erik for his lecture and to MIT for sharing this excellent course!!! I can't wait to see the next DP classes!!!

[@siddharthdyavanapalli4617]

What a guy he is!!! Just to say been through a lot of mit stuff but he really is matching the skills of all other professors in that place

[@sujaa1000]

I am 55 years old and whenever I watch a lecture from MIT or Stanford, I so wish I could study there! I feel so mesmerized.

[@thinhnguyenvan7003]

Me too, and i am 29. Just graduated at 27 as a Mechatronic Engineerer and now do IT job

[@JanacMeena]

I've always thought of Dynamic Programming as this big, scary, complicated concept that would take forever to understand. Erik helped me understand it easily.

[@mearaftadewos8508]

Teachers tend to make the same impressions they had about a course on us when its their turn.

[@hana-ci6ss]

I've been watching many video professor teaching in KZbin....Professor Erik Demaine definitely the best one that I've ever saw...His teaching style really on top.

[@free-palestine000]

i love love love Erik's teaching style. usually, i feel pretty dumb when i don't get a concept when my professor or other youtube videos explain but Erik makes it very.....accessible, and relatable.

[@JayShankarpure]

Really a great video 🙌 Have fallen in love with recursion and DP with this video , Great work Erik Sir 👏

[@sixpooltube]

My favorite DP lecture yet!

[@carolinewilhelm7672]

Great lecture! Thank you for making this topic accessible - in all senses of the word

[@lx4302]

My goal: binge watch mit course playlist for the whole day :)

[@espritgaronne9975]

Great teaching! I had an algorithmic challenge this year, the solution was dynamic programming.

[@rakeshjoshi2306]

this guy is awesome . very clear explaination..

[@kmishy]

After following 2-3 lectures of him, He became my favorite prof

[@meditating010]

Eriks lectures are awesome... pretty decent walkthrough of dynamic programming.

[@ChrisLeeX]

@26:36: "Man, so many typos" - How I feel whenever I'm coding up on the board.

[@justinwmusic]

26:10 "So is this clear what this is doing? I think so." [Then corrects the 2nd, 3rd, 4th and 5th typos that prevented comprehension. :) ]

[@dackerman123]

This man is absolutely brilliant. Perhaps too brilliant for anyone in the lecture hall to understand, and certainly for me.

[@srinivastadinada3571]

Best tutorial on Dynamic programming I have ever seen :)

[@misteralagiz4003]

that's genius, Erik has got some great teaching skillz, man

[@TheMasonX23]

artem alagizov He's amazing! He's the youngest professor in MIT history, and earned his PhD when he only 20...

[@jn3750]

Former child prodigy!

[@MahdiZouch]

I really love how they teach, everything becomes interesting..., i wish i could study full time there :D

[@MahdiZouch]

ha ha same here

[@TheMasonX23]

Mahdi Zouch A large part of what makes them the best institution (in my opinion, at least one of the best though) is the quality of their instructors. And yes, I too would love to attend MIT if/when I make enough money, but at least until then they've got a lot of their courses available :)

[@deepakmeena3874]

yeah IIT bombay has great quality of students but quality of teaching method is very poor that's why they are so low in rankings

[@saipanda893]

Deepak Meena yeah bro.

[@hiddenblade999]

Best lecture on the subject i've seen on youtube

[@denisthamrin3449]

I gotta say, this video series helps me to understand how to solve DP problem easily. For people trying to learn how to solve DP , what I recommend to do is to just watch all the series to understand the pattern (observe) then try to solve some other problems on internet (practise).

[@katkong281]

If this guy was my professor I might have not dropped out! We need ppl like him thank you for sharing this

[@cluckhead1913]

Turn on, tune in, and drop out! There is so much more to life than sitting in a room listening to someone drone on about nothing. Be your own man!

[@coronaweeks4577]

@@cluckhead1913 maybe so but his passion is infectious…I wish I had a real passion for something instead of slogging through life

[@c.j.dylanxu153]

the best algorithm lecture i have ever seen! I prefer this "old-style" lecture using blackboard and chalks so much without any bullshit slides and whiteboards

[@veden310]

I hate white boards too )

[@sitalsitoula6536]

I love them...Easy to use

[@ashokbudha4777]

i hate slides

[@mdyousufali3793]

Slides hide the incompetency of a teacher.

[@nguyenhoanglong420]

yeah ! OLD-STYLE BLACK BOARD AND CHALKS BRING ALOT OF MEMORY TOO !

[@vijayalagappan6119]

Lovely course! Awesome Profs.. I came here looking for help in trying to solve DP problems.. got glued to this video and the other videos in the course.. heard of the Double Rainbow for the first time and googled for it.. came to know about the Double Rainbow guy.. and read that he became one with the rainbow ... in the times of this pandemic.. strange how times fly by! ...and how time flies by!

[@ScoobieSwisher7413]

I am amazed just at his writing on the board at the very beginning .

[@victorrice4549]

Very great lecture, super helpful. Thank you MIT.

[@arsh99119]

thearshblog.blogspot.com/2022/09/beating-youtube-algorithm.html how to beat KZbin Algorithm 👆

[@karannchew2534]

Love the giant eraser. Must be quite satisfiying using it.

[@haneulkim4902]

Amazing lecture, Erik can really explain. Thanks!

[@SrSKaTiZSrS]

I felt like this was a great way to explain dynamic programming. My course work on it in college was fine enough for me to understand it but this is a much easier better way to explain it to learn from.

[@socrat33z]

Thank you sooooo much! A lot easier than learning from the economic side.

[@prashantsingh1096]

I wish I could have Professor like him in my bachelor's and master's .

[@jainamkhakhra3898]

I absolutely love that chalk... Kind of gets you interested in the topic...

[@AleksandarIvanovicTV]

Absolutely love this guy, what a great lecturer

[@sandipanaich]

23:22, hey... even I like the recursion based top-down approach over bottom up. Btw, very good teacher.

[@tv..6531]

# for def fibonacci(n): r = [1, 0] for i in range(1, n+1): a = r[0] b = r[1] c = a + b r[0] = b r[1] = c return r[1] if __name__=="__main__": for n in range(1, 101): print(n, ": ", fibonacci(n))

[@mpataki]

at 47:06 Eric mentions, we need to make the graph acyclic, we don't need to make the graph acyclic, we can mark the vertex which is being computed (which is in the call stack) as present in an infinite distance and use it's TAB[v]

[@platzhersch]

geniously simply explained! thank you! helped me a lot!

[@anniekelly3698]

way better than my current algorithms professor...thanks!

[@adityasinghaswal4923]

Unsurprised

[@ji3g4j6jo3p]

I wish my prof taught my algo class was this excited about any of the algos, instead of sitting there hating life

[@KimMorgan-hc3rk]

Don't know much about Dynamic Programming but do note the professor's unique and rather attractive chalkboard writing! And being a dapper hand myself at it, I salute you Prof!

[@martinp.2537]

What a great person. He is making a very complicated topic at least a bit understandable for me :)

[@user-sd7hh8ek1c]

Dynamic programming the my favrourite thing in the world... in algorithms.

[@eddiesneeh4266]

This instructor ROCKS!

[@dr.strangelove9815]

What an excellent explanation and professor. Very well done!

[@bman2549]

I am amazed there are actually ppl who understand and can make sense of this.

[@thecodingjournal4726]

When I came to know about the name story of DP, I was also very excited. If anyone has done the course algorithmic toolbox, you might also be knowing this story!

[@muzamilzaman7463]

Algorithm toolbox ❤

[@mindsauce3]

You kept on scrolling anyway, didn't you? ;)

[@drayc290]

Bruh moment

[@Webexplr]

Bruh

[@JesusGomez-uo5ei]

yes :(

[@ceragoguztuzun8528]

Duuude

[@Chris_Lan]

rekt lmao

[@user-jc4lt1vk2g]

0:00~31:00 DP 설명. - 벨만 포드 알고리즘 만든 벨만이 다이나믹 프로그래밍이라는 이름을 창시했다. 벨만이 다이나믹 프로그래밍이라고 이름 지은 이유는 별 뜻 없고 '그냥 멋있어 보여서'다. - 하나의 문제를 완전탐색으로 재귀적으로 풀 때 그 문제의 부분문제(Subproblems)의 답을 재활용하는 기법이 메모아이제이션(Memoization)이다. 재귀로 문제를 풀면 exponential 시간이 소요되는데, 한 번 풀었던 문제의 답을 메모로 적어 놓고 다시 필요할 때 반환함으로써 그 문제 풀이는 O(1) 시간밖에 걸리지 않는다. - DP에는 재귀적인 구현인 하향식 접근(Top-down approach) 외에도 반복문을 이용한 상향식 접근(Bottom-up approach) 방법이 있다. 상향식은 함수 호출을 적게 하니까 스택을 적게 먹는 장점이 있겠지. - 시간복잡도: 부분 문제의 갯수 * 부분 문제를 푸는 데에 걸리는 시간(메모 해 놔서 재귀 안 걸리는 데는 O(1)임) ---------- 31:00 이후 최단 경로 설명. - 오랫동안 잊고 있었던 그래프 이론이 등장함. / - 정점(vertex, node), 간선(edge, link, line) - 차수(degree): 어떤 정점의 간선의 수. / - DAG (Directed Acyclic Graph): 비순환 유향(방향) 그래프 - 내차수(indegree): 방향 그래프에서 어떤 정점으로 들어오는 간선(incoming edge)의 수. - 외차수(outdegree): 방향 그래프에서 어떤 정점에서 나가는 간선(outgoing edge)의 수. - 위상 정렬(Topological sorting): 선수과목 구조처럼 보이게 DAG을 왼쪽에서 오른쪽으로 한 방향으로 죽 늘어 놓는 것. - 출발점에서 나가는 간선을 기준으로 부분문제를 만들 수도 있고 목적지로 들어오는 간선을 기준으로 부분문제를 만들 수도 있음 - 사이클이 있는 그래프에서 시간복잡도는 무한대임. - DAG의 경우 시간복잡도: O(V+E) - DP 쓰려면 부분 문제는 비순환이어야 한다.

[@GeoRevilo]

the camera work is really good on this

[@NitRoGenSmile]

shortest paths 31:00

[@ludvikjerabek]

This guy is a fantastic teacher with great humor.

[@dathyr1]

I used to know all this in my past life. This is way beyond my comprehension. But a great teacher.

[@lloydangelo2315]

I love how he much he could do with his mind. I'm a theoretical physics major and computer science

[@rizanamatya9191]

Algorithms Professor wearing a Bob Dylan T-shirt! Awesome!

[@christoskettenis880]

I love those boards and chalk!

[@marvinlessknown3702]

Good job camera person. You're following the material and zooming in when relevant!

[@thiagoaugustomartins3482]

Great class ! Thanks for uploading that.

[@DeivitMV]

Extra credit for the Bob Dylan T-shirt! :D

[@scemat]

Warmed my heart when I noticed it!

[@astralchemistry8732]

34:49 The moment you stumble upon life-advice in part 19 of the programming lecture you've been watching.

[@AaAaAaA-mm7cc]

Just leaving this here for someone to remind me in a couple of years, I regretted majoring in information systems instead of CS, and I cannot change my degree, but I will work hard and learn most of the needed stuff online in order to be able to learn more cs stuff on top of finishing my degree and getting a nice job that i enjoy in the future, through hard work and determination!

[@paulthomann5544]

If we assume that the fibonacci calculator function will be called many times for different numbers in essentially random order, the 'top-down' memoized version has an advantage: When called repeatedly with the same n, it takes linear time the first time, but constant time afterwards (if the dictionary persists between calls). In the bottom-up version, changes are required to achieve the same: if n

[@mariotaz]

Great lecture!

[@RobertLugg]

Good talk. My sharpest criticism: I’ll bet he goes though a lot of chalk pressing as hard as he does.

[@johnnyisleet]

This refers to Python dictionaries which are amortized O(1) lookup time. As are most programming languages implementation of a key value store ie. Map, Hashtable, Associative Arrays, etc.

[@tonyjames1980]

Wonderful lecture, Thanks for explaining the term.

[@zeronothinghere9334]

Dynamic Programming AKA "Careful Brute Force" XDD I lost it. Sooo accurate

[@elias8294]

I love that he writes all of his code in python

[@adityamhatre4684]

Because Python Code is Closest to Pseudo Codes

[@MuhFerdiansyahscience]

General it 's very awesome for explanation about dynamic programming. I hope it very much very better in the for you Professor, I think videos it after old, but i hope you keep smart about explanation about physics, example dynamic programming

[@weignerg]

27:00 Bottom up can be optimized further by hard coding the first two results manually the itterate without a conditional and saving the result right away. When done looping, return the last result.

[@weignerg]

int n = 100; Map fib = new HashMap(); int i; for(i= -1; i

[@weignerg]

30:00 import java.math.BigInteger; public class FibonacciExample{ public static void main(String args[]){ int fibSequenceNum = 107; int n = fibSequenceNum/2; BigInteger[] fib = {BigInteger.valueOf(0),BigInteger.valueOf(1)}; for(int i = 0; i < n; i++) { fib[0] = fib[0].add(fib[1]); fib[1] = fib[0].add(fib[1]); } System.out.println(fib[fibSequenceNum%2]); }}

[@eddychoo7820]

@@weignerg 8ulk777uuu6

[@Paco1337]

I'm looking at this now,same as I was looking at math class when I was a kid

[@JackLe1127]

This video makes me wanna apply for MIT

[@darogajee3286]

just collect enough millions..

[@DarkLordAli95]

and very exceptional grades.

[@BrajeshKumar-ez8zs]

Jack Le you can... But you will be kicked out

[@mlst3rg]

just write #blacklivesmatter on your application

[@lambda494]

I went there (Course 8, 2012). Trust me, getting in is the easy part. All the top students at their high schools become more or less average there. You learn a ton but it will wring the life out of you too if you aren't careful. You may spare yourself and just enjoy OCW!

[@shenbagachella1041]

I'm in love with this guy.. btw nice lecture..

[@jaykay7932]

recursive functions alone is a concept which takes some adjusting too. In over 14 years of writing code in manufacturing i've yet to find a place for them. I get its for a situation where you'd want the end result without printing the path to the result on screen (ie a loop) but ive never really needed it in my line. I can imagine it with 3d graphics though

[@deanasplund3444]

Didn't get the part after '"Let's start a brand new thing called dynamic programming, yah......"

[@geekoist]

The last remark that the algorithm can be modified to compute weights of shortest paths in graphs (possibly with cycles) was rushed a little bit. Can someone explain it in better terms? Thanks :)

[@jakethewoz]

Love this guy's handwriting.

[@brooklyna007]

This teacher is amazing!! I loved it!