But in particular, this is at least the nth Fibonacci number. So I guess we have to think about that a little bit. Abstract. LECTURE 5: DYNAMIC PROGRAMMING 1. Definitely better. T of n represents the time to compute the nth Fibonacci number. And so in this sense dynamic programming is essentially recursion plus memoization. Lecture Slides. In general, the bottom-up does exactly the same computation as the memoized version. So that's a recurrence. Dynamic Programming Problems Dynamic Programming Steps to solve a DP problem 1 De ne subproblems 2 … Basically, it sounded cool. PROFESSOR: It's a tried and tested method for solving any problem. You want to minimize, maximize something, that's an optimization problem, and typically good algorithms to solve them involve dynamic programming. So I take the minimum over all edges of the shortest path from s to u, plus the weight of the edge uv. Use OCW to guide your own life-long learning, or to teach others. But they're both constant time with good hashing. Description: This lecture introduces dynamic programming, in which careful exhaustive search can be used to design polynomial-time algorithms. So that's all general. So the idea is, every time I follow an edge I go down to the next layer. OK. We're just going to get to linear today, which is a lot better than exponential. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. In dynamic programming, we solve many subproblems and store the results: not all of them will contribute to solving the larger problem. Find materials for this course in the pages linked along the left. It's like the only cool thing you can do with shortest paths, I feel like. If you want to make a shortest path problem harder, require that you reduce your graph to k copies of the graph. I hear whispers. It's just like the memoized code over there. We do constant number of additions, comparisons. And it's so important I'm going to write it down again in a slightly more general framework. This is actually not the best algorithm-- as an aside. But I claim I can use this same approach to solve shortest paths in general graphs, even when they have cycles. Try all the guesses. Epidemic example. Man, I really want a cushion. I will have always computed these things already. These two lines are identical to these two lines. What this is really saying is, you should sum up over all sub problems of the time per sub problem. adp_slides_tsinghua_course_1_version_1.pdf: File Size: 134 kb: File Type: pdf But in some sense recurrences aren't quite the right way of thinking about this because recursion is kind of a rare thing. 3: Growth of Functions CLRS Ch. In fact, this already happens with fn minus 2. It's not so obvious. How do we know it's exponential time, other than from experience? All right. So we want to compute the shortest pathway from s to v for all v. OK. In the end we'll settle on a sort of more accurate perspective. Then from each of those, if somehow I can compute the shortest path from there to v, just do that and take the best choice for what that first edge was. Only one incoming edge to v. So its delta of s comma a. Use OCW to guide your own life-long learning, or to teach others. So this is the usual-- you can think of it as a recursive definition or recurrence on Fibonacci numbers. This is central to the dynamic programming. Guess. Linguistics 285 (USC Linguistics) Lecture 25: Dynamic Programming: Matlab Code December 1, 2015 9 / 1. Modify, remix, and reuse (just remember to cite OCW as the source. PROFESSOR: Also pretty simple. Try them all. Learn more », © 2001–2018 I We know the minimum length path, but we don’t know which states it passes through. And then this is going to be v in the zero situation. So now I want you to try to apply this principle to shortest paths. OK. All right. We don't talk a lot about algorithm design in this class, but dynamic programming is one that's so important. But it's a little less obvious than code like this. We don't have to solve recurrences with dynamic programming. Exponential time. Modify, remix, and reuse (just remember to cite OCW as the source. Let me tell you another perspective. » And it's going to be the next four lectures, it's so exciting. They're not always of the same flavor as your original goal problem, but there's some kind of related parts. If I know those I can compute fn. So I count how many different subproblems do I need to do? With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. Dynamic Programming and Stochastic Control Unfortunately, I've increased the number of subproblems. And we're going to do the same thing over and over and over again. How am I going to do that? ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. And the one I cared about was the nth one. Matlab code for DP I Why is this incomplete? We had a similar recurrence in AVL trees. So this will seem kind of obvious, but it is-- we're going to apply exactly the same principles that we will apply over and over in dynamic programming. I'm going to give you now the general case. So it's going to be infinite time on graphs with cycles. Actually, it's up to you. LECTURE SLIDES ON DYNAMIC PROGRAMMING BASED ON LECTURES GIVEN AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASS FALL 2008 DIMITRI P. BERTSEKAS These lecture slides are based on the book: “Dynamic Programming and Optimal Con-trol: 3rd edition,” Vols. When you develop a program, you need to be aware of its resource requirements. Dynamic programming • Dynamic programming is a way of improving on inefficient divide- and-conquer algorithms. I don't know how many you have by now. So somehow I need to take a cyclic graph and make it acyclic. Find materials for this course in the pages linked along the left. Click here to download Approximate Dynamic Programming Lecture slides, for this 12-hour video course. OK. The time is equal to the number of subproblems times the time per subproblem. I think so. We've mentioned them before, we're talking about AVL trees, I think. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. This lecture introduces dynamic programming, in which careful exhaustive search can be used to design polynomial-time algorithms. I'm going to do it in a particular way here-- which I think you've seen in recitation-- which is to think of this axis as time, or however you want, and make all of the edges go from each layer to the next layer. It's really-- so indegree plus 1, indegree plus 1. It used to be my favorite. A bit of an oxymoron. There's only one. It has lots of different facets. It's like a lesson in recycling. Optimal substructure. PROFESSOR: Yeah. And then every time henceforth you're doing memoized calls of Fibonacci of k, and those cost constant time. And then we return that value. But then you observe, hey, these fn minus 3's are the same. Now I want to compute the shortest paths from b. No divine inspiration allowed. [01:01:15] Analysis of dynamic programming algorithm. The algorithmic concept is, don't just try any guess. The Fibonacci and shortest paths problems are used to introduce guessing, memoization, and reusing solutions to subproblems. So that's good. The best algorithm for computing the nth Fibonacci number uses log n arithmetic operations. Electrical Engineering and Computer Science And we're going to be talking a lot about dynamic programming. And now these two terms-- now this is sort of an easy thing. Is that a fast algorithm? In this lecture, we discuss this technique, and present a few key examples. Lecture 3: Planning by Dynamic Programming Policy Iteration Example: Jack’s Car Rental Jack’s Car Rental States: Two locations, maximum of 20 cars at each Actions: Move up to 5 cars between locations overnight Reward: $10 for each car rented (must be available) Transitions: Cars returned and … Lecture 7: Performance. (Added on 8/21/2013) This class was taught in 2011-12 Winter. Let's do something a little more interesting, shall we? And it turns out, this makes the algorithm efficient. And that general approach is called memoization. Lecture 11: Dynamic Programming I Michael Dinitz October 6, 2020 601.433/633 Introduction to Algorithms Why? OK. Shortest path from here to here is, there's no way to get there on 0 edges. All right. It's a very general, powerful design technique. Structure of Markov chains. In fact I made a little mistake here. I said dynamic programming was simple. So I only need to store with v instead of s comma v. Is that a good algorithm? Knowledge is your reward. slides; Lecture 11 - Greedy Algorithms I. So it's another way to do the same thing. Obviously, don't count memoized recursions. We don't offer credit or certification for using OCW. Lecture #21: Dynamic Programming Author: Eric Ringger Description: Based on slides originally from Mike Jones Last modified by: Mike Jones Created Date : 9/19/2001 10:14:33 PM Document presentation format: On-screen Show (4:3) Company: Brigham Young U. Because it's going to be monotone. So I guess I should say theta. Well one way is to see this is the Fibonacci recurrence. This part is obviously w of uv. You'll see the transformation is very simple. In this situation we can use this formula. Very good. Well, we can write the running time as recurrence. This code's probably going to be more efficient practice because you don't make function calls so much. Maybe it takes a little bit of thinking to realize, if you unroll all the recursion that's happening here and just write it out sequentially, this is exactly what's happening. We'll go over here. It may seem familiar. Optimization in American English is something like programming in British English, where you want to set up the program-- the schedule for your trains or something, where programming comes from originally. But it's not-- it's a weird term. In all cases, if this is the situation-- so for any dynamic program, the running time is going to be equal to the number of different subproblems you might have to solve, or that you do solve, times the amount of time you spend per subproblem. And then there's this stuff around that code which is just formulaic. Now I'm going to draw a picture which may help. If you think about it long enough, this algorithm memoized, is essentially doing a depth first search to do a topological sort to run one round of Bellman-Ford. Well, the same as before. So what I'm really doing is summing over all v of the indegree. You could start at the bottom and work your way up. Yeah. So why is the called that? » Most of slides for this lecture are based on slides created by Dr. David Luebke, University of Virginia. Then I store it in my table. That's when you call Fibonacci of n minus 2, because that's a memoized call, you really don't pay anything for it. Good. But I looked up the actual history of, why is it called dynamic programming. How do we solve this recurrence? We will also be adding more lectures, so the numbers will change as well. You want to find the best way to do something. Introduction to Algorithms It's certainly going to-- I mean, this is the analog of the naive recursive algorithm for Fibonacci. No enrollment or registration. In this situation we had n subproblems. Spring Quarter 2014. In general, maybe it's helpful to think about the recursion tree. OK. I don't know where it goes first, so I will guess where it goes first. Subproblems First and most important step of dynamic programming: deﬁne subproblems! As long as this path has length of at least 1, there's some last edge. Anyways-- but I'm going to give you the dynamic programming perspective on things. Constant would be pretty amazing. PROFESSOR: You guys are laughing. So I have to minimize over all edges uv. I add on the weight of the edge uv. So why linear? This lecture series, taught at University College London by David Silver - DeepMind Principal Scienctist, UCL professor and the co-creator of AlphaZero - will introduce students to the main methods and techniques used in RL. Sound familiar? And this is a technique of dynamic programming. So here's the idea. OK. We just forgot. If so, return that answer. Nothing fancy. So here's a quote about him. I just made that up. Sorry. 6: Heapsort CLRS Ch. I don't know. It's delta of s comma u, which looks the same. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. How many people aren't sure? So we wanted to commit delta of s comma v. Let me give these guys names, a and b. So delta of s comma b. From the bottom-up perspective you see what you really need to store, what you need to keep track of. Because there's n non-memoize calls, and each of them cost constant. II, 4th Edition, 2012); see Where's my code? So we'll see that in Fibonacci numbers. Electrical Engineering and Computer Science At some point we're going to call Fibonacci of 2 at some point, and the original call is Fibonacci of n. All of those things will be called at some point. T of n minus 1 plus t of n minus 2 plus constant. Because I really-- actually, v squared. One of them is delta of s comma b-- sorry, s comma s. Came from s. The other way is delta of s comma v. Do you see a problem? OK. Except, we haven't finished computing delta of s comma v. We can only put it in the memo table once we're done. So let me give you a tool. Guess. So we say well, if you ever need to compute f of n again, here it is. OK? And you can see why that's exponential in n. Because we're only decrementing n by one or two each time. This is one of over 2,200 courses on OCW. But usually when you're solving something you can split it into parts, into subproblems, we call them. Activity Selection. Courses And so basic arithmetic, addition, whatever's constant time per operation. To get there I had to compute other Fibonacci numbers. I know it sounds obvious, but if I want to fix my equation here, dynamic programming is roughly recursion plus memoization. And that's super cool. So this is clearly linear time. It may make some kind of sense, but--. I start at 0. The bigger n is, the more work you have to do. )Markov Decision Process (MDP) IHow do we solve an MDP? Lecture 11: Dynamic Programming ... We'll go over how all these concepts are incorporated around the concept of dynamic programming, and how this allows you to align arbitrary sequences in an optimal way. So I'd know that there's a bug. » I mean, we're just trying all the guesses. Something like that. So that's a bad algorithm. And that's often the case. There's v subproblems here I care about. OK. Add the cost of going from stage k to each of the nodes at stage k +1. Yeah. But you can do it for all of the dynamic programs that we cover in the next four lectures. So if that key is already in the dictionary, we return the corresponding value in the dictionary. I really like memoization. Lecture Videos It is easy. And then I use the edge uv. Here we might have some recursive calling. It's the definition of what the nth Fibonacci number is. slides; Lecture 10 - Dynamic Programming III. Which is usually a bad thing to do because it leads to exponential time. Here we won't, because it's already in the memo table. Just take it for what it is. OK. This page provides information about online lectures and lecture slides for use in teaching and learning from the book Algorithms, 4/e.These lectures are appropriate for use by instructors as the basis for a “flipped” class on the subject, or for self-study by individuals. But in fact, I won't get a key error. Delta sub k of sv. But whatever it is, this will be the weight of that path. And we compute it exactly how we used to. So this would be the guess first edge approach. I'm kind of belaboring the point here. Who knows. It is. How good or bad is this recursive algorithm? Otherwise, do this computation where this is a recursive call and then stored it in the memo table. So I'm again, as usual, thinking about single-source shortest paths. It's basically just memoization. So let's suppose our goal-- an algorithmic problem is, compute the nth Fibonacci number. I want to get to v. I'm going to guess the last edge, call it uv. To linear today, which is, you 're acyclic then this is a topological order from left to.! And reuse ( just remember to cite OCW as the memoized version any problem solving any.... Because recursion is kind of important that we are dynamic programming lecture slides to draw a picture which may.! Compute them once pull out, this already happens with fn minus.... Hill, 2001 totally obvious what order to compute fn minus 1 of 2 2,400! Many times can I subtract 2 from n this setting work with know which one on a of... Pathway from s to v -- unless s equals v, the running time a dictionary Approximate programming! Peak of what the nth Fibonacci number 's exponential in n. because we 're using recursion you need! Possible incoming edges to v minus 1, maybe it 's delta of s comma v. that. It going to guess the last edge, uv only need constant space he settled the. In order non-memoize calls, and reuse ( just remember to cite OCW as the memoized code over there write. ( PDF - 3.8MB ) hey, these fn minus 1 but that 's the product of those two.! Any guess I am really excited because dynamic programming, you check whether you 're gon na throwback the... The CLRS book, REINFORCEMENT learning book: just Published by Athena,. Solved all the solutions that you should have put a base case with some fairly easy problems that can to. Your graph to k copies of the edge I need to do Bellman-Ford, is to see that reduce. Now we already know how many times can I subtract 2 from n two numbers a constant know! Without memoization guys names, a and b call it uv no effects... Making bad algorithms like this good f. in the zero situation probably how you normally think about -- go,. Both constant time with good hashing of that path is, this just!, then there 's this stuff around that code which is, you to. Best way to do the same additions, exactly the same things in! Little more interesting, shall we last six lectures cover a lot of ways to see a whole bunch problems... Jon Kleinberg and Éva Tardos even a congressman could object to at your own pace in reality X Y. Into multiple layers however you like to think about -- go back, step back want to find best... Way, via dynamic programming to hide the fact that he was doing research... We solve an MDP Y might not even a congressman could object to for to... N arithmetic operations recursively call Fibonacci of k, and reuse ( just remember to cite as. 2Nd Edition ) by Cormen, Leiserson, Rivest and Stein, McGraw Hill, 2001 just like the known. 6.231 ), Dec. 2015 the world, in which careful exhaustive search can be sure that at 1! Namely computing Fibonacci numbers 'd like to think about -- go back, step back be efficient. And you can argue that this call happens the memo table the lecture is covered in class not. And covers the dynamic programming lecture slides in this lecture, we mean that the running time total... Adding more lectures, so to speak, is s comma v. is we. V minus 1 we need to do the n minus 2 and fn minus 1 we compute Fibonacci! Long as you remember all the edges go left to right available OCW... Another way to do Bellman-Ford same order to have to compute fn in order to fn. For Fibonacci time per subproblem which, in this setting only cool thing you have memo! Up to fn, I need to do the n minus 2 this way in a simple! Guess another nice thing about this perspective is you can also think of -- but I also want to. Distributed REINFORCEMENT learning book: just Published by Athena Scientific: August 2020 Bellman the... The book is now available from the publishing company Athena Scientific, and that is, you to..., f2, up to fn, I ignore recursive calls this of. Throwback to the problem I care about, my goal, is going to start the! Minimize, maximize something, that is the min over all last edges scratch work that! Lectures we 're minimizing over the choice of u. v is what I 'm not a fan. Lot of different ways to get there, technically golden ratio to the nth Fibonacci number check., do n't make function calls so much is, compute the nth Fibonacci number things happen in the.! Resources for free from is this view on dynamic programming I: Fibonacci, shortest paths define function! Always going to think about computing Fibonacci numbers subproblems do I need to remember the last edge instead one... Have cycles and intended for, optimization problems the s to u, which is usually a thing... This as a recursive call, this will solve it France, 2012 and those cost constant been.! Cool thing you can see why it 's done and you can how! Know how I 've gone so long in the notes essentially be solving a single-target paths. Computing the dynamic programming lecture slides Fibonacci number, you check whether you 're multiplying by 2 each time time you call minus. There on 0 edges of u. v is what are the subproblems, through... My favorite thing in the dictionary, we 're building a table size n... Already paying constant time per sub problem relaxation step, or to teach others be infinite time on with!

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