字幕表 動画を再生する 英語字幕をプリント The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOHN GUTTAG: All right, welcome to the 60002, or if you were in 600, the second half of 600. I'm John Guttag. Let me start with a few administrative things. What's the workload? There are problem sets. They'll all be programming problems much in the style of 60001. And the goal-- really twofold. 60001 problem sets were mostly about you learning to be a programmer. A lot of that carries over. No one learns to be a programmer in half a semester. So a lot of it is to improve your skills, but also there's a lot more, I would say, conceptual, algorithmic material in 60002, and the problem sets are designed to help cement that as well as just to give you programming experience. Finger exercises, small things. If they're taking you more than 15 minutes, let us know. They really shouldn't, and they're generally designed to help you learn a single concept, usually a programming concept. Reading assignments in the textbooks, I've already posted the first reading assignment, and essentially they should provide you a very different take on the same material we're covering in lectures and recitations. We've tried to choose different examples for lectures and from the textbooks for the most part, so you get to see things in two slightly different ways. There'll be a final exam based upon all of the above. All right, prerequisites-- experience writing object-oriented programs in Python, preferably Python 3.5. Familiarity with concepts of computational complexity. You'll see even in today's lecture, we'll be assuming that. Familiarity with some simple algorithms. If you took 60001 or you took the 60001 advanced standing exam, you'll be fine. Odds are you'll be fine anyway, but that's the safest way to do it. So the programming assignments are going to be a bit easier, at least that's what students have reported in the past, because they'll be more focused on the problem to be solved than on the actual programming. The lecture content, more abstract. The lectures will be-- and maybe I'm speaking euphemistically-- a bit faster paced. So hang on to your seats. And the course is really less about programming and more about dipping your toe into the exotic world of data science. We do want you to hone your programming skills. There'll be a few additional bits of Python. Today, for example, we'll talk about lambda abstraction. Inevitably, some comments about software engineering, how to structure your code, more emphasis in using packages. Hopefully it will go a little bit smoother than in the last problem set in 60001. And finally, it's the old joke about programming that somebody walks up to a taxi driver in New York City and says, "I'm lost. How do I get to Carnegie Hall?" The taxi driver turns to the person and says, "practice, practice, practice." And that's really the only way to learn to program is practice, practice, practice. The main topic of the course is what I think of as computational models. How do we use computation to understand the world in which we live? What is a model? To me I think of it as an experimental device that can help us to either understand something that has happened, to sort of build a model that explains phenomena we see every day, or a model that will allow us to predict the future, something that hasn't happened. So you can think of, for example, a climate change model. We can build models that sort of explain how the climate has changed over the millennia, and then we can build probably a slightly different model that might predict what it will be like in the future. So essentially what's happening is science is moving out of the wet lab and into the computer. Increasingly, I'm sure you all see this-- those of you who are science majors-- an increasing reliance on computation rather than traditional experimentation. As we'll talk about, traditional experimentation is and will remain important, but now it has to really be supplemented by computation. We'll talk about three kinds of models-- optimization models, statistical models, and simulation models. So let's talk first about optimization models. An optimization model is a very simple thing. We start with an objective function that's either to be maximized or minimized. So for, example, if I'm going from New York to Boston, I might want to find a route by car or plane or train that minimizes the total travel time. So my objective function would be the number of minutes spent in transit getting from a to b. We then often have to layer on top of that objective function a set of constraints, sometimes empty, that we have to obey. So maybe the fastest way to get from New York to Boston is to take a plane, but I only have $100 to spend. So that option is off the table. So I have the constraints there on the amount of money I can spend. Or maybe I have to be in Boston before 5:00 PM and while the bus would get me there for $15, it won't get me there before 5:00. And so maybe what I'm left with is driving, something like that. So objective function, something you're either minimizing or maximizing, and a set of constraints that eliminate some solutions. And as we'll see, there's an asymmetry here. We handle these two things differently. We use these things all the time. I commute to work using Waze, which essentially is solving-- not very well, I believe-- an optimization problem to minimize my time from home to here. When you travel, maybe you log into various advisory programs that try and optimize things for you. They're all over the place. Today you really can't avoid using optimization algorithm as you get through life. Pretty abstract. Let's talk about a specific optimization problem called the knapsack problem. The first time I talked about the knapsack problem I neglected to show a picture of a knapsack, and I was 10 minutes into it before I realized most of the class had no idea what a knapsack was. It's what we old people used to call a backpack, and they used to look more like that than they look today. So the knapsack problem involves-- usually it's told in terms of a burglar who breaks into a house and wants to steal a bunch of stuff but has a knapsack that will only hold a finite amount of stuff that he or she wishes to steal. And so the burglar has to solve the optimization problem of stealing the stuff with the most value while obeying the constraint that it all has to fit in the knapsack. So we have an objective function. I'll get the most for this when I fence it. And a constraint, it has to fit in my backpack. And you can guess which of these might be the most valuable items here. So here is in words, written words what I just said orally. There's more stuff than you can carry, and you have to choose which stuff to take and which to leave behind. I should point out that there are two variants of it. There's the 0/1 knapsack problem and the continuous. The 0/1 would be illustrated by something like this. So the 0/1 knapsack problem means you either take the object or you don't. I take that whole gold bar or I take none of it. The continuous or so-called fractional knapsack problem says I can take pieces of it. So maybe if I take in my gold bar and shaved it into gold dust, I then can say, well, the whole thing won't fit in, but I can fit in a path, part of it. The continuous knapsack problem is really boring. It's easy to solve. How do you think you would solve the continuous problem? Suppose you had over here a pile of gold and a pile of silver and a pile of raisins, and you wanted to maximize your value. Well, you'd fill up your knapsack with gold until you either ran out of gold or ran out of space. If you haven't run out of space, you'll now put silver in until you run out of space. If you still haven't run out of space, well, then you'll take as many raisins as you can fit in. But you can solve it with what's called a greedy algorithm, and we'll talk much more about this as we go forward. Where you take the best thing first as long as you can and then you move on to the next thing. As we'll see, the 0/1 knapsack problem is much more complicated because once you make a decision, it will affect the future decisions. Let's look at an example, and I should probably warn you, if you're hungry, this is not going to be a fun lecture. So here is my least favorite because I always want to eat more than I'm supposed to eat. So the point is typically knapsack problems are not physical knapsacks but some conceptual idea. So let's say that I'm allowed 1,500 calories of food, and these are my options. I have to go about deciding, looking at this food-- and it's interesting, again, there's things showing up on your screen that are not showing up on my screen, but they're harmless, things like how my mouse works. Anyway, so I'm trying to take some fraction of this food, and it can't add up to more than 1,500 calories. The problem might be that once I take something that's 1,485 calories, I can't take anything else, or maybe 1,200 calories and everything else is more than 300. So once I take one thing, it constrains possible solutions. A greedy algorithm, as we'll see, is not guaranteed to give me the best answer. Let's look at a formalization of it. So each item is represented by a pair, the value of the item and the weight of the item. And let's assume the knapsack can accommodate items with the total weight of no more than w. I apologize for the short variable names, but they're easier to fit on a slide. Finally, we're going to have a vector l of length n representing the set of available items. This is assuming we have n items to choose from. So each element of the vector represents an item. So those are the items we have. And then another vector v is going to indicate whether or not an item was taken. So essentially I'm going to use a binary number to represent the set of items I choose to take. For item three say, if bit three is zero I'm not taking the item. If bit three is one, then I am taking the item. So it just shows I can now very nicely represent what I've done by a single vector of zeros and ones. Let me pause for a second. Does anyone have any questions about this setup? It's important to get this setup because what we're going to see now depends upon that setting in your head. So I've kind of used mathematics to describe the backpack problem. And that's typically the way we deal with these optimization problems. We start with some informal description, and then we translate them into a mathematical representation. So here it is. We're going to try and find a vector v that maximizes the sum of V sub i times I sub i.