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  • We're going to get started. Handouts are the by the door if

  • anybody didn't pick one up. My name is Charles Leiserson.

  • I will be lecturing this course this term, Introduction to

  • Algorithms, with Erik Demaine. In addition,

  • this is an SMA course, a Singapore MIT Alliance course

  • which will be run in Singapore by David Hsu.

  • And so all the lectures will be videotaped and made available on

  • the Web for the Singapore students, as well as for MIT

  • students who choose to watch them on the Web.

  • If you have an issue of not wanting to be on the videotape,

  • you should sit in the back row. OK?

  • Otherwise, you will be on it. There is a video recording

  • policy, but it seems like they ran out.

  • If anybody wants to see it, people, if they could just sort

  • of pass them around maybe a little bit, once you're done

  • reading it, or you can come up. I did secure one copy.

  • Before we get into the content of the course,

  • let's briefly go over the course information because there

  • are some administrative things that we sort of have to do.

  • As you can see, this term we have a big staff.

  • Take a look at the handout here.

  • Including this term six TAs, which is two more TAs than we

  • normally get for this course. That means recitations will be

  • particularly small. There is a World Wide Web page,

  • and you should bookmark that and go there regularly because

  • that is where everything will be distributed.

  • Email. You should not be emailing

  • directly to, even though we give you our email addresses,

  • to the individual members of the staff.

  • You should email us generally. And the reason is you will get

  • much faster response. And also, for any

  • communications, generally we like to monitor

  • what the communications are so it's helpful to have emails

  • coming to everybody on the course staff.

  • As I mentioned, we will be doing distance

  • learning this term. And so you can watch lectures

  • online if you choose to do that. I would recommend,

  • for people who have the opportunity to watch,

  • to come live. It's better live.

  • You get to interact. There's an intangible that

  • comes with having it live. In fact, in addition to the

  • videos, I meet weekly with the Singapore students so that they

  • have a live session as well. Prerequisites.

  • The prerequisites for this course are 6.042,

  • which is Math for Computer Science, and 6.001.

  • You basically need discrete mathematics and probability,

  • as well as programming experience to take this course

  • successfully. People do not have that

  • background should not be in the class.

  • We will be checking prerequisites.

  • If you have any questions, please come to talk to us after

  • class. Let's see.

  • Lectures are here. For SMA students,

  • they have the videotapes and they will also have a weekly

  • meeting. Students must attend a one-hour

  • recitation session each week. There will be new material

  • presented in the recitation. Unlike the lectures,

  • they will not be online. Unlike the lectures,

  • there will not be lecture notes distributed for the recitations

  • in general. And, yet, there will be

  • material there that is directly on the exams.

  • And so every term we say oh, when did you cover that?

  • That was in recitation. You missed that one.

  • So, recitations are mandatory. And, in particular,

  • also let me just mention your recitation instructor is the one

  • who assigns your final grade. So we have a grade meeting and

  • keep everybody normal, but your recitation has the

  • final say on your grade. Handouts.

  • Handouts are available on the course Web page.

  • We will not generally, except for this one,

  • first handout, be bringing handouts to class.

  • Textbook is this book, Introduction to Algorithms.

  • MIT students can get it any of the local bookstores,

  • including the MIT Coop. There is also a new online

  • service that provides textbooks. You can also get a discount if

  • you buy it at the MIT Press Bookstore.

  • There is a coupon in the MIT Student Telephone Directory for

  • a discount on MIT Press books. And you can use that to

  • purchase this book at a discount.

  • Course website. This is the course website.

  • It links to the Stellar website, which is where,

  • actually, everything will be kept.

  • And SMA students have their own website.

  • Some students find this course particularly challenges so we

  • will have extra help. We will post weekly office

  • hours on the course website for the TAs.

  • And then as an experiment this term, we are going to offer

  • homework labs for this class. What a homework lab is,

  • is it's a place and a time you can go where other people in the

  • course will go to do homework. And there will be typically two

  • TAs who staff the lab. And so, as you're working on

  • your homework, you can get help from the TAs

  • if you need it. And it's generally a place,

  • we're going to schedule those, and they will be on the course

  • calendar for where it is and when it is that they will be

  • held, but usually Sundays 2:00 to 4:00 pm, or else it will be

  • some evening. I think the first one is an

  • evening, right? Near to when the homework is

  • due. Your best bet is try to do the

  • homework in advance of the homework lab.

  • But then, if you want extra help, if you want to talk over

  • your solutions with people because as we will talk about

  • problem sets you can solve in collaboration with other people

  • in the class. In addition,

  • there are several peer assistance programs.

  • Also the office of Minority Education has an assistance

  • program, and those usually get booked up pretty quickly.

  • If you're interested in those, good idea to make an

  • appointment to get there and get help soon.

  • The homework labs, I hope a lot of people will try

  • that out. We've never done this.

  • I don't know of any other course.

  • Do other people know of courses at MIT that have done this?

  • 6.011 did it, OK.

  • Good. And was it successful in that

  • class? It never went,

  • OK. Good.

  • [LAUGHTER] We will see. If it's not paying off then we

  • will just return to ordinary office hours for those TAs,

  • but I think for some students that is a good opportunity.

  • If you wish to be registered in this course, you must sign up on

  • the course Web page. So, that is requirement one.

  • It must be done today. You will find it difficult to

  • pass the course if you are not in the class.

  • And you should notify your TA if you decide to drop so that we

  • can get you off and stop the mailings, stop the spam.

  • And you should register today before 7:00 PM.

  • And then we're going to email your recitation assignment to

  • you before Noon tomorrow. And if you don't receive this

  • information by Thursday Noon, please send us an email to the

  • course staff generally, not to me individually,

  • saying that you didn't receive your recitation assignment.

  • And so if you haven't received it by Thursday Noon you want to.

  • I think generally they are going to send them out tonight

  • or at least by tomorrow morning. Yeah.

  • OK. SMA students don't have to

  • worry about this. Problem sets.

  • We have nine problem sets that we project will be assigned

  • during the semester. A couple things about problem

  • sets. Homeworks won't generally be

  • accepted, if you have extenuating circumstances you

  • should make prior arrangements with your recitation instructor.

  • In fact, almost all of the administrative stuff,

  • you shouldn't come to me to ask and say can I hand in something

  • late? You should be talking to your

  • recitation instructor. You can read the other things

  • about the form, but let me just mention that

  • there are exercises that should be solved but not handed in as

  • well to give you drill on the material.

  • I highly recommend you doing the exercises.

  • They both test your understanding of the material,

  • and exercises have this way of finding themselves on quizzes.

  • You're often asked to describe algorithms.

  • And here is a little outline of what you can use to describe an

  • algorithm. The grading policy is something

  • that somehow I cover. And always every term there are

  • at least a couple of students who pretend like I never showed

  • them this. If you skip problems it has a

  • nonlinear effect on your grade. Nonlinear, OK?

  • If you don't skip any problems, no effect on your grade.

  • If you skip one problem, a hundredth of a letter grade,

  • we can handle that. But two problems it's a tenth.

  • And, as you see, by the time you have skipped

  • like five letter grades, it is already five problems.

  • This is not problem sets, by the way.

  • This is problems, OK?

  • You're down a third of a letter grade.

  • And if you don't do nine or more, so that's typically about

  • three to four problem sets, you don't pass the class.

  • I always have some students coming at the end of the year

  • saying oh, I didn't do any of my problems.

  • Can you just pass me because I did OK on the exams?

  • Answer no, a very simple answer because we've said it upfront.

  • So, the problem sets are an integral part of the course.

  • Collaboration policy. This is extremely important so

  • everybody pay attention. If you are asleep now wake up.

  • Like that's going to wake anybody up, right?

  • [LAUGHTER] The goal of homework.

  • Professor Demaine and my philosophy is that the goal of

  • homework is to help you learn the material.

  • And one way of helping to learn is not to just be stuck and

  • unable to solve something because then you're in no better

  • shape when the exam roles around, which is where we're

  • actually evaluating you. So, you're encouraged to

  • collaborate. But there are some commonsense

  • things about collaboration. If you go and you collaborate

  • to the extent that all you're doing is getting the information

  • from somebody else, you're not learning the

  • material and you're not going to do well on the exams.

  • In our experience, students who collaborate

  • generally do better than students who work alone.

  • But you owe it to yourself, if you're going to work in a

  • study group, to be prepared for your study group meeting.

  • And specifically you should spend a half an hour to 45

  • minutes on each problem before you go to group so you're up to

  • speed and you've tried out your ideas.

  • And you may have solutions to some, you may be stuck on some

  • other ones, but at least you applied yourself to it.

  • After 30 to 45 minutes, if you cannot get the problem,

  • just sitting there and banging your head against it makes no

  • sense. It's not a productive use of

  • your time. And I know most of you have

  • issues with having time on your hands, right?

  • Like it's not there. So, don't go banging your head

  • against problems that are too hard or where you don't

  • understand what's going on or whatever.

  • That's when the study group can help out.

  • And, as I mentioned, we'll have homework labs which

  • will give you an opportunity to go and do that and coordinate

  • with other students rather than necessarily having to form your

  • own group. And the TAs will be there.

  • If your group is unable to solve the problem then talk to

  • other groups or ask your recitation instruction.

  • And, that's how you go about solving them.

  • Writing up the problem sets, however, is your individual

  • responsibility and should be done alone.

  • You don't write up your problem solutions with other students,

  • you write them up on your own. And you should on your problem

  • sets, because this is an academic place,

  • we understand that the source of academic information is very

  • important, if you collaborated on solutions you should write a

  • list of the collaborators. Say I worked with so and so on

  • this solution. It does not affect your grade.

  • It's just a question of being scholarly.

  • It is a violation of this policy to submit a problem

  • solution that you cannot orally explain to a member of the

  • course staff. You say oh, well,

  • my write-up is similar to that other person's.

  • I didn't copy them. We may ask you to orally

  • explain your solution. If you are unable,

  • according to this policy, the presumption is that you

  • cheated. So, do not write up stuff that

  • you don't understand. You should be able to write up

  • the stuff that you understand. Understand why you're putting

  • down what you're putting down. If it isn't obvious,

  • no collaboration whatsoever is permitted on exams.

  • Exams is when we evaluate you. And now we're not interested in

  • evaluating other people, we're interested in evaluating

  • you. So, no collaboration on exams.

  • We will have a take-home exam for the second quiz.

  • You should look at the schedule.

  • If there are problems with the schedule of that,

  • we want to know early. And we will give you more

  • details about the collaboration in the lecture on Monday,

  • November 28th. Now, generally,

  • the lectures here, they're mandatory and you have

  • to know them, but I know that some people say

  • gee, 9:30 is kind of early, especially on a Monday or

  • whatever. It can be kind of early to get

  • up. However, on Monday,

  • November 28th, you fail the exam if you do not

  • show up to lecture on time. That one day you must show up.

  • Any questions about that? That one day you must show up

  • here, even if you've been watching them on the Web.

  • And generally, if you think you have

  • transgressed, the best is to come to us to

  • talk about it. We can usually work something

  • out. It's when we find somebody has

  • transgressed from a third-party or from obvious analyses that we

  • do with homeworks, that's when things get messy.

  • So, if you think, for some reason or other,

  • oh, I may have done something wrong, please come and talk to

  • us. We actually were students once,

  • too, albeit many years ago. Any questions?

  • So, this class has great material.

  • Fabulous material. And it's really fun,

  • but you do have to work hard. Let's talk content.

  • This is the topic of the first part of the course.

  • The first part of the course is focused on analysis.

  • The second part of the course is focused on design.

  • Before you can do design, you have to master a bunch of

  • techniques for analyzing algorithms.

  • And then you'll be in a position to design algorithms

  • that you can analyze and that which are efficient.

  • The analysis of algorithm is the theoretical study --

  • -- of computer program performance --

  • -- and resource usage. And a particular focus on

  • performance. We're studying how to make

  • things fast. In particular,

  • computer programs. We also will discover and talk

  • about other resources such as communication,

  • such as memory, whether RAM memory or disk

  • memory. There are other resources that

  • we may care about, but predominantly we focus on

  • performance. Because this is a course about

  • performance, I like to put things in perspective a little

  • bit by starting out and asking, in programming,

  • what is more important than performance?

  • If you're in an engineering situation and writing code,

  • writing software, what's more important than

  • performance? Correctness.

  • Good. OK.

  • What else? Simplicity can be.

  • Very good. Yeah.

  • Maintainability often much more important than performance.

  • Cost. And what type of cost are you

  • thinking? No, I mean cost of what?

  • We're talking software here, right?

  • What type of cost do you have in mind?

  • There are some costs that are involved when programming like

  • programmer time. So, programmer time is another

  • thing also that might be. Stability.

  • Robustness of the software. Does it break all the time?

  • What else?

  • Come on. We've got a bunch of engineers

  • here. A lot of things.

  • How about features? Features can be more important.

  • Having a wider collection of features than your competitors.

  • Functionality. Modularity.

  • Is it designed in a way where you can make changes in a local

  • part of the code and you don't have to make changes across the

  • code in order to affect a simple change in the functionality?

  • There is one big one which definitely, especially in the

  • `90s, was like the big thing in computers.

  • The big thing. Well, security actually.

  • Good. I don't even have that one

  • down. Security is excellent.

  • That's actually been more in the 2000.

  • Security has been far more important often than

  • performance. Scalability has been important,

  • although scalability, in some sense,

  • is performance related. But, yes, scalability is good.

  • What was the big breakthrough and why do people use Macintosh

  • rather than Windows, those people who are of that

  • religion? User-friendliness.

  • Wow. If you look at the number of

  • cycles of computers that went into user-friendliness in the

  • `90s, it grew from almost nothing to where it's now the

  • vast part of the computation goes into user-friendly.

  • So, all those things are more important than performance.

  • This is a course on performance.

  • Then you can say OK, well, why do we bother and why

  • study algorithms and performance if it's at the bottom of the

  • heap? Almost always people would

  • rather have these other things than performance.

  • You go off and you say to somebody, would I rather have

  • performance or more user-friendliness?

  • It's almost always more important than performance.

  • Why do we care then? Yeah?

  • That wasn't user-friendly. Sometimes performance is

  • correlated with user-friendliness,

  • absolutely. Nothing is more frustrating

  • than sitting there waiting, right?

  • So, that's a good reason. What are some other reasons

  • why? Sometimes they have real-time

  • constraints so they don't actually work unless they

  • perform adequately. Yeah?

  • Hard to get, well, we don't usually quantify

  • user-friendliness so I'm not sure, but I understand what

  • you're saying. He said we don't get

  • exponential performance improvements in

  • user-friendliness. We often don't get that in

  • performance either, by the way.

  • [LAUGHTER] Sometimes we do, but that's good.

  • There are several reasons that I think are important.

  • Once is that often performance measures the line between the

  • feasible and the infeasible. We have heard some of these

  • things. For example,

  • when there are real-time requirements,

  • if it's not fast enough it's simply not functional.

  • Or, if it uses too much memory it's simply not going to work

  • for you. And, as a consequence,

  • what you find is algorithms are on the cutting edge of

  • entrepreneurship. If you're talking about just

  • re-implementing stuff that people did ten years ago,

  • performance isn't that important at some level.

  • But, if you're talking about doing stuff that nobody has done

  • before, one of the reasons often that they haven't done it is

  • because it's too time-consuming. Things don't scale and so

  • forth. So, that's one reason,

  • is the feasible versus infeasible.

  • Another thing is that algorithms give you a language

  • for talking about program behavior, and that turns out to

  • be a language that has been pervasive through computer

  • science, is that the theoretical language is what gets adopted by

  • all the practitioners because it's a clean way of thinking

  • about things. A good way I think about

  • performance, and the reason it's on the bottom of the heap,

  • is sort of like performance is like money, it's like currency.

  • You say what good does a stack of hundred dollar bills do for

  • you? Would you rather have food or

  • water or shelter or whatever? And you're willing to pay those

  • hundred dollar bills, if you have hundred dollar

  • bills, for that commodity. Even though water is far more

  • important to your living. Well, similarly,

  • performance is what you use to pay for user-friendliness.

  • It's what you pay for security. And you hear people say,

  • for example, that I want greater

  • functionality, so people will program in Java,

  • even though it's much slower than C, because they'll say it

  • costs me maybe a factor of three or something in performance to

  • program in Java. But Java is worth it because

  • it's got all these object-oriented features and so

  • forth, exception mechanisms and so on.

  • And so people are willing to pay a factor of three in

  • performance. So, that's why you want

  • performance because you can use it to pay for these other things

  • that you want. And that's why,

  • in some sense, it's on the bottom of the heap,

  • because it's the universal thing that you quantify.

  • Do you want to spend a factor of two on this or spend a factor

  • of three on security, et cetera?

  • And, in addition, the lessons generalize to other

  • resource measures like communication,

  • like memory and so forth. And the last reason we study

  • algorithm performance is it's tons of fun.

  • Speed is always fun, right?

  • Why do people drive fast cars, race horses,

  • whatever? Rockets, et cetera,

  • why do we do that? Because speed is fun.

  • Ski. Who likes to ski?

  • I love to ski. I like going fast on those

  • skis. It's fun.

  • Hockey, fast sports, right?

  • We all like the fast sports. Not all of us,

  • I mean. Some people say he's not

  • talking to me. OK, let's move on.

  • That's sort of a little bit of a notion as to why we study

  • this, is that it does, in some sense,

  • form a common basis for all these other things we care

  • about. And so we want to understand

  • how can we generate money for ourselves in computation?

  • We're going to start out with a very simple problem.

  • It's one of the oldest problems that has been studied in

  • algorithms, is the problem of sorting.

  • We're going to actually study this for several lectures

  • because sorting contains many algorithmic techniques.

  • The sorting problem is the following.

  • We have a sequence a_1, a_2 up to a_n of numbers as

  • input. And our output is a permutation

  • of those numbers.

  • A permutation is a rearrangement of the numbers.

  • Every number appears exactly once in the rearrangement such

  • that, I sometimes use a dollar sign to mean "such that," a_1 is

  • less than or equal to a_2 prime. Such that they are

  • monotonically increasing in size.

  • Take a bunch of numbers, put them in order.

  • Here's an algorithm to do it. It's called insertion sort.

  • And we will write this algorithm in what we call

  • pseudocode. It's sort of a programming

  • language, except it's got English in there often.

  • And it's just a shorthand for writing for being precise.

  • So this sorts A from 1 to n. And here is the code for it.

  • This is what we call pseudocode.

  • And if you don't understand the pseudocode then you should ask

  • questions about any of the notations.

  • You will start to get used to it as we go on.

  • One thing is that in the pseudocode we use indentation,

  • where in most languages they have some kind of begin-end

  • delimiters like curly braces or something in Java or C,

  • for example. We just use indentation.

  • The whole idea of the pseudocode is to try to get the

  • algorithms as short as possible while still understanding what

  • the individual steps are. In practice,

  • there actually have been languages that use indentation

  • as a means of showing the nesting of things.

  • It's generally a bad idea, because if things go over one

  • page to another, for example,

  • you cannot tell what level of nesting it is.

  • Whereas, with explicit braces it's much easier to tell.

  • So, there are reasons why this is a bad notation if you were

  • doing software engineering. But it's a good one for us

  • because it just keeps things short and makes fewer things to

  • write down. So, this is insertion sort.

  • Let's try to figure out a little bit what this does.

  • It basically takes an array A and at any point the thing to

  • understand is, we're setting basically,

  • we're running the outer loop from j is 2 to n,

  • and the inner loop that starts at j minus 1 and then goes down

  • until it's zero. Basically, if we look at any

  • point in the algorithm, we essentially are looking at

  • some element here j. A of j, the jth element.

  • And what we do essentially is we pull a value out here that we

  • call the key. And at this point the important

  • thing to understand, and we'll talk more about this

  • in recitation on Friday, is that there is an invariant

  • that is being maintained by this loop each time through.

  • And the invariant is that this part of the array is sorted.

  • And the goal each time through the loop is to increase,

  • is to add one to the length of the things that are sorted.

  • And the way we do that is we pull out the key and we just

  • copy values up like this. And keep copying up until we

  • find the place where this key goes, and then we insert it in

  • that place. And that's why it's called

  • insertion sort. We just sort of move the

  • things, copy the things up until we find where it goes,

  • and then we put it into place. And now we have it from A from

  • one to j is sorted, and now we can work on j plus

  • one. Let's give an example of that.

  • Imagine we are doing 8, 2, 4, 9, 3, 6.

  • We start out with j equals 2. And we figure out that we want

  • to insert it there. Now we have 2,

  • 8, 4, 9, 3, 6. Then we look at the four and

  • say oh, well, that goes over here.

  • We get 2, 4, 8, 9, 3, 6 after the second

  • iteration of the outer loop. Then we look at 9 and discover

  • immediately it just goes right there.

  • Very little work to do on that step.

  • So, we have exactly the same output after that iteration.

  • Then we look at the 3 and that's going to be inserted over

  • there. 2, 3, 4, 8, 9,

  • 6. And finally we look at the 6

  • and that goes in there. 2, 3, 4, 6, 8,

  • 9. And at that point we are done.

  • Question?

  • The array initially starts at one, yes.

  • A[1...n], OK? So, this is the insertion sort

  • algorithm. And it's the first algorithm

  • that we're going to analyze. And we're going to pull out

  • some tools that we have from our math background to help to

  • analyze it. First of all,

  • let's take a look at the issue of running time.

  • The running time depends, of this algorithm depends on a

  • lot of things. One thing it depends on is the

  • input itself. For example,

  • if the input is already sorted --

  • -- then insertion sort has very little work to do.

  • Because every time through it's going to be like this case.

  • It doesn't have to shuffle too many guys over because they're

  • already in place. Whereas, in some sense,

  • what's the worst case for insertion sort?

  • If it is reverse sorted then it's going to have to do a lot

  • of work because it's going to have to shuffle everything over

  • on each step of the outer loop. In addition to the actual input

  • it depends, of course, on the input size.

  • Here, for example, we did six elements.

  • It's going to take longer if we, for example,

  • do six times ten to the ninth elements.

  • If we were sorting a lot more stuff, it's going to take us a

  • lot longer. Typically, the way we handle

  • that is we are going to parameterize things in the input

  • size. We are going to talk about time

  • as a function of the size of things that we are sorting so we

  • can look at what is the behavior of that.

  • And the last thing I want to say about running time is

  • generally we want upper bonds on the running time.

  • We want to know that the time is no more than a certain

  • amount. And the reason is because that

  • represents a guarantee to the user.

  • If I say it's not going to run, for example,

  • if I tell you here's a program and it won't run more than three

  • seconds, that gives you real information about how you could

  • use it, for example, in a real-time setting.

  • Whereas, if I said here's a program and it goes at least

  • three seconds, you don't know if it's going to

  • go for three years. It doesn't give you that much

  • guarantee if you are a user of it.

  • Generally we want upper bonds because it represents a

  • guarantee to the user.

  • There are different kinds of analyses that people do.

  • The one we're mostly going to focus on is what's called

  • worst-case analysis. And this is what we do usually

  • where we define T of n to be the maximum time on any input of

  • size n. So, it's the maximum input,

  • the maximum it could possibly cost us on an input of size n.

  • What that does is, if you look at the fact that

  • sometimes the inputs are better and sometimes they're worse,

  • we're looking at the worst case of those because that's the way

  • we're going to be able to make a guarantee.

  • It always does something rather than just sometimes does

  • something. So, we're looking at the

  • maximum. Notice that if I didn't have

  • maximum then T(n) in some sense is a relation,

  • not a function, because the time on an input of

  • size n depends on which input of size n.

  • I could have many different times, but by putting the

  • maximum at it, it turns that relation into a

  • function because there's only one maximum time that it will

  • take. Sometimes we will talk about

  • average case. Sometimes we will do this.

  • Here T of n is then the expected time over all inputs of

  • size n. It's the expected time.

  • Now, if I talk about expected time, what else do I need to say

  • here? What does that mean,

  • expected time? I'm sorry.

  • Raise your hand. Expected inputs.

  • What does that mean, expected inputs?

  • I need more math. What do I need by expected time

  • here, math? You have to take the time of

  • every input and then average them, OK.

  • That's kind of what we mean by expected time.

  • Good. Not quite.

  • I mean, what you say is completely correct,

  • except is not quite enough. Yeah?

  • It's the time of every input times the probability that it

  • will be that input. It's a way of taking a weighted

  • average, exactly right. How do I know what the

  • probability of every input is? How do I know what the

  • probability a particular input occurs is in a given situation?

  • I don't. I have to make an assumption.

  • What's that assumption called? What kind of assumption do I

  • have to meet? I need an assumption --

  • -- of the statistical distribution of inputs.

  • Otherwise, expected time doesn't mean anything because I

  • don't know what the probability of something is.

  • In order to do probability, you need some assumptions and

  • you've got to state those assumptions clearly.

  • One of the most common assumptions is that all inputs

  • are equally likely. That's called the uniform

  • distribution. Every input of size n is

  • equally likely, that kind of thing.

  • But there are other ways that you could make that assumption,

  • and they may not all be true. This is much more complicated,

  • as you can see. Fortunately,

  • all of you have a strong probability background.

  • And so we will not have any trouble addressing these

  • probabilistic issues of dealing with expectations and such.

  • If you don't, time to go and say gee,

  • maybe I should take that Probability class that is a

  • prerequisite for this class. The last one I am going to

  • mention is best-case analysis. And this I claim is bogus.

  • Bogus. No good.

  • Why is best-case analysis bogus?

  • Yeah? The best-case probably doesn't

  • ever happen. Actually, it's interesting

  • because for the sorting problem, the most common things that get

  • sorted are things that are already sorted interestingly,

  • or at least almost sorted. For example,

  • one of the most common things that are sorted is check numbers

  • by banks. They tend to come in,

  • in the same order that they are written.

  • They're sorting things that are almost always sorted.

  • I mean, it's good. When upper bond,

  • not lower bound? Yeah, you want to make a

  • guarantee. And so why is this not a

  • guarantee? You're onto something there,

  • but we need a little more precision here.

  • How can I cheat? Yeah?

  • Yeah, you can cheat. You cheat.

  • You take any slow algorithm that you want and just check for

  • some particular input, and if it's that input,

  • then you say immediately yeah, OK, here is the answer.

  • And then it's got a good best-case.

  • But I didn't tell you anything about the vast majority of what

  • is going on. So, you can cheat with a slow

  • algorithm that works fast on some input.

  • It doesn't really do much for you so we normally don't worry

  • about that. Let's see.

  • What is insertion sorts worst-case time?

  • Now we get into some sort of funny issues.

  • First of all, it sort of depends on the

  • computer you're running on. Whose computer,

  • right? Is it a big supercomputer or is

  • it your wristwatch? They have different

  • computational abilities. And when we compare algorithms,

  • we compare them typically for relative speed.

  • This is if you compared two algorithms on the same machine.

  • You could argue, well, it doesn't really matter

  • what the machine is because I will just look at their relative

  • speed. But, of course,

  • I may also be interested in absolute speed.

  • Is one algorithm actually better no matter what machine

  • it's run on?

  • And so this kind of gets sort of confusing as to how I can

  • talk about the worst-case time of an algorithm of a piece of

  • software when I am not talking about the hardware because,

  • clearly, if I had run on a faster machine,

  • my algorithms are going to go faster.

  • So, this is where you get the big idea of algorithms.

  • Which is why algorithm is such a huge field,

  • why it spawns companies like Google, like Akamai,

  • like Amazon. Why algorithmic analysis,

  • throughout the history of computing, has been such a huge

  • success, is our ability to master and to be able to take

  • what is apparently a really messy, complicated situation and

  • reduce it to being able to do some mathematics.

  • And that idea is called asymptotic analysis.

  • And the basic idea of asymptotic analysis is to ignore

  • machine-dependent constants --

  • -- and, instead of the actual running time,

  • look at the growth of the running time.

  • So, we don't look at the actual running time.

  • We look at the growth. Let's see what we mean by that.

  • This is a huge idea. It's not a hard idea,

  • otherwise I wouldn't be able to teach it in the first lecture,

  • but it's a huge idea. We are going to spend a couple

  • of lectures understanding the implications of that and will

  • basically be doing it throughout the term.

  • And if you go on to be practicing engineers,

  • you will be doing it all the time.

  • In order to do that, we adopt some notations that

  • are going to help us. In particular,

  • we will adopt asymptotic notation.

  • Most of you have seen some kind of asymptotic notation.

  • Maybe a few of you haven't, but mostly you should have seen

  • a little bit. The one we're going to be using

  • in this class predominantly is theta notation.

  • And theta notation is pretty easy notation to master because

  • all you do is, from a formula,

  • just drop low order terms and ignore leading constants.

  • For example, if I have a formula like 3n^3 =

  • 90n^2 - 5n + 6046, I say, well,

  • what low-order terms do I drop? Well, n^3 is a bigger term n^2

  • than. I am going to drop all these

  • terms and ignore the leading constant, so I say that's

  • Theta(n^3). That's pretty easy.

  • So, that's theta notation. Now, this is an engineering way

  • of manipulating theta notation. There is actually a

  • mathematical definition for this, which we are going to talk

  • about next time, which is a definition in terms

  • of sets of functions. And, you are going to be

  • responsible, this is both a math and a computer science

  • engineering class. Throughout the course you are

  • going to be responsible both for mathematical rigor as if it were

  • a math course and engineering commonsense because it's an

  • engineering course. We are going to be doing both.

  • This is the engineering way of understanding what you do,

  • so you're responsible for being able to do these manipulations.

  • You're also going to be responsible for understanding

  • the mathematical definition of theta notion and of its related

  • O notation and omega notation. If I take a look as n

  • approached infinity, a Theta(n^2) algorithm always

  • beats, eventually, a Theta(n^3) algorithm.

  • As n gets bigger, it doesn't matter what these

  • other terms were if I were describing the absolute precise

  • behavior in terms of a formula. If I had a Theta(n^2)

  • algorithm, it would always be faster for sufficiently large n

  • than a Theta(n^3) algorithm. It wouldn't matter what those

  • low-order terms were. It wouldn't matter what the

  • leading constant was. This one will always be faster.

  • Even if you ran the Theta(n^2) algorithm on a slow computer and

  • the Theta(n^3) algorithm on a fast computer.

  • The great thing about asymptotic notation is it

  • satisfies our issue of being able to compare both relative

  • and absolute speed, because we are able to do this

  • no matter what the computer platform.

  • On different platforms we may get different constants here,

  • machine-dependent constants for the actual running time,

  • but if I look at the growth as the size of the input gets

  • larger, the asymptotics generally won't change.

  • For example, I will just draw that as a

  • picture. If I have n on this axis and

  • T(n) on this axis. This may be,

  • for example, a Theta(n^3) algorithm and this

  • may be a Theta(n^2) algorithm. There is always going to be

  • some point n_o where for everything larger the Theta(n^2)

  • algorithm is going to be cheaper than the Theta(n^3) algorithm

  • not matter how much advantage you give it at the beginning in

  • terms of the speed of the computer you are running on.

  • Now, from an engineering point of view, there are some issues

  • we have to deal with because sometimes it could be that that

  • n_o is so large that the computers aren't big enough to

  • run the problem. That's why we,

  • nevertheless, are interested in some of the

  • slower algorithms, because some of the slower

  • algorithms, even though they may not asymptotically be slower,

  • I mean asymptotically they will be slower.

  • They may still be faster on reasonable sizes of things.

  • And so we have to both balance our mathematical understanding

  • with our engineering commonsense in order to do good programming.

  • So, just having done analysis of algorithms doesn't

  • automatically make you a good programmer.

  • You also need to learn how to program and use these tools in

  • practice to understand when they are relevant and when they are

  • not relevant. There is a saying.

  • If you want to be a good program, you just program ever

  • day for two years, you will be an excellent

  • programmer. If you want to be a world-class

  • programmer, you can program every day for ten years,

  • or you can program every day for two years and take an

  • algorithms class. Let's get back to what we were

  • doing, which is analyzing insertion sort.

  • We are going to look at the worse-case.

  • Which, as we mentioned before, is when the input is reverse

  • sorted. The biggest element comes first

  • and the smallest last because now every time you do the

  • insertion you've got to shuffle everything over.

  • You can write down the running time by looking at the nesting

  • of loops. What we do is we sum up.

  • What we assume is that every operation, every elemental

  • operation is going to take some constant amount of time.

  • But we don't have to worry about what that constant is

  • because we're going to be doing asymptotic analysis.

  • As I say, the beautify of the method is that it causes all

  • these things that are real distinctions to sort of vanish.

  • We sort of look at them from 30,000 feet rather than from

  • three millimeters or something. Each of these operations is

  • going to sort of be a basic operation.

  • One way to think about this, in terms of counting

  • operations, is counting memory references.

  • How many times do you actually access some variable?

  • That's another way of sort of thinking about this model.

  • When we do that, well, we're going to go through

  • this loop, j is going from 2 to n, and then we're going to add

  • up the work that we do within the loop.

  • We can sort of write that in math as summation of j equals 2

  • to n. And then what is the work that

  • is going on in this loop? Well, the work that is going on

  • in this loop varies, but in the worst case how many

  • operations are going on here for each value of j?

  • For a given value of j, how much work goes on in this

  • loop? Can somebody tell me?

  • Asymptotically. It's j times some constant,

  • so it's theta j. So, there is theta j work going

  • on here because this loop starts out with i being j minus 1,

  • and then it's doing just a constant amount of stuff for

  • each step of the value of i, and i is running from j minus

  • one down to zero. So, we can say that is theta j

  • work that is going on. Do people follow that?

  • OK. And now we have a formula we

  • can evaluate. What is the evaluation?

  • If I want to simplify this formula, what is that equal to?

  • Sorry. In the back there.

  • Yeah. OK. That's just Theta(n^2),

  • good. Because when you're saying is

  • the sum of consecutive numbers, you mean what?

  • What's the mathematic term we have for that so we can

  • communicate? You've got to know these things

  • so you can communicate. It's called what type of

  • sequence? It's actually a series,

  • but that's OK. What type of series is this

  • called? Arithmetic series,

  • good. Wow, we've got some sharp

  • people who can communicate. This is an arithmetic series.

  • You're basically summing 1 + 2 + 3 + 4, some constants in

  • there, but basically it's 1 + 2 + 3 + 4 + 5 + 6 up to n.

  • That's Theta(n^2). If you don't know this math,

  • there is a chapter in the book, or you could have taken the

  • prerequisite. Erythematic series.

  • People have this vague recollection.

  • Oh, yeah. Good.

  • Now, you have to learn these manipulations.

  • We will talk about a bit next time, but you have to learn your

  • theta manipulations for what works with theta.

  • And you have to be very careful because theta is a weak

  • notation. A strong notation is something

  • like Leibniz notation from calculus where the chain rule is

  • just canceling two things. It's just fabulous that you can

  • cancel in the chain rule. And Leibniz notation just

  • expresses that so directly you can manipulate.

  • Theta notation is not like that.

  • If you think it is like that you are in trouble.

  • You really have to think of what is going on under the theta

  • notation. And it is more of a descriptive

  • notation than it is a manipulative notation.

  • There are manipulations you can do with it, but unless you

  • understand what is really going on under the theta notation you

  • will find yourself in trouble. And next time we will talk a

  • little bit more about theta notation.

  • Is insertion sort fast?

  • Well, it turns out for small n it is moderately fast.

  • But it is not at all for large n.

  • So, I am going to give you an algorithm that is faster.

  • It's called merge sort. I wonder if I should leave

  • insertion sort up. Why not.

  • I am going to write on this later, so if you are taking

  • notes, leave some space on the left.

  • Here is merge sort of an array A from 1 up to n.

  • And it is basically three steps.

  • If n equals 1 we are done. Sorting one element,

  • it is already sorted. All right.

  • Recursive algorithm. Otherwise, what we do is we

  • recursively sort A from 1 up to the ceiling of n over 2.

  • And the array A of the ceiling of n over 2 plus one up to n.

  • So, we sort two halves of the input.

  • And then, three, we take those two lists that we

  • have done and we merge them.

  • And, to do that, we use a merge subroutine which

  • I will show you.

  • The key subroutine here is merge, and it works like this.

  • I have two lists. Let's say one of them is 20.

  • I am doing this in reverse order.

  • I have sorted this like this. And then I sort another one.

  • I don't know why I do it this order, but anyway.

  • Here is my other list. I have my two lists that I have

  • sorted. So, this is A[1] to A[|n/2|]

  • and A[|n/2|+1] to A[n] for the way it will be called in this

  • program. And now to merge these two,

  • what I want to do is produce a sorted list out of both of them.

  • What I do is first observe where is the smallest element of

  • any two lists that are already sorted?

  • It's in one of two places, the head of the first list or

  • the head of the second list. I look at those two elements

  • and say which one is smaller? This one is smaller.

  • Then what I do is output into my output array the smaller of

  • the two. And I cross it off.

  • And now where is the next smallest element?

  • And the answer is it's going to be the head of one of these two

  • lists. Then I cross out this guy and

  • put him here and circle this one.

  • Now I look at these two guys. This one is smaller so I output

  • that and circle that one. Now I look at these two guys,

  • output 9. So, every step here is some

  • fixed number of operations that is independent of the size of

  • the arrays at each step. Each individual step is just me

  • looking at two elements and picking out the smallest and

  • advancing some pointers into the array so that I know where the

  • current head of that list is. And so, therefore,

  • the time is order n on n total elements.

  • The time to actually go through this and merge two lists is

  • order n. We sometimes call this linear

  • time because it's not quadratic or whatever.

  • It is proportional to n, proportional to the input size.

  • It's linear time. I go through and just do this

  • simple operation, just working up these lists,

  • and in the end I have done essentially n operations,

  • order n operations each of which cost constant time.

  • That's a total of order n time. Everybody with me?

  • OK. So, this is a recursive

  • program. We can actually now write what

  • is called a recurrence for this program.

  • The way we do that is say let's let the time to sort n elements

  • to be T(n). Then how long does it take to

  • do step one?

  • That's just constant time. We just check to see if n is 1,

  • and if it is we return. That's independent of the size

  • of anything that we are doing. It just takes a certain number

  • of machine instructions on whatever machine and we say it

  • is constant time. We call that theta one.

  • This is actually a little bit of an abuse if you get into it.

  • And the reason is because typically in order to say it you

  • need to say what it is growing with.

  • Nevertheless, we use this as an abuse of the

  • notation just to mean it is a constant.

  • So, that's an abuse just so people know.

  • But it simplifies things if I can just write theta one.

  • And it basically means the same thing.

  • Now we recursively sort these two things.

  • How can I describe that? The time to do this,

  • I can describe recursively as T of ceiling of n over 2 plus T of

  • n minus ceiling of n over 2. That is actually kind of messy,

  • so what we will do is just be sloppy and write 2T(n/2).

  • So, this is just us being sloppy.

  • And we will see on Friday in recitation that it is OK to be

  • sloppy. That's the great thing about

  • algorithms. As long as you are rigorous and

  • precise, you can be as sloppy as you want.

  • [LAUGHTER] This is sloppy because I didn't worry about

  • what was going on, because it turns out it doesn't

  • make any difference. And we are going to actually

  • see that that is the case. And, finally,

  • I have to merge the two sorted lists which have a total of n

  • elements. And we just analyze that using

  • the merge subroutine. And that takes us to theta n

  • time.

  • That allows us now to write a recurrence for the performance

  • of merge sort.

  • Which is to say that T of n is equal to theta 1 if n equals 1

  • and 2T of n over 2 plus theta of n if n is bigger than 1.

  • Because either I am doing step one or I am doing all steps one,

  • two and three. Here I am doing step one and I

  • return and I am done. Or else I am doing step one,

  • I don't return, and then I also do steps two

  • and three. So, I add those together.

  • I could say theta n plus theta 1, but theta n plus theta 1 is

  • just theta n because theta 1 is a lower order term than theta n

  • and I can throw it away. It is either theta 1 or it is

  • 2T of n over 2 plus theta n. Now, typically we won't be

  • writing this. Usually we omit this.

  • If it makes no difference to the solution of the recurrence,

  • we will usually omit constant base cases.

  • In algorithms, it's not true generally in

  • mathematics, but in algorithms if you are running something on

  • a constant size input it takes constant time always.

  • So, we don't worry about what this value is.

  • And it turns out it has no real impact on the asymptotic

  • solution of the recurrence. How do we solve a recurrence

  • like this? I now have T of n expressed in

  • terms of T of n over 2. That's in the book and it is

  • also in Lecture 2. We are going to do Lecture 2 to

  • solve that, but in the meantime what I am going to do is give

  • you a visual way of understanding what this costs,

  • which is one of the techniques we will elaborate on next time.

  • It is called a recursion tree technique.

  • And I will use it for the actual recurrence that is almost

  • the same 2T(n/2), but I am going to actually

  • explicitly, because I want you to see where it occurs,

  • plus some constant times n where c is a constant greater

  • than zero. So, we are going to look at

  • this recurrence with a base case of order one.

  • I am just making the constant in here, the upper bound on the

  • constant be explicit rather than implicit.

  • And the way you do a recursion tree is the following.

  • You start out by writing down the left-hand side of the

  • recurrence. And then what you do is you say

  • well, that is equal to, and now let's write it as a

  • tree. I do c of n work plus now I am

  • going to have to do work on each of my two children.

  • T of n over 2 and T of n over 2.

  • If I sum up what is in here, I get this because that is what

  • the recurrence says, T(n)=2T(n/2)+cn.

  • I have 2T(n/2)+cn. Then I do it again.

  • I have cn here. I now have here cn/2.

  • And here is cn/2. And each of these now has a

  • T(n/4).

  • And these each have a T(n/4). And this has a T(n/4).

  • And I keep doing that, the dangerous dot,

  • dot, dots. And, if I keep doing that,

  • I end up with it looking like this.

  • And I keep going down until I get to a leaf.

  • And a leaf, I have essentially a T(1).

  • That is T(1). And so the first question I ask

  • here is, what is the height of this tree?

  • Yeah. It's log n.

  • It's actually very close to exactly log n because I am

  • starting out at the top with n and then I go to n/2 and n/4 and

  • all the way down until I get to 1.

  • The number of halvings of n until I get to 1 is log n so the

  • height here is log n. It's OK if it is constant times

  • log n. It doesn't matter.

  • How many leaves are in this tree, by the way?

  • How many leaves does this tree have?

  • Yeah. The number of leaves,

  • once again, is actually pretty close.

  • It's actually n. If you took it all the way

  • down. Let's make some simplifying

  • assumption. n is a perfect power of 2,

  • so it is an integer power of 2. Then this is exactly log n to

  • get down to T(1). And then there are exactly n

  • leaves, because the number of leaves here, the number of nodes

  • at this level is 1, 2, 4, 8.

  • And if I go down height h, I have 2 to the h leaves,

  • 2 to the log n, that is just n.

  • We are doing math here, right?

  • Now let's figure out how much work, if I look at adding up

  • everything in this tree I am going to get T(n),

  • so let's add that up. Well, let's add it up level by

  • level. How much do we have in the

  • first level? Just cn.

  • If I add up the second level, how much do I have?

  • cn. How about if I add up the third

  • level? cn.

  • How about if I add up all the leaves?

  • Theta n. It is not necessarily cn

  • because the boundary case may have a different constant.

  • It is actually theta n, but cn all the way here.

  • If I add up the total amount, that is equal to cn times log

  • n, because that's the height, that is how many cn's I have

  • here, plus theta n. And this is a higher order term

  • than this, so this goes away, get rid of the constants,

  • that is equal to theta(n lg n). And theta(n lg n) is

  • asymptotically faster than theta(n^2).

  • So, merge sort, on a large enough input size,

  • is going to beat insertion sort.

  • Merge sort is going to be a faster algorithm.

  • Sorry, you guys, I didn't realize you couldn't

  • see over there. You should speak up if you

  • cannot see. So, this is a faster algorithm

  • because theta(n lg n) grows more slowly than theta(n^2).

  • And merge sort asymptotically beats insertion sort.

  • Even if you ran insertion sort on a supercomputer,

  • somebody running on a PC with merge sort for sufficient large

  • input will clobber them because actually n^2 is way bigger than

  • n log n once you get the n's to be large.

  • And, in practice, merge sort tends to win here

  • for n bigger than, say, 30 or so.

  • If you have a very small input like 30 elements,

  • insertion sort is a perfectly decent sort to use.

  • But merge sort is going to be a lot faster even for something

  • that is only a few dozen elements.

  • It is going to actually be a faster algorithm.

  • That's sort of the lessons, OK?

  • Remember that to get your recitation assignments and

  • attend recitation on Friday. Because we are going to be

  • going through a bunch of the things that I have left on the

  • table here. And see you next Monday.

We're going to get started. Handouts are the by the door if

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Lec 1|MIT 6.046J / 18.410J アルゴリズム入門 (SMA 5503), Fall 2005 (Lec 1 | MIT 6.046J / 18.410J Introduction to Algorithms (SMA 5503), Fall 2005)

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