Lec 14 | MIT 6.00 Introduction to Computer Science and Programming, Fall 2008

Lec 14 | MIT 6.00 Introduction to Computer Science and Programming, Fall 2008


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MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: At the end of the
lecture on Tuesday, a number of people asked me questions,
asked Professor Grimson questions, which made it clear
that I had been less than clear on at least a few things,
so I want to come back and revisit a couple of the
things we talked about at the end of the lecture. You’ll remember that I had drawn
this decision tree, in part because it’s an important
concept I want you to understand, the concept of
decision trees, and also to illustrate, I hope, visually,
some things related to dynamic programming. So we had in that decision tree,
is we had the weight vector, and I just given a very
simple one [5,3,2], and we had a very simple value
vector, [9,7,8]. And then the way we drew the
tree, was we started at the top, and said all right, we’re
going to first look at item number 2, which was the third
item in our list of items, of course, and say that we had
five pounds left of weight that our knapsack that could
hold, and currently had a value of 0. And then we made a decision, to
not put that last item in the backpack, and said if we
made that decision, the next item we had to consider was
item 1, we still had five pounds available, and we still
had a weight 0 available. Now I, said the next item to
consider is item 1, but really what I meant is, 1 and all of
the items proceeding it in the list. This is my shorthand for
saying the list up to and including items sub 1,
kind of a normal way to think about it. And then we finish building
the tree, left first step first, looking at all the no
branches, 0,5,0 and then we were done, that was
one branch. We then backed up, and said
let’s look at a yes, we’ll include item number 1. Well, what happens here, if
we’ve included that, it uses up all the available weight,
and gave us the value of 9. STUDENT: [UNINTELLIGIBLE] PROFESSOR: Pardon? STUDENT: — want to be off
the bottom branch. PROFESSOR: Yup, Off by 1. Yeah, I wanted to come off
this branch, because I’ve backtrack just 1, thank you. And then I backtrack up
to this branch, and from here we got 0,2,7. And I’m not going to draw the
rest of the tree for you here, because I drew it last time, and
you don’t need to see the whole tree. The point I wanted to make is
that for every node, except the leaves, the leaves are the
bottom of a tree in this case, computer scientists are weird,
right, they draw trees where the root is at the top, and the
leaves are at the bottom. And I don’t know why, but since
time immemorial that is the way computer scientists
have drawn trees. That’s why we’re not biologists,
I guess. We don’t understand
these things. But what I want you to notice is
that for each node, except the leaves, the solution for
that node can be computed from the solutions from
it’s children. So in order to look at the
solution of this node, I choose one of the solutions of
it’s children, a or b, is the best solution if I’m here, and
of course this is the better of the 2 solutions. If I look at this node, I get to
choose its solution as the better of the solution for
this node, and this node. All the way up to the top, where
when I have to choose the best solution to the whole
problem, it’s either the best solution to the left node,
or the best solution to the right node. This happens to be a binary
decision tree. There’s nothing magic about
there being only two nodes, for the knapsack problem, that’s
just the way it works out, but there are other
problems where there might be multiple decisions to make, more
than a or yes or no, but it’s always the case here that
I have what we last time talked about as what? Optimal sub structure. As I defined it last time, it
means that I can solve a problem by finding the optimal
solution to smaller sub problems. Classic divide and
conquer that we’ve seen over and over again in the term. Take a hard problem, say well,
I can solve it by solving 2 smaller problems and combine
their solution, and this case, the combining is choosing
the best, it’s a or b. So then, I went directly from
that way of thinking about the problem, to this
straightforward, at the top of the slide here, also at the
top of your handout, both yesterday and today, a
straightforward implementation of max val, that basically
just did this. And as you might have guessed,
when you’re doing this sort of thing, recursion is a very
natural way to implement it. We then ran this, demonstrated
that it got the right answer on problems that were small
enough that we knew with the right answer was, ran it on a
big problem, got what we hoped was the right answer, but we had
no good way to check it in our heads, but noticed it
took a long time to run. And then we asked ourselves,
why did it take so long to run? And when we turned on the print
statement, what we saw is because it was doing
the same thing over and over again. Because we had a lot of the
sub-problems were the same. It was as if, when we went
through this search tree, we never remembered what we got
at the bottom, and we just re-computed things
over and over. So that led us to look at
memoization, the sort of key idea behind dynamic programming,
which says let’s remember the work we’ve done and
not do it all over again. We used a dictionary to
implement the memo, and that got us to the fast max val,
which got called from max val 0, because I wanted to make
sure I didn’t change the specification of max val by
introducing this memo that users shouldn’t know even
exists, because it’s part of the implementation, not part
of the problem statement. We did that, and all I did was
take the original code and keep track of what I’ve done,
and say have I computed this value before, if so, don’t
compute it again. And that’s the key idea that
you’ll see over and over again as you solve problems with
dynamic programming, is you say, have I already solved this
problem, if so, let me look up the answer. If I haven’t solved the problem,
let me solve it, and store the answer away
for later reference. Very simple idea, and typically
the beauty of dynamic programming as you’ve
seen here, is not only is the idea simple, even the
implementation is simple. There are a lot of complicated
algorithmic ideas, dynamic programming is not
one of them. Which is one of the reasons
we teach it here. The other reason we teach it
here, in addition to it being simple, is that it’s
incredibly useful. It’s probably among the most
useful ideas there is for solving complicated problems. All right, now let’s
look at it. So here’s the fast version, we
looked at it last time, I’m not going to bore you by going
through the details of it again, but we’ll run it. This was the big example we
looked at last time, where we had 30 items we could put in to
choose from, so when we do it exponentially, it looks like
it’s 2 to the 30, which is a big number, but when we ran
this, it found the answer, and it took only 1805 calls. Now I got really excited about
this because, to me it’s really amazing, that we’ve
taken a problem that is apparently exponential, and
solved it like that. And in fact, I could double
the size of the items to choose from, and it would
still run like. Eh – I’m not very good at
snapping my fingers — it would still run quickly. All right, so here’s the
question: have I found a way to solve an inherently
exponential problem in linear time. Because what we’ll see here,
and we saw a little of this last time, as I double the
size of the items, I only really roughly double
the running time. Quite amazing. So have I done that? Well, I wish I had, because then
I would be really famous, and my department head would
give me a big raise, and all sorts of wonderful things
would follow. But, I’m not famous, and I
didn’t solve that problem. What’s going on? Well this particular algorithm
takes roughly, and I’ll come back to the roughly question,
order (n,s) time, where n is the number of items in the list
and s, roughly speaking, is the size of the knapsack. We should also observe, that
it takes order and s space. Because it’s not free to
store all these values. So at one level what I’m doing
is trading time for space. It can run faster because
I’m using some space to save things. So in this case, we had 30 items
and the wait was 40, and, you know, this gives
us 1200 which is kind of where we were. And I’m really emphasizing kind
of here, because really what I’m using the available
size for, is as a proxy for the number of items that can
fit in the knapsack. Because the actual running time
of this, and the actual space of this algorithm, is
governed, interestingly enough, not by the size of the
problem alone, but by the size of the solution. And I’m going to come
back to that. So how long it takes to run is
related to how many items I end up being able to fit
into the knapsack. If you think about it,
this make sense. An entry is made in the memo
whenever an item, and an available size pair
is considered. As soon as the available size
goes to 0, I know I can’t enter any more items into
the memo, right? So the number of items I have to
remember is related to how many items I can fit
in the knapsack. And of course, the amount of
running time is exactly the number of things I
have to remember, almost exactly, right? So you can see if you think
about it abstractly, why the amount of work I have to do here
will be proportional to the number of items I can fit
in, that is to say, the size of the solution. This is not the way we’d like
to talk about complexity. When we talk about the order,
or big O, as we keep writing it, of a problem, we always
prefer to talk about it in terms of the size
of the problem. And that makes sense because in
general we don’t know the size of the solution until
we’ve solved it. So we’d much rather define big
O in terms of the inputs. What we have here is
what’s called a pseudo-polynomial algorithm. You remember a polynomial
algorithm is an algorithm that’s polynomial in the
size of the inputs. Here we have an algorithm that’s
polynomial in the size of the solution, hence
the qualifier pseudo. More formally, and again this is
not crucial to get all the details on this, if we think
about a numerical algorithm, a pseudo-polynomial algorithm
has running time that’s polynomial in the numeric
value of the input. I’m using a numeric example
because it’s easier to talk about it that way. So you might to look at, say,
an implementation of factorial, and say its running
time is proportional to the numerical value of the number
who’s factorial. If I’m computing factorial of 8,
I’ll do 8 operations, Right Factorial of 10, I’ll
do 10 operations. Now the key issue to think about
here, is that as we look at this kind of thing, what
we’ll see is that, if we look at a numeric value, we know that
that’s exponential number in the number of digits. So that’s the key thing to think
about, Right That you can take a problem, and
typically, when we’re actually formally looking at
computational complexity, big O, what we’ll define the in
terms of, is the size of the coding of the input. The number of bits required
to represent the input in the computer. And so when we say something is
exponential, we’re talking about in terms of the number
of bits required to represent it. Now why am I going through all
this, maybe I should use the word pseudo-theory? Only because I want you to
understand that when we start talking about complexity, it
can be really quite subtle. And you have to be very careful
to think about what you mean, or you can be very
surprised at how long something takes to run, or
how much space it uses. And you have to understand the
difference between, are you defining the performance in
terms of the size of the problem, or the size
of the solution. When you talk about the size
of the problem, what do you mean by that, is it the length
of an array, is it the size of the elements of the array,
and it can matter. So when we ask you to tell
us something about the efficiency, on for example a
quiz, we want you to be very careful not to just write
something like, order n squared, but to tell
us what n is. For example, the number of
elements in the list. But if you have a list of lists, maybe
it’s not just the number elements in the list, maybe
it depends upon what the elements are. So just sort of a warning to try
and be very careful as you think about these things,
all right. So I haven’t done magic, I’ve
given you a really fast way to solve a knapsack problem, but
it’s still exponential deep down in its heart,
in something. All right, in recitation you’ll
get a chance to look at yet another kind of problem that
can be solved by dynamic programming, there
are many of them. Before we leave the knapsack
problem though, I want to take a couple of minutes to
look at a slight variation of the problem. So let’s look at this one. Suppose I told you that not only
was there a limit on the total weight of the items
in the knapsack, but also on the volume. OK, if I gave you a box of
balloons, the fact that they didn’t weight anything wouldn’t
mean you couldn’t put, you could put lots of them
in the knapsack, right? Sometimes it’s the volume not
the weight that matters, sometimes it’s both. So how would we go about solving
this problem if I told you not only was there a maximum
weight, but there was a maximum volume. Well, we want to go back and
attack it exactly the way we attacked it the first time,
which was write some mathematical formulas. So you’ll remember that when we
looked at it, we said that the problem was to maximize the
sum from i equals 1 to n, of p sub i, x sub i, maybe it
should be 0 to n minus 1, but we won’t worry about that. And we had to do it subject to
the constraint that the sum from 1 to n of the weight sub i
times x sub i, remember x is 0 if it was in, 1 if it wasn’t,
was less than or equal to the cost, as I wrote it
this time, which was the maximum allowable weight. What do we do if we want to
add volume, is an issue? Does this change? Does the goal change? You’re answering. Answer out — no one else can
see you shake your head. STUDENT: No. PROFESSOR: No. The goal does not change, it’s
still the same goal. What changes? STUDENT: The constraints. PROFESSOR: Yeah, and you
don’t get another bar. The constraint has to change. I’ve added a constraint. And, what’s the constraint
I’ve added? Somebody else — yeah? STUDENT: You can’t exceed
the volume that the knapsack can hold. PROFESSOR: Right, but can
you state in this kind of formal way? STUDENT: [INAUDIBLE] PROFESSOR: — sum from
i equals 1 to n — STUDENT: [INAUDIBLE] PROFESSOR: Let’s say v sub i, x
sub i, is less than or equal to, we’ll write k for the
total allowable volume. Exactly. So the thing to notice here,
is it’s actually quite a simple little change
we’ve made. I’ve simply added this one extra
constraint, nice thing about thinking about it this
way is it’s easy to think about it, and what do you think
I’ll have to do if I want to go change the code? I’m not going to do it for you,
but what would I think about doing when I
change the code? Well, let’s look at the simple
version first, because it’s easier to look at. At the top. Well basically, all I’d have to
do is go through and find every place I checked the
constraint, and change it. To incorporate the
new constraint. And when I went to the dynamic
programming problem, what would I have to do,
what would change? The memo would have to change,
as well as the checks, right? Because now, I not only would
have to think about how much weight did I have available, but
I have to think about how much volume did I
have available. So whereas before, I had a
mapping from the item and the weight available, now I would
have to have it from a tuple of the weight and the volume. Very small changes. That’s one of the things I want
you to sort of understand as we look at algorithms, that
they’re very general, and once you’ve figured out how to solve
one problem, you can often solve another problem
by a very straightforward reduction of this
kind of thing. All right, any questions
about that. Yeah? STUDENT: I had a question about
what you were talking about just before. PROFESSOR: The
pseudo-polynomial? STUDENT: Yes. PROFESSOR: Ok. STUDENT: So, how do you come
to a conclusion as to which you should use then, if you can
determine the size based on solution, or based on input,
so how do you decide? PROFESSOR: Great question. So the question is, how do you
choose an algorithm, why would I choose to use a
pseudo-polynomial algorithm when I don’t know how big the
solution is likely to be, I think that’s one way
to think about it. Well, so if we think about the
knapsack problem, we can look at it, and we can ask ourselves,
well first of all we know that the brute force
exponential solution is going to be a loser if the number
of items is large. Fundamentally in this case, what
I could look at is the ratio of the number of items to
the size of the knapsack, say well, I’ve got lots items
to choose from, I probably won’t put them all in. But even if I did, it
would still only be 30 of them, right? It’s hard. Typically what we’ll discover
is the pseudo-polynomial algorithms are usually better,
and in this case, never worse. So this will never be worse than
the brute force one. if I get really unlucky, I end up
checking the same number of things, but I’d have to be
really, it’d have to be a very strange structure
to the problem. So it’s almost always the case
that, if you can find a solution that uses dynamic
programming, it will be better than the brute force, and
certainly not, well, maybe use more space, but not
use more time. But there is no magic, here, and
so the question you asked is a very good question. And it’s sometimes the case in
real life that you don’t know which is the better algorithm
on the data you’re actually going to be crunching. And you pays your money and you
takes your chances, right? And if the data is not what you
think it’s going to be, you may be wrong in your choice,
so you typically do have to spend some time thinking
about it, what’s the data going to actually
look like. Very good question. Anything else? All right, a couple of closing
points before we leave this, things I would like
you to remember. In dynamic programming, one of
the things that’s going on is we’re trading time for space. Dynamic programming is not
the only time we do that. We’ve solved a lot of problems
that way, in fact, by trading time for space. Table look-up, for example,
right, that if you’re going to have trig tables, you may want
to compute them all at once and then just look it up. So that’s one thing. Two, don’t be intimidated by
exponential problems. There’s a tendency for people to say, oh
this problem’s exponential, I can’t solve it. Well, I solve 2 or 3 exponential
problems before breakfast every day. You know things like, how to
find my way to the bathroom is inherently exponential, but I
manage to solve it anyway. Don’t be intimidated. Even though it is apparently
exponential, a lot of times you can actually solve
it much, much faster. Other issues. Three: dynamic programming
is broadly useful. Whenever you’re looking at a
problem that seems to have a natural recursive solution,
think about whether you can attack it with dynamic
programming. If you’ve got this optimal
substructure, and overlapping sub-problems, you can use
dynamic programming. So it’s good for knapsacks, it’s
good for shortest paths, it’s good for change-making,
it’s good for a whole variety of problems. And so keep it in
your toolbox, and when you have a hard problem to solve,
one of the first questions you should ask yourself is, can
I use dynamic programming? It’s great for string-matching
problems of a whole variety. It’s hugely useful. And finally, I want you to keep
in mind the whole concept of problem reduction. I started with this silly
description of a burglar, and said : Well this is really the
knapsack problem, and now I can go Google the knapsack
problem and find code to solve it. Any time you can reduce
something to a previously solved problem, that’s good. And this is a hugely important
lesson to learn. People tend not to realize that
the first question you should always ask yourself, is
this really just something that’s well-known in disguise? Is it a shortest path problem? Is it a nearest neighbor
problem? Is it what string is this
most similar to problem? There are scores of
well-understood problems, but only really scores, it’s not
thousands of them, and over time you’ll build up a
vocabulary of these problems, and when you see something in
your domain, be it physics or biology or anything else,
linguistics, the question you should ask is can I transform
this into an existing problem? Ok, double line. If there are no questions I’m
going to make a dramatic change in topic. We’re going to temporarily get
off of this more esoteric stuff, and go back to Python. And for the next, off and on for
the next couple of weeks, we’ll be talking about Python
and program organization. And what I want to be talking
about is modules of one sort, and of course that’s because
what we’re interested in is modularity. How do we take a complex
program, again, divide and conquer, I feel like a 1-trick
pony, I keep repeating the same thing over and
over again. Divide and conquer to make our
programs modular so we can write them a little piece at a
time and understand them a little piece at a time. Now I think of a module as a collection of related functions. We’ve already seen these, and
we’re going to refer to the functions using dot notation. We’ve been doing this all
term, right, probably somewhere around lecture 2, we
said import math, and then somewhere in our program we
wrote something like math dot sqrt of 11, or some
other number. And the good news was we didn’t
have to worry about how math did square root or anything
like that, we just got it and we used it. Now we have the dot notation
to avoid name conflicts. Imagine, for example, that in
my program I wrote something like import set, because
somebody had written a module that implements mathematical
sets, and somewhere else I’d written something like import
table, because someone had something that implemented
look-up tables of some sort, something like dictionaries,
for example. And then, I wanted to ask
something like membership. Is something in the set, or
is something in the table? Well, what I would have written
is something like table dot member. And then the element and
maybe the table. And the dot notation was used
to disambiguate, because I want the member operation
from table, not the member one from set. This was important because the
people who implemented table and set might never have met
each other, and so they can hardly have been expected not
to have used the same name somewhere by accident. Hence the use of the
dot notation. I now want to talk about a
particular kind of module, and those are the modules
that include classes or that are classes. This is a very important concept
as we’ll see, it’s why MIT calls things like 6.00
subjects, so that they don’t get confused with classes in
Python, something we really need to remember here. Now they can be used in
different ways, and they have been historically used
in different ways. In this subject we’re going to
emphasize using classes in the context of what’s called
object-oriented programming. And if you go look up at Python
books on the web, or Java books on the web, about 80%
of them will include the word object-oriented
in their title. Object-oriented Python
programming for computer games, or who knows what else. And we’re going to use this
object-oriented programming, typically to create something
called data abstractions. And over the next couple of
days, you’ll see what we mean by this in detail. A synonym for this is an
abstract data type. You’ll see both terms used on
the web, and the literature etc., and think of them as
synonyms. Now these ideas of classes, object-oriented
programming, data abstraction, are about 40 years old,
they’re not new ideas. But they’ve only been really
widely accepted in practice for 10 to 15 years. It was in the mid-70’s, people
began to write articles advocating this style of
programming, and actually building programming languages,
notably Smalltalk and Clue at MIT in fact, that
provided linguistic support for the ideas of data
abstraction and object-oriented programming. But it really wasn’t until, I
would say, the arrival of Java that object-oriented
programming caught the popular attention. And then Java, C++ ,
Python of, course. And today nobody advocates a
programming language that does not support it in some
sort of way. So what is this all about? What is an object in
object-oriented programming? An object is a collection
of data and functions. In particular functions that
operate on the data, perhaps on other data as well. The key idea here is to bind
together the data and the functions that operate on that
data as a single thing. Now typically that’s probably
not the way you’ve been thinking about things. When you think about an int or
a float or a dictionary or a list, you knew that there
were functions that operated on them. But when you pass a parameter
say, a list, you didn’t think that you were not only passing
the list, you were also passing the functions
that operate on the list. In fact you are. It often doesn’t matter, but
it sometimes really does. The advantage of that, is that
when you pass an object to another part of the program,
that part of the program also gets the ability to perform
operations on the object. Now when the only types we’re
dealing with are the built-in types, the ones that came with
the programming language, that doesn’t really matter. Because, well, the programming
language means everybody has access to those operations. But the key idea here is that
we’re going to be generating user-defined types, we’ll invent
new types, and as we do that we can’t assume that if
as we pass objects of that type around, that the
programming language is giving us the appropriate operations
on that type. This combining of data and
functions on that data is a very essence of object-oriented
programming. That’s really what defines it. And the word that’s often used
for that is encapsulation. Think of it as we got a capsule,
like a pill or something, and in that capsule
we’ve got data and a bunch of functions, which as we’ll
see are called methods. Don’t worry, it doesn’t matter
that they’re called methods, it’s a historical artifact. All right, so what’s
an example of this? Well, we could create a circle
object, that would store a representation of the circle
and also provide methods to operate on it, for example, draw
the circle on the screen, return the area of the circle,
inscribe it in a square, who knows what you want
to do with it. As we talk about this, as people
talk about this, in the context of our object-oriented
programming, they typically will talk about it in terms
of message pass, a message passing metaphor. I want to mention it’s just
a metaphor, just a way of thinking about it, it’s not
anything very deep here. So, the way people will talk
about this, is one object can pass a message to another
object, and the receiving object responds by executing
one of its methods on the object. So let’s think about lists. So if l is a list, I can
call something like s dot sort, l dot sort. You’ve seen this. This says, pass the object l
the message sort, and that message says find the method
sort, and apply it to the object l, in this case mutating
the object so that the elements are now
in sorted order. If c is a circle, I might write something like c dot area. And this would say, pass to
the object denoted by the variable c, the message area,
which says execute a method called area, and in this case
the method might return a float, rather than have
a side-effect. Now again, don’t get carried
away, I almost didn’t talk about this whole message-passing
paradigm, but it’s so pervasive in the
world I felt you needed to hear about it. But it’s nothing very deep, and
if you want to not think about messages, and just think
oh, c has a method area, a circle has a method area, and
c as a circle will apply it and do what it says, you won’t
get in any trouble at all. Now the next concept to think
about here, is the notion of an instance. So we’ve already thought about,
we create instances of types, so when we looked at
lists, and we looked at the notion of aliasing, we used the
word instance, and said this is 1 object, this is
another object, each of those objects is an instance of
type list. So now that gets us to a class. A class is a collection
of objects with characteristics in common. So you can think of class
list. What is the characteristic that all objects
of class list have in common, all instances of class
list? it’s the set of methods that can be applied to lists. Methods like sort, append,
other things. So you should think of all of
the built-in types we’ve talked about as actually just
built-in classes, like dictionaries, lists, etc. The beauty of being able to
define your own class is you can now extend the language. So if, for example, you’re in
the business, God forbid, of writing financial software
today, you might decide, I’d really like to have a class
called tanking stock, or bad mortgage, or something like
that or mortgage, right? Which would have a bunch of
operations, like, I won’t go into what they might be. But you’d like to write your
program not in terms of floats and ints and lists, but in terms
of mortgages, and CDOs, and all of the objects that you
read about in the paper, the types you read about. And so you get to build your
own special purpose programming language that helped
you solve your problems in biology or finance or
whatever, and we’ll pick up here again on Tuesday.

30 thoughts to “Lec 14 | MIT 6.00 Introduction to Computer Science and Programming, Fall 2008”

  1. and not only there is excellence of the profs and material, but the profs are so personable, likable characters. thank you MIT.

  2. @ultimateZigzag . Who cares if they teach on blackboards? As long as the students understand the lesson it doesn't matter…

  3. @arkiesfroak That is effectively true in Python too.
    The MIT professors are talking about the concept of a class and a object, but in practice, your definition is exactly what a Python class really is.

  4. Love the way they reduce bewildering terms to basic logical statements applied to simple postulates that anyone who concentrates can follow. just as with programming you can't assume anything is 'in memory' until you've put it there yourself. The same goes for human brains – you can't use the concepts of object, method, class etc in a students brain until you've used your teaching program to write those concepts into them. Complexity is just lots and lots and lots of simple stuff.

  5. Don't you love how we have all these "real" computer geeks who are PC fanboys who make claims like "people who know about computers use PC's." And here you have an MIT computer science professor using a Mac. Priceless. Macs are the best.

  6. IMHO, I would not have the creative flow encumbered by the details of complexity in my code. Let your code flow – like an outline to a book, at first – then worry about the complexity and optimization details once you have something you like.

  7. While there is much discussion about the details of complexity, I am not getting a sense that a frame of time has been given for when these tools and methods are most useful, and how their utility varies over the course of the application development lifecycle.

  8. Would it not be possible to determine a complexity factor or Order by running a series of pre-established data types and set size designed to characterize the program complexity?

  9. @4:45 , The scientists have made tree and its roots.
    Roots always divide and go below. So rather than thinking as leaves , it is better to think as roots.

    Btw, Im much grateful for the lectures provided, it is the best ones available.

  10. Correct me if I'm wrong but technically don't the bits needed to represent a number grow logarithmic-ally ? For example 2^n needs n+1 bits to be represented.

  11. O (ns) is quadratic not exponential. So he made an exponential problem a quadratic one, why did he say its still exponential.

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